problem solving method of characteristics

The Method of Characteristics with Applications to Conservation Laws

Scott a. sarra ( e-mail ), marshall university.

In this article and its accompanying applet, I introduce the method of characteristics for solving first order partial differential equations (PDEs). First, the method of characteristics is used to solve first order linear PDEs. Next, I apply the method to a first order nonlinear problem, an example of a conservation law , and I discuss why the method may break down for nonlinear problems. I examine difficulties that appear in the nonlinear case, and I introduce the mathematical resolutions of these problems. Concepts dealing with the solutions of nonlinear hyperbolic conservation laws are introduced and explored graphically. However, detailed explanations of such concepts may take up entire textbooks, and the concepts are not explored in detail or with rigor in this short paper. I include References to more detailed material.

Intended Audience

This material is appropriate for undergraduate students in a partial differential equations class, as well as for undergraduate (or graduate) students in mathematics or other sciences who desire a brief and graphical introduction to the solutions of nonlinear hyperbolic conservation laws or to the method of characteristics for first order hyperbolic partial differential equations.

1 - The Method of Characteristics 1.1 - General Strategy

2 - Special Case: b ( x , t ) = 1 and c ( x , t ) = 0

2.1 - Constant Coefficient Advection Equation 2.2 - Variable Coefficient Advection Equation

3 - Conservation Laws

3.1 - Inviscid Burgers' Equation 3.2 - Numerical Methods for Conservation Laws 3.3 - Inviscid Burgers' equation example problems 3.3.1 - Riemann problem - shock 3.3.2 - Riemann problem - rarefaction 3.3.3 - Sine 3.3.4 - N wave 3.3.5 - Single hump

4 - The Shock Applet

5 - References and Links

Note: For a "printer-friendly" version of this article (all in one browser page), click here .

Method of characteristics

Branko Ćurgus.

Example 1. Let $a \in [0,+\infty).$ Find the solution of the initial value problem \begin{align*} \frac{dx}{dt} & = 1 \\ \frac{dy}{dt} & = -x \\ \end{align*} subject to the initial conditions \begin{align*} x(0) & = a \\ y(0) & = 0 \\ \end{align*} Solution. The solution of this system is a pair of functions $x(t) = t+a,$ $y(t) = (-1/2)t^2 - a t,$ with $t \in \mathbb{R}.$ These are the parametric equations of the parabola with the equation $y = (-1/2)(x^2 - a^2).$ With $a \geq 0$ this is a family of the parabolas which are obtained by the vertical shift of the parabola $y=(-1/2)x^2$ up by $(1/2)a^2.$
Example 2. Let $a \in (0,+\infty).$ Find the solution of the initial value problem \begin{align*} \frac{dx}{dt} & = x \\ \frac{dy}{dt} & = 2x - y \\ \end{align*} subject to the initial conditions \begin{align*} x(0) & = a \\ y(0) & = 0 \\ \end{align*} Solution. The solution of this system is a pair of functions $x(t) = a \exp(t),$ $y(t) = a \exp(t) - a \exp(-t),$ with $t \in \mathbb{R}.$ These are the parametric equations of one branch of the hyperbola with the equation $y =x - a^2(1/x)$ where $x\gt 0.$ With all $a \gt 0$ we get a family of functions picture to come.
Assume that a smooth function $z = u(x,y)$ is a solution of \eqref{eq:pde}. Let $x_0,y_0 \in \mathbb R$ and set $z_0 = u(x_0,y_0).$ Then the point $P_0 = (x_0,y_0,z_0)$ is a point on the graph of the function $z = u(x,y)$. From a multivariable calculus course we know that the vector \begin{equation*} \Bigl\langle 1, 0, u_x(x_0,y_0) \Bigr\rangle \end{equation*} is tangent to the curve \[ \bigl\{ \bigl( x, y_0, u(x,y_0) \bigr) : x_0 -\epsilon \lt x \lt x_0 + \epsilon \bigr\} \quad (\text{where} \quad \epsilon \gt 0 \quad \text{is small}) \] at the point $P_0$. Similarly, the vector \begin{equation*} \Bigl\langle 0, 1, u_y(x_0,y_0) \Bigr\rangle \end{equation*} is tangent to the curve \[ \bigl\{ \bigl( x_0, y, u(x_0,y) \bigr) : y_0 -\epsilon \lt y \lt y_0 + \epsilon \bigr\} \quad (\text{where} \quad \epsilon \gt 0 \quad \text{is small}) \] at the point $P_0$.
Thus both vectors \begin{equation*} \Bigl\langle 1, 0, u_x(x_0,y_0) \Bigr\rangle \quad \text{and} \quad \Bigl\langle 0, 1, u_y(x_0,y_0) \Bigr\rangle \end{equation*} are tangent to the graph of the function $u$ at the point $P_0$. The reasoning from the previous item applies to an arbitrary point $P = \bigl(x,y,u(x,y)\bigr)$ on the graph of the function $u.$ Therefore the vecors \begin{equation} \label{eq:tvxy} \Bigl\langle 1, 0, u_x(x,y) \Bigr\rangle \quad \text{and} \quad \Bigl\langle 0, 1, u_y(x,y) \Bigr\rangle \end{equation} are tangent to the graph of $u$ at the point $P$. Two vectors in \eqref{eq:tvxy} are linearly independent. (This follows from the definition of linear independence. The only way to get the zero vector $\langle 0,0,0 \rangle$ as a linear combination of the vectors in \eqref{eq:tvxy} is with both coefficients equal to $0.$) Therefore, the tangent plane to the graph of $u$ at the point $P$ is spanned by the vectors in \eqref{eq:tvxy}. That is, a vector is tangent to the graph of the function $u$ at the point $P$ if and only if it is a linear combination of the vectors in \eqref{eq:tvxy}.
Since we assume that the function $u$ is a solution of \eqref{eq:pde} we have that \eqref{eq:pde} holds for points $(x,y)$ in the domain of $u$. That is we have \begin{equation} \label{eq:pdee} A\bigl(x,y,u(x,y)\bigr) \, u_x(x,y) + B\bigl(x,y,u(x,y)\bigr) \, u_y(x,y) = C\bigl(x,y,u(x,y)\bigr). \end{equation}
The following step might look a little bit like a Deus ex machina action. But it is like that sometimes in math. Consider the vector field \begin{equation} \label{eq:vf} (x,y,z) \mapsto \Bigl\langle A(x,y,z), B(x,y,z), C(x,y,z) \Bigr\rangle. \end{equation} Next consider the vectors in this field that have tails at the points on the graph of the function $u,$ that is the vectors \begin{equation} \label{eq:vfu} \Bigl\langle A\bigl(x,y,u(x,y)\bigr), B\bigl(x,y,u(x,y)\bigr), C\bigl(x,y,u(x,y)\bigr) \Bigr\rangle. \end{equation} Next we use \eqref{eq:pdee} to establish a connection between the vector in \eqref{eq:vfu} and the vectors in \eqref{eq:tvxy}. But, for the sake of brevity of the notation we set $P = \bigl(x,y,u(x,y)\bigr)$ and write \eqref{eq:pdee} as \begin{equation*} A(P) \, u_x(x,y) + B(P) \, u_y(x,y) = C(P) \end{equation*} and \eqref{eq:vfu} as \begin{equation*} \Bigl\langle A(P), B(P), C(P) \Bigr\rangle. \end{equation*} Now we calculate \begin{align*} \Bigl\langle A(P), B(P), C(P) \Bigr\rangle & = \Bigl\langle A(P), B(P), A(P) \, u_x(x,y) + B(P) \, u_y(x,y) \Bigr\rangle \\ & = \Bigl\langle A(P), 0, A(P) \, u_x(x,y) \Bigr\rangle + \Bigl\langle 0, B(P), B(P)\, u_y(x,y) \Bigr\rangle \\ & = A(P) \Bigl\langle 1, 0, u_x(x,y) \Bigr\rangle + B(P) \Bigl\langle 0, 1, u_y(x,y) \Bigr\rangle. \end{align*} Notice that for the first equality in the preceding calculation we used \eqref{eq:pdee} and for the second and the third we used the vector algebra. To celebrate what we calculated we write: \begin{equation} \label{eq:vfut} \Bigl\langle A(P), B(P), C(P) \Bigr\rangle = A(P) \bigl\langle 1, 0, u_x(x,y) \bigr\rangle + B(P) \bigl\langle 0, 1, u_y(x,y) \bigr\rangle. \end{equation} Equality \eqref{eq:vfut} states that the vector $\Bigl\langle A(P), B(P), C(P) \Bigr\rangle$ is a linear combination of the vectors in \eqref{eq:tvxy}. Therefore, the vector $\Bigl\langle A(P), B(P), C(P) \Bigr\rangle$ is tangent to the graph of the function $u.$
The conclusion from the preceding item is the following: To solve \eqref{eq:pde} subject to \eqref{eq:ic} we have to find a surface which is tangent to the vector field \eqref{eq:vf} and passes through the curve \begin{equation}\label{eq:ic-curve} \Bigl\{ \bigl(\xi, 0, f(\xi) \bigr) : \xi \in \mathbb R \Bigr\}. \end{equation} The reason why I use $\xi$ instead of $x$ will be clear in the next section.
Assume that a smooth function $z = u(x,y)$ is a solution of \eqref{eq:pde}. Let $x_0,y_0 \in \mathbb R$ and set $z_0 = u(x_0,y_0).$ Then the point $P_0 = (x_0,y_0,z_0)$ is a point on the graph of the function $z = u(x,y)$. From a multivariable calculus course we know that the vector \begin{equation} \label{eq:tvx} \bigl\langle 1, 0, u_x(x_0,y_0) \bigr\rangle \end{equation} is tangent to the curve \[ \bigl\{ \bigl( x, y_0, u(x,y_0) \bigr) : x_0 -\epsilon \lt x \lt x_0 + \epsilon \bigr\} \quad (\text{where} \quad \epsilon \gt 0 \quad \text{is small}) \] at the point $P_0$. Similarly, the vector \begin{equation} \label{eq:tvy} \bigl\langle 0, 1, u_y(x_0,y_0) \bigr\rangle \end{equation} is tangent to the curve \[ \bigl\{ \bigl( x_0, y, u(x_0,y) \bigr) : y_0 -\epsilon \lt y \lt y_0 + \epsilon \bigr\} \quad (\text{where} \quad \epsilon \gt 0 \quad \text{is small}) \] at the point $P_0$. Thus both vectors \eqref{eq:tvx} and \eqref{eq:tvy} are tangent to the graph of $z = u(x,y)$ at the point $P_0$. Consequently the cross product of the vectors \eqref{eq:tvy} and \eqref{eq:tvx}, that is the vector \begin{equation} \label{eq:nv} \bigl\langle u_x(x_0,y_0), u_y(x_0,y_0), - 1 \bigr\rangle, \end{equation} is orthogonal to the graph of $z = u(x,y)$ at the point $P_0$.
Since we assume that $z= u(x,y)$ is a solution of \eqref{eq:pde}, \eqref{eq:pde} holds for the values $x = x_0$, $y= y_0$ and $z_0 = u(x_0,y_0)$. That is we have \begin{equation*} A(x_0,y_0,z_0) \, u_x(x_0,y_0) + B(x_0,y_0,z_0) \, u_y(x_0,y_0) = C(x_0,y_0,z_0) \end{equation*} The last identity can be written as a dot product of two vectors: \[ \Bigl\langle A(x_0,y_0,z_0), B(x_0,y_0,z_0), C(x_0,y_0,z_0) \Bigr\rangle \cdot \bigl\langle u_x(x_0,y_0), u_y(x_0,y_0), - 1 \bigr\rangle = 0. \] Hence the vector \begin{equation} \label{eq:vvf} \Bigl\langle A(x_0,y_0,z_0), B(x_0,y_0,z_0), C(x_0,y_0,z_0) \Bigr\rangle \end{equation} is orthogonal to the vector in \eqref{eq:nv}. Since the vector in \eqref{eq:nv} is orthogonal to the graph of $z = u(x,y)$ at the point $P_0$, we conclude that the vector \eqref{eq:vvf} is tangent to the graph of $z = u(x,y)$ at the point $P_0$.
The point $P_0 = (x_0,y_0,z_0)$, where $z_0 = u(x_0,y_0)$, in the previous item was an arbitrary point on the graph of $z = u(x,y)$. Therefore, the previous item shows that the vector field \begin{equation} \label{eq:vf} \Bigl\langle A(x,y,z), B(x,y,z), C(x,y,z) \Bigr\rangle \end{equation} is tangent to the surface $z= u(x,y).$ Thus, to solve \eqref{eq:pde} subject to \eqref{eq:ic} we have to find a surface which is tangent to the vector field \eqref{eq:vf} and passes through the curve \begin{equation}\label{eq:ic-curve} \bigl\{ \bigl(\xi, 0, f(\xi) \bigr) : \xi \in \mathbb R \bigr\}. \end{equation} The reason why I use $\xi$ instead of $x$ will be clear in the next section.
There is a simpler way of seeing that the vector in \eqref{eq:vf} is tangent to the surface $z= u(x,y).$ Abbreviate the vector in \eqref{eq:vf} as $\bigl\langle A, B, C \bigr\rangle$ and the partial differential equation in \eqref{eq:pde} as $Au_x+Bu_y=C$. Then we calculate \begin{align*} \bigl\langle A, B, C \bigr\rangle & = \Bigl\langle A, B, Au_x+Bu_y \Bigr\rangle \\ & = \bigl\langle A, 0, Au_x \bigr\rangle + \bigl\langle 0, B, B u_y \bigr\rangle \\ & = A \bigl\langle 1, 0, u_x \bigr\rangle + B \bigl\langle 0, 1, u_y \bigr\rangle. \end{align*} Thus, the vector $\bigl\langle A, B, C \bigr\rangle$ is a linear combination of the vectors $\bigl\langle 1, 0, u_x \bigr\rangle$ and $\bigl\langle 0, 1, u_y \bigr\rangle.$ Since the last two vectors are in the tangent plane of the surface $z= u(x,y)$ their linear combination is also in the tangent plane to this surface.
From the previous section we learned that to solve \eqref{eq:pde} we need to find a surface which is tangent to the vector field \eqref{eq:vf} and passes through the curve \eqref{eq:ic-curve}.
Now we recall that studying systems of ordinary differential equations we learned how to find a curve \[ \Bigl\{ \bigl(X(s), Y(s), Z(s) \bigr) : s \geq 0 \Bigr\} \] which is tangent to the vector field \eqref{eq:vf} and starts at the given point \[ \bigl(X(0), Y(0), Z(0) \bigr). \] Such curve is called an integral curve of the vector field \eqref{eq:vf}. It is obtained by solving the system of ordinary differential equations \begin{equation} \label{eq:sys} \begin{array}{l} \dfrac{dX}{ds} = A(X,Y,Z) \\ \dfrac{dY}{ds} = B(X,Y,Z) \\ \dfrac{dZ}{ds} = C(X,Y,Z) \end{array} \end{equation} subject to the initial conditions \begin{equation}\label{eq:sys-ic} \begin{array}{l} X(0) = \xi \\ Y(0) = 0 \\ Z(0) = f(\xi) \end{array} \end{equation} The ordinary differential equations in \eqref{eq:sys} are called the characteristic equations for partial differential equation \eqref{eq:pde}. These initial conditions are chosen so that the integral curves that we find start at the curve \eqref{eq:ic-curve}.
Assume that we can solve the system of ODEs \eqref{eq:sys} subject to \eqref{eq:sys-ic}. The solution is a triple of functions $(X, Y, Z)$ which depends on two variables $s$ and $\xi$: \begin{equation*} \begin{split} x &= X(s,\xi) \\ y & = Y(s,\xi) \\ z &= Z(s,\xi) \\ \end{split} \end{equation*} For a fixed $\xi$ with $s$ being the independent variable the preceding three equations represent a curve in $xyz$-space. These curves are called the characteristics of the partial differential equation \eqref{eq:pde} subject to the initial condition \eqref{eq:ic}. Now we can vary $\xi.$ By varying $\xi$ we get a set of curves in $xyz$-space that all start at points on the curve \eqref{eq:ic-curve}. The union of all the these curves is the surface which passes through the curve in \eqref{eq:ic-curve} and is tangent to the vector field \eqref{eq:vf}. This surface is the graph of the solution that we are seeking. However, we still do not have the algebraic formula for the solution since the surface is given by its parametric equations depending on $s$ and $\xi.$
Another set of curves that plays important role for the partial differential eqation \eqref{eq:pde} is the set of projected characteristics . The projected characteristics are the curves in $xy$-plane. The significance of the projected characteristics is as follows: From the equations of the characteristics, it is often "easy" to deduce how the solution $u(x,y)$ behaves along the projected characteristics.
There is one more step to find the formula for the function $z= u(x,y)$. We need to solve the system \begin{equation*} \begin{split} x &= X(s,\xi) \\ y & = Y(s,\xi) \end{split} \end{equation*} for $s$ and $\xi$ in terms of $x$ and $y$. Solving this system will lead to two functions \begin{equation*} \begin{split} s &= S(x,y) \\ \xi & = \Xi(x,y). \end{split} \end{equation*} Now the desired function $u(x,y)$ is \[ u(x,y) = Z\bigl( S(x,y), \Xi(x,y) \bigr). \]
This Mathematica notebook contains the examples below worked out in Mathematica. This notebook has been written in Mathematica 8. It should run well in later versions as well. Here is a pdf print-out of this notebook so that you can view it when Mathematica is not available.
The characteristic equations are: \begin{equation*} \begin{array}{l} \dfrac{dX}{ds} = Y \\ \dfrac{dY}{ds} = 3 \\ \dfrac{dZ}{ds} = -Z \end{array} \end{equation*} subject to the initial conditions \begin{equation*} \begin{array}{l} X(0) = \xi \\ Y(0) = 0 \\ Z(0) = \cos(\xi) \end{array} \end{equation*}
The solution of the above system is \begin{equation*} \begin{array}{l} X(s,\xi) = \frac{3}{2}s^2 +\xi \\ Y(s,\xi) = 3s \\ Z(s,\xi) = e^{-s} \cos(\xi) \end{array} \end{equation*}
Now solve the system \begin{equation*} \begin{array}{l} x = \frac{3}{2}s^2 +\xi \\ y = 3s \end{array} \end{equation*} for $s$ and $\xi$ to get \begin{equation*} \begin{array}{l} s = \frac{1}{3} y \\ \xi = x - \frac{1}{6} y^2. \end{array} \end{equation*}
Finally, substitute the formulas for $s$ and $xi$ into $Z(s,\xi) = (\cos \xi)e^{-s}$ to get \[ u(x,y) = e^{-y/3} \cos\!\left(x - \frac{1}{6} y^2\!\right) \] Clearly the domain of $u$ is the entire plane ${\mathbb R}^2$.
The characteristic equations are \begin{equation*} \begin{array}{l} \dfrac{dX}{ds} = 2 \\ \dfrac{dY}{ds} = 1 \\ \dfrac{dZ}{ds} = Z^2 \end{array} \end{equation*} subject to the initial conditions \begin{equation*} \begin{array}{l} X(0) = \xi \\ Y(0) = 0 \\ Z(0) = \cos(\xi) \end{array} \end{equation*}
The solution of the above system is \begin{equation*} \begin{array}{l} X(s,\xi) = 2s +\xi \\ Y(s,\xi) = s \\ Z(s,\xi) = \dfrac{\cos \xi}{1- s \cos \xi } \end{array} \end{equation*}
Now solve the system \begin{equation*} \begin{array}{l} x = 2s +\xi \\ y = s \end{array} \end{equation*} for $s$ and $\xi$ to get \begin{equation*} \begin{array}{l} s = y \\ \xi = x - 2 y. \end{array} \end{equation*}
Finally, substitute the formulas for $s$ and $\xi$ into $Z(s,\xi) = \dfrac{\cos \xi}{1- s \cos \xi }$ to get \[ u(x,y) = \frac{\cos(x-2y)}{1- y \cos(x-2y)} \]

problem solving method of characteristics

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This problem is solved in the Mathematica notebook MoC_Burgers_eq1_v12.nb . I printed this notebook as a pdf file MoC_Burgers_eq1_v12.pdf , so that you can read it without access to Mathematica.
My thinking about Burgers' Equation is on the website Burgers' Equation .
Solve the following first order PDE \begin{equation*} y\, u_x + u_y = 0 \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \in \mathbb{R}. \end{equation*} You can try $f(x) = x$, $f(x) = x^2$, $f(x) = \cos x$ as some specific examples.
Solve the following first order PDE \begin{equation*} u_x - x u_y = 0 \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \gt 0. \end{equation*} You can try $f(x) = x$, $f(x) = x^2$, $f(x) = \cos x$ as some specific examples.
Solve the following first order PDE \begin{equation*} y\, u_x - x u_y = 0 \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \gt 0. \end{equation*} You can try $f(x) = x^2$, $f(x) = (\sin x)^2$, $f(x) = \cos x$ as some specific examples.
Solve the following first order PDE \begin{equation*} u_x + x u_y = u \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the boundary condition \begin{equation*} u(0,y) = g(y) \quad \text{for} \quad x \in \mathbb R. \end{equation*} You can try $g(y) = y$, $g(y) = \cos y$ as some specific examples.
Solve the following first order PDE \begin{equation*} y\, u_x - x u_y = u \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = f(x) \quad \text{for} \quad x \geq 0. \end{equation*} You can try $f(x) = x^2$, $f(x) = (\sin x)^2$, as some specific examples.
Solve the following first order PDE \begin{equation*} x u_x + y\, u_y = u \quad \text{in} \quad \mathbb R^2 \end{equation*} subject to the condition \begin{equation*} u\bigl(\cos \theta, \sin \theta\bigr) = 1 \quad \text{for} \quad 0 \leq \theta \lt 2 \pi. \end{equation*}
Solve the following first order PDE \begin{equation*} y u_x + u_y = x \quad \text{in} \quad \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\} \end{equation*} subject to the boundary condition \begin{equation*} u(x,0) = x^2 \quad \text{for} \quad x \in \mathbb R. \end{equation*}
For an arbitrary differentiable function $f$ find the solution $u(x,y)$ of the above problem and determine its maximum domain $U\subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$.
Under which condition on $f$ the solution $u(x,y)$ will be defined on $\bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$.
Does there exist an open and convex set $U \subseteq \bigl\{(x,y) \in \mathbb R^2 : y \gt 0 \bigr\}$ whose boundary is the $x$-axis such that the above problem has a solution defined on $U$?
Can you modify the boxed formula in the statement of the problem in such a way that your explorations in part a. lead to the solution of such a modified problem? State explicitly the problem and its solution.
Solve the following first order PDE \begin{equation*} (1+x^2) u_x + 2 x y \, u_y = 0 \quad \text{in} \quad {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(0,y) = g(y) \quad \text{for all} \quad y \in \mathbb R. \end{equation*}
Solve the following first order PDE \begin{equation*} (1+x^2) u_x - 2 x y \, u_y = 0 \quad \text{in} \quad {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(0,y) = g(y) \quad \text{for all} \quad y \in \mathbb R. \end{equation*}
Solve the following first order PDE \begin{equation*} x \, u_x(x,y) + y \, u_y(x,y) = 2 u(x,y) \ln \bigl(\bigr) u(x,y) \quad \text{in} \quad (x,y) \in {\mathbb R}^2 \end{equation*} subject to the condition \begin{equation*} u(x, 1) = e^{x^2-1} \quad \text{for all} \quad x \in \mathbb R. \end{equation*}
Solve the given PDE. Notice that the solution will be a piecewise defined function with the domain \[ \bigl\{ (x,y) \in {\mathbb R}^2 : x \geq 0, \ y \geq 0 \bigr\}. \]
Prove that the partial derivatives $u_x(x,y)$ ang $u_y(x,y)$ of the solution $u(x,y)$ provided in the preceding item are defined on the open quadrant \[ \bigl\{ (x,y) \in {\mathbb R}^2 : x \gt 0, \ y \gt 0 \bigr\} \] and satify the given PDE on this open quadrant.
Illustrate the solution in Mathematica with the functions \[ f(x) = h(x) = \cos x \quad x \in [0,+\infty). \] and \[ f(x) = - \sin x, \qquad h(y) = c \sin y \quad x,y \in [0,+\infty). \] For $c \gt 0$, consider several different values, for example $c \in \{1/2, 1, 2\}.$
Use the method of characteristics to solve the given initial value problem.
What is the largest domain in which the solution of this problem is defined?
Give an explicit formula for the solution $u(x,y)$.
Think of the variable $y$ as being time. Explain how the solution $u(x,y)$ evolves in time. That is, consider $u(x,y)$ as a function of $x$ and explain how this function changes with increasing $y \gt 1$ and decreasing time $y \lt 1$. Please pay special attention to the maximums, minimums, and zero values of the function $u(x,y)$ at a specific time $y$ with $y$ a large positive number and $y$ a small positive number.

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Critical thinking and problem-solving, jump to: , what is critical thinking, characteristics of critical thinking, why teach critical thinking.

Teaching Strategies to Help Promote Critical Thinking Skills

References and Resources

When examining the vast literature on critical thinking, various definitions of critical thinking emerge. Here are some samples:

"Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action" (Scriven, 1996).
"Most formal definitions characterize critical thinking as the intentional application of rational, higher order thinking skills, such as analysis, synthesis, problem recognition and problem solving, inference, and evaluation" (Angelo, 1995, p. 6).
"Critical thinking is thinking that assesses itself" (Center for Critical Thinking, 1996b).
"Critical thinking is the ability to think about one's thinking in such a way as 1. To recognize its strengths and weaknesses and, as a result, 2. To recast the thinking in improved form" (Center for Critical Thinking, 1996c).

Perhaps the simplest definition is offered by Beyer (1995) : "Critical thinking... means making reasoned judgments" (p. 8). Basically, Beyer sees critical thinking as using criteria to judge the quality of something, from cooking to a conclusion of a research paper. In essence, critical thinking is a disciplined manner of thought that a person uses to assess the validity of something (statements, news stories, arguments, research, etc.).

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Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking, "Ambiguity and doubt serve a critical-thinking function and are a necessary and even a productive part of the process" (p. 56).

Another characteristic of critical thinking identified by many sources is metacognition. Metacognition is thinking about one's own thinking. More specifically, "metacognition is being aware of one's thinking as one performs specific tasks and then using this awareness to control what one is doing" (Jones & Ratcliff, 1993, p. 10 ).

In the book, Critical Thinking, Beyer elaborately explains what he sees as essential aspects of critical thinking. These are:

Dispositions: Critical thinkers are skeptical, open-minded, value fair-mindedness, respect evidence and reasoning, respect clarity and precision, look at different points of view, and will change positions when reason leads them to do so.
Criteria: To think critically, must apply criteria. Need to have conditions that must be met for something to be judged as believable. Although the argument can be made that each subject area has different criteria, some standards apply to all subjects. "... an assertion must... be based on relevant, accurate facts; based on credible sources; precise; unbiased; free from logical fallacies; logically consistent; and strongly reasoned" (p. 12).
Argument: Is a statement or proposition with supporting evidence. Critical thinking involves identifying, evaluating, and constructing arguments.
Reasoning: The ability to infer a conclusion from one or multiple premises. To do so requires examining logical relationships among statements or data.
Point of View: The way one views the world, which shapes one's construction of meaning. In a search for understanding, critical thinkers view phenomena from many different points of view.
Procedures for Applying Criteria: Other types of thinking use a general procedure. Critical thinking makes use of many procedures. These procedures include asking questions, making judgments, and identifying assumptions.

Oliver & Utermohlen (1995) see students as too often being passive receptors of information. Through technology, the amount of information available today is massive. This information explosion is likely to continue in the future. Students need a guide to weed through the information and not just passively accept it. Students need to "develop and effectively apply critical thinking skills to their academic studies, to the complex problems that they will face, and to the critical choices they will be forced to make as a result of the information explosion and other rapid technological changes" (Oliver & Utermohlen, p. 1 ).

As mentioned in the section, Characteristics of Critical Thinking , critical thinking involves questioning. It is important to teach students how to ask good questions, to think critically, in order to continue the advancement of the very fields we are teaching. "Every field stays alive only to the extent that fresh questions are generated and taken seriously" (Center for Critical Thinking, 1996a ).

Beyer sees the teaching of critical thinking as important to the very state of our nation. He argues that to live successfully in a democracy, people must be able to think critically in order to make sound decisions about personal and civic affairs. If students learn to think critically, then they can use good thinking as the guide by which they live their lives.

Teaching Strategies to Help Promote Critical Thinking

The 1995, Volume 22, issue 1, of the journal, Teaching of Psychology , is devoted to the teaching critical thinking. Most of the strategies included in this section come from the various articles that compose this issue.

CATS (Classroom Assessment Techniques): Angelo stresses the use of ongoing classroom assessment as a way to monitor and facilitate students' critical thinking. An example of a CAT is to ask students to write a "Minute Paper" responding to questions such as "What was the most important thing you learned in today's class? What question related to this session remains uppermost in your mind?" The teacher selects some of the papers and prepares responses for the next class meeting.
Cooperative Learning Strategies: Cooper (1995) argues that putting students in group learning situations is the best way to foster critical thinking. "In properly structured cooperative learning environments, students perform more of the active, critical thinking with continuous support and feedback from other students and the teacher" (p. 8).
Case Study /Discussion Method: McDade (1995) describes this method as the teacher presenting a case (or story) to the class without a conclusion. Using prepared questions, the teacher then leads students through a discussion, allowing students to construct a conclusion for the case.
Using Questions: King (1995) identifies ways of using questions in the classroom:
Reciprocal Peer Questioning: Following lecture, the teacher displays a list of question stems (such as, "What are the strengths and weaknesses of...). Students must write questions about the lecture material. In small groups, the students ask each other the questions. Then, the whole class discusses some of the questions from each small group.
Reader's Questions: Require students to write questions on assigned reading and turn them in at the beginning of class. Select a few of the questions as the impetus for class discussion.
Conference Style Learning: The teacher does not "teach" the class in the sense of lecturing. The teacher is a facilitator of a conference. Students must thoroughly read all required material before class. Assigned readings should be in the zone of proximal development. That is, readings should be able to be understood by students, but also challenging. The class consists of the students asking questions of each other and discussing these questions. The teacher does not remain passive, but rather, helps "direct and mold discussions by posing strategic questions and helping students build on each others' ideas" (Underwood & Wald, 1995, p. 18 ).
Use Writing Assignments: Wade sees the use of writing as fundamental to developing critical thinking skills. "With written assignments, an instructor can encourage the development of dialectic reasoning by requiring students to argue both [or more] sides of an issue" (p. 24).
Written dialogues: Give students written dialogues to analyze. In small groups, students must identify the different viewpoints of each participant in the dialogue. Must look for biases, presence or exclusion of important evidence, alternative interpretations, misstatement of facts, and errors in reasoning. Each group must decide which view is the most reasonable. After coming to a conclusion, each group acts out their dialogue and explains their analysis of it.
Spontaneous Group Dialogue: One group of students are assigned roles to play in a discussion (such as leader, information giver, opinion seeker, and disagreer). Four observer groups are formed with the functions of determining what roles are being played by whom, identifying biases and errors in thinking, evaluating reasoning skills, and examining ethical implications of the content.
Ambiguity: Strohm & Baukus advocate producing much ambiguity in the classroom. Don't give students clear cut material. Give them conflicting information that they must think their way through.
Angelo, T. A. (1995). Beginning the dialogue: Thoughts on promoting critical thinking: Classroom assessment for critical thinking. Teaching of Psychology, 22(1), 6-7.
Beyer, B. K. (1995). Critical thinking. Bloomington, IN: Phi Delta Kappa Educational Foundation.
Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Cooper, J. L. (1995). Cooperative learning and critical thinking. Teaching of Psychology, 22(1), 7-8.
Jones, E. A. & Ratcliff, G. (1993). Critical thinking skills for college students. National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA. (Eric Document Reproduction Services No. ED 358 772)
King, A. (1995). Designing the instructional process to enhance critical thinking across the curriculum: Inquiring minds really do want to know: Using questioning to teach critical thinking. Teaching of Psychology, 22 (1) , 13-17.
McDade, S. A. (1995). Case study pedagogy to advance critical thinking. Teaching Psychology, 22(1), 9-10.
Oliver, H. & Utermohlen, R. (1995). An innovative teaching strategy: Using critical thinking to give students a guide to the future.(Eric Document Reproduction Services No. 389 702)
Robertson, J. F. & Rane-Szostak, D. (1996). Using dialogues to develop critical thinking skills: A practical approach. Journal of Adolescent & Adult Literacy, 39(7), 552-556.
Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
Strohm, S. M., & Baukus, R. A. (1995). Strategies for fostering critical thinking skills. Journalism and Mass Communication Educator, 50 (1), 55-62.
Underwood, M. K., & Wald, R. L. (1995). Conference-style learning: A method for fostering critical thinking with heart. Teaching Psychology, 22(1), 17-21.
Wade, C. (1995). Using writing to develop and assess critical thinking. Teaching of Psychology, 22(1), 24-28.

On the Internet

Carr, K. S. (1990). How can we teach critical thinking. Eric Digest. [On-line]. Available HTTP: http://ericps.ed.uiuc.edu/eece/pubs/digests/1990/carr90.html
The Center for Critical Thinking (1996). Home Page. Available HTTP: http://www.criticalthinking.org/University/
Ennis, Bob (No date). Critical thinking. [On-line], April 4, 1997. Available HTTP: http://www.cof.orst.edu/cof/teach/for442/ct.htm
Montclair State University (1995). Curriculum resource center. Critical thinking resources: An annotated bibliography. [On-line]. Available HTTP: http://www.montclair.edu/Pages/CRC/Bibliographies/CriticalThinking.html
No author, No date. Critical Thinking is ... [On-line], April 4, 1997. Available HTTP: http://library.usask.ca/ustudy/critical/
Sheridan, Marcia (No date). Internet education topics hotlink page. [On-line], April 4, 1997. Available HTTP: http://sun1.iusb.edu/~msherida/topics/critical.html

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14.3 Problem Solving and Decision Making in Groups

Learning objectives.

Discuss the common components and characteristics of problems.
Explain the five steps of the group problem-solving process.
Describe the brainstorming and discussion that should take place before the group makes a decision.
Compare and contrast the different decision-making techniques.
Discuss the various influences on decision making.

Although the steps of problem solving and decision making that we will discuss next may seem obvious, we often don’t think to or choose not to use them. Instead, we start working on a problem and later realize we are lost and have to backtrack. I’m sure we’ve all reached a point in a project or task and had the “OK, now what?” moment. I’ve recently taken up some carpentry projects as a functional hobby, and I have developed a great respect for the importance of advanced planning. It’s frustrating to get to a crucial point in building or fixing something only to realize that you have to unscrew a support board that you already screwed in, have to drive back to the hardware store to get something that you didn’t think to get earlier, or have to completely start over. In this section, we will discuss the group problem-solving process, methods of decision making, and influences on these processes.

Group Problem Solving

The problem-solving process involves thoughts, discussions, actions, and decisions that occur from the first consideration of a problematic situation to the goal. The problems that groups face are varied, but some common problems include budgeting funds, raising funds, planning events, addressing customer or citizen complaints, creating or adapting products or services to fit needs, supporting members, and raising awareness about issues or causes.

Problems of all sorts have three common components (Adams & Galanes, 2009):

An undesirable situation. When conditions are desirable, there isn’t a problem.
A desired situation. Even though it may only be a vague idea, there is a drive to better the undesirable situation. The vague idea may develop into a more precise goal that can be achieved, although solutions are not yet generated.
Obstacles between undesirable and desirable situation. These are things that stand in the way between the current situation and the group’s goal of addressing it. This component of a problem requires the most work, and it is the part where decision making occurs. Some examples of obstacles include limited funding, resources, personnel, time, or information. Obstacles can also take the form of people who are working against the group, including people resistant to change or people who disagree.

Discussion of these three elements of a problem helps the group tailor its problem-solving process, as each problem will vary. While these three general elements are present in each problem, the group should also address specific characteristics of the problem. Five common and important characteristics to consider are task difficulty, number of possible solutions, group member interest in problem, group member familiarity with problem, and the need for solution acceptance (Adams & Galanes, 2009).

Task difficulty. Difficult tasks are also typically more complex. Groups should be prepared to spend time researching and discussing a difficult and complex task in order to develop a shared foundational knowledge. This typically requires individual work outside of the group and frequent group meetings to share information.
Number of possible solutions. There are usually multiple ways to solve a problem or complete a task, but some problems have more potential solutions than others. Figuring out how to prepare a beach house for an approaching hurricane is fairly complex and difficult, but there are still a limited number of things to do—for example, taping and boarding up windows; turning off water, electricity, and gas; trimming trees; and securing loose outside objects. Other problems may be more creatively based. For example, designing a new restaurant may entail using some standard solutions but could also entail many different types of innovation with layout and design.
Group member interest in problem. When group members are interested in the problem, they will be more engaged with the problem-solving process and invested in finding a quality solution. Groups with high interest in and knowledge about the problem may want more freedom to develop and implement solutions, while groups with low interest may prefer a leader who provides structure and direction.
Group familiarity with problem. Some groups encounter a problem regularly, while other problems are more unique or unexpected. A family who has lived in hurricane alley for decades probably has a better idea of how to prepare its house for a hurricane than does a family that just recently moved from the Midwest. Many groups that rely on funding have to revisit a budget every year, and in recent years, groups have had to get more creative with budgets as funding has been cut in nearly every sector. When group members aren’t familiar with a problem, they will need to do background research on what similar groups have done and may also need to bring in outside experts.
Need for solution acceptance. In this step, groups must consider how many people the decision will affect and how much “buy-in” from others the group needs in order for their solution to be successfully implemented. Some small groups have many stakeholders on whom the success of a solution depends. Other groups are answerable only to themselves. When a small group is planning on building a new park in a crowded neighborhood or implementing a new policy in a large business, it can be very difficult to develop solutions that will be accepted by all. In such cases, groups will want to poll those who will be affected by the solution and may want to do a pilot implementation to see how people react. Imposing an excellent solution that doesn’t have buy-in from stakeholders can still lead to failure.

Group problem solving can be a confusing puzzle unless it is approached systematically.

Muness Castle – Problem Solving – CC BY-SA 2.0.

Group Problem-Solving Process

There are several variations of similar problem-solving models based on US American scholar John Dewey’s reflective thinking process (Bormann & Bormann, 1988). As you read through the steps in the process, think about how you can apply what we learned regarding the general and specific elements of problems. Some of the following steps are straightforward, and they are things we would logically do when faced with a problem. However, taking a deliberate and systematic approach to problem solving has been shown to benefit group functioning and performance. A deliberate approach is especially beneficial for groups that do not have an established history of working together and will only be able to meet occasionally. Although a group should attend to each step of the process, group leaders or other group members who facilitate problem solving should be cautious not to dogmatically follow each element of the process or force a group along. Such a lack of flexibility could limit group member input and negatively affect the group’s cohesion and climate.

Step 1: Define the Problem

Define the problem by considering the three elements shared by every problem: the current undesirable situation, the goal or more desirable situation, and obstacles in the way (Adams & Galanes, 2009). At this stage, group members share what they know about the current situation, without proposing solutions or evaluating the information. Here are some good questions to ask during this stage: What is the current difficulty? How did we come to know that the difficulty exists? Who/what is involved? Why is it meaningful/urgent/important? What have the effects been so far? What, if any, elements of the difficulty require clarification? At the end of this stage, the group should be able to compose a single sentence that summarizes the problem called a problem statement . Avoid wording in the problem statement or question that hints at potential solutions. A small group formed to investigate ethical violations of city officials could use the following problem statement: “Our state does not currently have a mechanism for citizens to report suspected ethical violations by city officials.”

Step 2: Analyze the Problem

During this step a group should analyze the problem and the group’s relationship to the problem. Whereas the first step involved exploring the “what” related to the problem, this step focuses on the “why.” At this stage, group members can discuss the potential causes of the difficulty. Group members may also want to begin setting out an agenda or timeline for the group’s problem-solving process, looking forward to the other steps. To fully analyze the problem, the group can discuss the five common problem variables discussed before. Here are two examples of questions that the group formed to address ethics violations might ask: Why doesn’t our city have an ethics reporting mechanism? Do cities of similar size have such a mechanism? Once the problem has been analyzed, the group can pose a problem question that will guide the group as it generates possible solutions. “How can citizens report suspected ethical violations of city officials and how will such reports be processed and addressed?” As you can see, the problem question is more complex than the problem statement, since the group has moved on to more in-depth discussion of the problem during step 2.

Step 3: Generate Possible Solutions

During this step, group members generate possible solutions to the problem. Again, solutions should not be evaluated at this point, only proposed and clarified. The question should be what could we do to address this problem, not what should we do to address it. It is perfectly OK for a group member to question another person’s idea by asking something like “What do you mean?” or “Could you explain your reasoning more?” Discussions at this stage may reveal a need to return to previous steps to better define or more fully analyze a problem. Since many problems are multifaceted, it is necessary for group members to generate solutions for each part of the problem separately, making sure to have multiple solutions for each part. Stopping the solution-generating process prematurely can lead to groupthink. For the problem question previously posed, the group would need to generate solutions for all three parts of the problem included in the question. Possible solutions for the first part of the problem (How can citizens report ethical violations?) may include “online reporting system, e-mail, in-person, anonymously, on-the-record,” and so on. Possible solutions for the second part of the problem (How will reports be processed?) may include “daily by a newly appointed ethics officer, weekly by a nonpartisan nongovernment employee,” and so on. Possible solutions for the third part of the problem (How will reports be addressed?) may include “by a newly appointed ethics commission, by the accused’s supervisor, by the city manager,” and so on.

Step 4: Evaluate Solutions

During this step, solutions can be critically evaluated based on their credibility, completeness, and worth. Once the potential solutions have been narrowed based on more obvious differences in relevance and/or merit, the group should analyze each solution based on its potential effects—especially negative effects. Groups that are required to report the rationale for their decision or whose decisions may be subject to public scrutiny would be wise to make a set list of criteria for evaluating each solution. Additionally, solutions can be evaluated based on how well they fit with the group’s charge and the abilities of the group. To do this, group members may ask, “Does this solution live up to the original purpose or mission of the group?” and “Can the solution actually be implemented with our current resources and connections?” and “How will this solution be supported, funded, enforced, and assessed?” Secondary tensions and substantive conflict, two concepts discussed earlier, emerge during this step of problem solving, and group members will need to employ effective critical thinking and listening skills.

Decision making is part of the larger process of problem solving and it plays a prominent role in this step. While there are several fairly similar models for problem solving, there are many varied decision-making techniques that groups can use. For example, to narrow the list of proposed solutions, group members may decide by majority vote, by weighing the pros and cons, or by discussing them until a consensus is reached. There are also more complex decision-making models like the “six hats method,” which we will discuss later. Once the final decision is reached, the group leader or facilitator should confirm that the group is in agreement. It may be beneficial to let the group break for a while or even to delay the final decision until a later meeting to allow people time to evaluate it outside of the group context.

Step 5: Implement and Assess the Solution

Implementing the solution requires some advanced planning, and it should not be rushed unless the group is operating under strict time restraints or delay may lead to some kind of harm. Although some solutions can be implemented immediately, others may take days, months, or years. As was noted earlier, it may be beneficial for groups to poll those who will be affected by the solution as to their opinion of it or even to do a pilot test to observe the effectiveness of the solution and how people react to it. Before implementation, groups should also determine how and when they would assess the effectiveness of the solution by asking, “How will we know if the solution is working or not?” Since solution assessment will vary based on whether or not the group is disbanded, groups should also consider the following questions: If the group disbands after implementation, who will be responsible for assessing the solution? If the solution fails, will the same group reconvene or will a new group be formed?

Once a solution has been reached and the group has the “green light” to implement it, it should proceed deliberately and cautiously, making sure to consider possible consequences and address them as needed.

Jocko Benoit – Prodigal Light – CC BY-NC-ND 2.0.

Certain elements of the solution may need to be delegated out to various people inside and outside the group. Group members may also be assigned to implement a particular part of the solution based on their role in the decision making or because it connects to their area of expertise. Likewise, group members may be tasked with publicizing the solution or “selling” it to a particular group of stakeholders. Last, the group should consider its future. In some cases, the group will get to decide if it will stay together and continue working on other tasks or if it will disband. In other cases, outside forces determine the group’s fate.

“Getting Competent”

Problem Solving and Group Presentations

Giving a group presentation requires that individual group members and the group as a whole solve many problems and make many decisions. Although having more people involved in a presentation increases logistical difficulties and has the potential to create more conflict, a well-prepared and well-delivered group presentation can be more engaging and effective than a typical presentation. The main problems facing a group giving a presentation are (1) dividing responsibilities, (2) coordinating schedules and time management, and (3) working out the logistics of the presentation delivery.

In terms of dividing responsibilities, assigning individual work at the first meeting and then trying to fit it all together before the presentation (which is what many college students do when faced with a group project) is not the recommended method. Integrating content and visual aids created by several different people into a seamless final product takes time and effort, and the person “stuck” with this job at the end usually ends up developing some resentment toward his or her group members. While it’s OK for group members to do work independently outside of group meetings, spend time working together to help set up some standards for content and formatting expectations that will help make later integration of work easier. Taking the time to complete one part of the presentation together can help set those standards for later individual work. Discuss the roles that various group members will play openly so there isn’t role confusion. There could be one point person for keeping track of the group’s progress and schedule, one point person for communication, one point person for content integration, one point person for visual aids, and so on. Each person shouldn’t do all that work on his or her own but help focus the group’s attention on his or her specific area during group meetings (Stanton, 2009).

Scheduling group meetings is one of the most challenging problems groups face, given people’s busy lives. From the beginning, it should be clearly communicated that the group needs to spend considerable time in face-to-face meetings, and group members should know that they may have to make an occasional sacrifice to attend. Especially important is the commitment to scheduling time to rehearse the presentation. Consider creating a contract of group guidelines that includes expectations for meeting attendance to increase group members’ commitment.

Group presentations require members to navigate many logistics of their presentation. While it may be easier for a group to assign each member to create a five-minute segment and then transition from one person to the next, this is definitely not the most engaging method. Creating a master presentation and then assigning individual speakers creates a more fluid and dynamic presentation and allows everyone to become familiar with the content, which can help if a person doesn’t show up to present and during the question-and-answer section. Once the content of the presentation is complete, figure out introductions, transitions, visual aids, and the use of time and space (Stanton, 2012). In terms of introductions, figure out if one person will introduce all the speakers at the beginning, if speakers will introduce themselves at the beginning, or if introductions will occur as the presentation progresses. In terms of transitions, make sure each person has included in his or her speaking notes when presentation duties switch from one person to the next. Visual aids have the potential to cause hiccups in a group presentation if they aren’t fluidly integrated. Practicing with visual aids and having one person control them may help prevent this. Know how long your presentation is and know how you’re going to use the space. Presenters should know how long the whole presentation should be and how long each of their segments should be so that everyone can share the responsibility of keeping time. Also consider the size and layout of the presentation space. You don’t want presenters huddled in a corner until it’s their turn to speak or trapped behind furniture when their turn comes around.

Of the three main problems facing group presenters, which do you think is the most challenging and why?
Why do you think people tasked with a group presentation (especially students) prefer to divide the parts up and have members work on them independently before coming back together and integrating each part? What problems emerge from this method? In what ways might developing a master presentation and then assigning parts to different speakers be better than the more divided method? What are the drawbacks to the master presentation method?

Decision Making in Groups

We all engage in personal decision making daily, and we all know that some decisions are more difficult than others. When we make decisions in groups, we face some challenges that we do not face in our personal decision making, but we also stand to benefit from some advantages of group decision making (Napier & Gershenfeld, 2004). Group decision making can appear fair and democratic but really only be a gesture that covers up the fact that certain group members or the group leader have already decided. Group decision making also takes more time than individual decisions and can be burdensome if some group members do not do their assigned work, divert the group with self-centered or unproductive role behaviors, or miss meetings. Conversely, though, group decisions are often more informed, since all group members develop a shared understanding of a problem through discussion and debate. The shared understanding may also be more complex and deep than what an individual would develop, because the group members are exposed to a variety of viewpoints that can broaden their own perspectives. Group decisions also benefit from synergy, one of the key advantages of group communication that we discussed earlier. Most groups do not use a specific method of decision making, perhaps thinking that they’ll work things out as they go. This can lead to unequal participation, social loafing, premature decisions, prolonged discussion, and a host of other negative consequences. So in this section we will learn some practices that will prepare us for good decision making and some specific techniques we can use to help us reach a final decision.

Brainstorming before Decision Making

Before groups can make a decision, they need to generate possible solutions to their problem. The most commonly used method is brainstorming, although most people don’t follow the recommended steps of brainstorming. As you’ll recall, brainstorming refers to the quick generation of ideas free of evaluation. The originator of the term brainstorming said the following four rules must be followed for the technique to be effective (Osborn, 1959):

Evaluation of ideas is forbidden.
Wild and crazy ideas are encouraged.
Quantity of ideas, not quality, is the goal.
New combinations of ideas presented are encouraged.

To make brainstorming more of a decision-making method rather than an idea-generating method, group communication scholars have suggested additional steps that precede and follow brainstorming (Cragan & Wright, 1991).

Do a warm-up brainstorming session. Some people are more apprehensive about publicly communicating their ideas than others are, and a warm-up session can help ease apprehension and prime group members for task-related idea generation. The warm-up can be initiated by anyone in the group and should only go on for a few minutes. To get things started, a person could ask, “If our group formed a band, what would we be called?” or “What other purposes could a mailbox serve?” In the previous examples, the first warm up gets the group’s more abstract creative juices flowing, while the second focuses more on practical and concrete ideas.
Do the actual brainstorming session. This session shouldn’t last more than thirty minutes and should follow the four rules of brainstorming mentioned previously. To ensure that the fourth rule is realized, the facilitator could encourage people to piggyback off each other’s ideas.
Eliminate duplicate ideas. After the brainstorming session is over, group members can eliminate (without evaluating) ideas that are the same or very similar.
Clarify, organize, and evaluate ideas. Before evaluation, see if any ideas need clarification. Then try to theme or group ideas together in some orderly fashion. Since “wild and crazy” ideas are encouraged, some suggestions may need clarification. If it becomes clear that there isn’t really a foundation to an idea and that it is too vague or abstract and can’t be clarified, it may be eliminated. As a caution though, it may be wise to not throw out off-the-wall ideas that are hard to categorize and to instead put them in a miscellaneous or “wild and crazy” category.

Discussion before Decision Making

The nominal group technique guides decision making through a four-step process that includes idea generation and evaluation and seeks to elicit equal contributions from all group members (Delbecq & Ven de Ven, 1971). This method is useful because the procedure involves all group members systematically, which fixes the problem of uneven participation during discussions. Since everyone contributes to the discussion, this method can also help reduce instances of social loafing. To use the nominal group technique, do the following:

Silently and individually list ideas.
Create a master list of ideas.
Clarify ideas as needed.
Take a secret vote to rank group members’ acceptance of ideas.

During the first step, have group members work quietly, in the same space, to write down every idea they have to address the task or problem they face. This shouldn’t take more than twenty minutes. Whoever is facilitating the discussion should remind group members to use brainstorming techniques, which means they shouldn’t evaluate ideas as they are generated. Ask group members to remain silent once they’ve finished their list so they do not distract others.

During the second step, the facilitator goes around the group in a consistent order asking each person to share one idea at a time. As the idea is shared, the facilitator records it on a master list that everyone can see. Keep track of how many times each idea comes up, as that could be an idea that warrants more discussion. Continue this process until all the ideas have been shared. As a note to facilitators, some group members may begin to edit their list or self-censor when asked to provide one of their ideas. To limit a person’s apprehension with sharing his or her ideas and to ensure that each idea is shared, I have asked group members to exchange lists with someone else so they can share ideas from the list they receive without fear of being personally judged.

During step three, the facilitator should note that group members can now ask for clarification on ideas on the master list. Do not let this discussion stray into evaluation of ideas. To help avoid an unnecessarily long discussion, it may be useful to go from one person to the next to ask which ideas need clarifying and then go to the originator(s) of the idea in question for clarification.

During the fourth step, members use a voting ballot to rank the acceptability of the ideas on the master list. If the list is long, you may ask group members to rank only their top five or so choices. The facilitator then takes up the secret ballots and reviews them in a random order, noting the rankings of each idea. Ideally, the highest ranked idea can then be discussed and decided on. The nominal group technique does not carry a group all the way through to the point of decision; rather, it sets the group up for a roundtable discussion or use of some other method to evaluate the merits of the top ideas.

Specific Decision-Making Techniques

Some decision-making techniques involve determining a course of action based on the level of agreement among the group members. These methods include majority, expert, authority, and consensus rule. Table 14.1 “Pros and Cons of Agreement-Based Decision-Making Techniques” reviews the pros and cons of each of these methods.

Majority rule is a simple method of decision making based on voting. In most cases a majority is considered half plus one.

Becky McCray – Voting – CC BY-NC-ND 2.0.

Majority rule is a commonly used decision-making technique in which a majority (one-half plus one) must agree before a decision is made. A show-of-hands vote, a paper ballot, or an electronic voting system can determine the majority choice. Many decision-making bodies, including the US House of Representatives, Senate, and Supreme Court, use majority rule to make decisions, which shows that it is often associated with democratic decision making, since each person gets one vote and each vote counts equally. Of course, other individuals and mediated messages can influence a person’s vote, but since the voting power is spread out over all group members, it is not easy for one person or party to take control of the decision-making process. In some cases—for example, to override a presidential veto or to amend the constitution—a super majority of two-thirds may be required to make a decision.

Minority rule is a decision-making technique in which a designated authority or expert has final say over a decision and may or may not consider the input of other group members. When a designated expert makes a decision by minority rule, there may be buy-in from others in the group, especially if the members of the group didn’t have relevant knowledge or expertise. When a designated authority makes decisions, buy-in will vary based on group members’ level of respect for the authority. For example, decisions made by an elected authority may be more accepted by those who elected him or her than by those who didn’t. As with majority rule, this technique can be time saving. Unlike majority rule, one person or party can have control over the decision-making process. This type of decision making is more similar to that used by monarchs and dictators. An obvious negative consequence of this method is that the needs or wants of one person can override the needs and wants of the majority. A minority deciding for the majority has led to negative consequences throughout history. The white Afrikaner minority that ruled South Africa for decades instituted apartheid, which was a system of racial segregation that disenfranchised and oppressed the majority population. The quality of the decision and its fairness really depends on the designated expert or authority.

Consensus rule is a decision-making technique in which all members of the group must agree on the same decision. On rare occasions, a decision may be ideal for all group members, which can lead to unanimous agreement without further debate and discussion. Although this can be positive, be cautious that this isn’t a sign of groupthink. More typically, consensus is reached only after lengthy discussion. On the plus side, consensus often leads to high-quality decisions due to the time and effort it takes to get everyone in agreement. Group members are also more likely to be committed to the decision because of their investment in reaching it. On the negative side, the ultimate decision is often one that all group members can live with but not one that’s ideal for all members. Additionally, the process of arriving at consensus also includes conflict, as people debate ideas and negotiate the interpersonal tensions that may result.

Table 14.1 Pros and Cons of Agreement-Based Decision-Making Techniques

Decision-Making Technique	Pros	Cons
Majority rule
Minority rule by expert
Minority rule by authority
Consensus rule

“Getting Critical”

Six Hats Method of Decision Making

Edward de Bono developed the Six Hats method of thinking in the late 1980s, and it has since become a regular feature in decision-making training in business and professional contexts (de Bono, 1985). The method’s popularity lies in its ability to help people get out of habitual ways of thinking and to allow group members to play different roles and see a problem or decision from multiple points of view. The basic idea is that each of the six hats represents a different way of thinking, and when we figuratively switch hats, we switch the way we think. The hats and their style of thinking are as follows:

White hat. Objective—focuses on seeking information such as data and facts and then processes that information in a neutral way.
Red hat. Emotional—uses intuition, gut reactions, and feelings to judge information and suggestions.
Black hat. Negative—focuses on potential risks, points out possibilities for failure, and evaluates information cautiously and defensively.
Yellow hat. Positive—is optimistic about suggestions and future outcomes, gives constructive and positive feedback, points out benefits and advantages.
Green hat. Creative—tries to generate new ideas and solutions, thinks “outside the box.”
Blue hat. Philosophical—uses metacommunication to organize and reflect on the thinking and communication taking place in the group, facilitates who wears what hat and when group members change hats.

Specific sequences or combinations of hats can be used to encourage strategic thinking. For example, the group leader may start off wearing the Blue Hat and suggest that the group start their decision-making process with some “White Hat thinking” in order to process through facts and other available information. During this stage, the group could also process through what other groups have done when faced with a similar problem. Then the leader could begin an evaluation sequence starting with two minutes of “Yellow Hat thinking” to identify potential positive outcomes, then “Black Hat thinking” to allow group members to express reservations about ideas and point out potential problems, then “Red Hat thinking” to get people’s gut reactions to the previous discussion, then “Green Hat thinking” to identify other possible solutions that are more tailored to the group’s situation or completely new approaches. At the end of a sequence, the Blue Hat would want to summarize what was said and begin a new sequence. To successfully use this method, the person wearing the Blue Hat should be familiar with different sequences and plan some of the thinking patterns ahead of time based on the problem and the group members. Each round of thinking should be limited to a certain time frame (two to five minutes) to keep the discussion moving.

This decision-making method has been praised because it allows group members to “switch gears” in their thinking and allows for role playing, which lets people express ideas more freely. How can this help enhance critical thinking? Which combination of hats do you think would be best for a critical thinking sequence?
What combinations of hats might be useful if the leader wanted to break the larger group up into pairs and why? For example, what kind of thinking would result from putting Yellow and Red together, Black and White together, or Red and White together, and so on?
Based on your preferred ways of thinking and your personality, which hat would be the best fit for you? Which would be the most challenging? Why?

Influences on Decision Making

Many factors influence the decision-making process. For example, how might a group’s independence or access to resources affect the decisions they make? What potential advantages and disadvantages come with decisions made by groups that are more or less similar in terms of personality and cultural identities? In this section, we will explore how situational, personality, and cultural influences affect decision making in groups.

Situational Influences on Decision Making

A group’s situational context affects decision making. One key situational element is the degree of freedom that the group has to make its own decisions, secure its own resources, and initiate its own actions. Some groups have to go through multiple approval processes before they can do anything, while others are self-directed, self-governing, and self-sustaining. Another situational influence is uncertainty. In general, groups deal with more uncertainty in decision making than do individuals because of the increased number of variables that comes with adding more people to a situation. Individual group members can’t know what other group members are thinking, whether or not they are doing their work, and how committed they are to the group. So the size of a group is a powerful situational influence, as it adds to uncertainty and complicates communication.

Access to information also influences a group. First, the nature of the group’s task or problem affects its ability to get information. Group members can more easily make decisions about a problem when other groups have similarly experienced it. Even if the problem is complex and serious, the group can learn from other situations and apply what it learns. Second, the group must have access to flows of information. Access to archives, electronic databases, and individuals with relevant experience is necessary to obtain any relevant information about similar problems or to do research on a new or unique problem. In this regard, group members’ formal and information network connections also become important situational influences.

The urgency of a decision can have a major influence on the decision-making process. As a situation becomes more urgent, it requires more specific decision-making methods and types of communication.

Judith E. Bell – Urgent – CC BY-SA 2.0.

The origin and urgency of a problem are also situational factors that influence decision making. In terms of origin, problems usually occur in one of four ways:

Something goes wrong. Group members must decide how to fix or stop something. Example—a firehouse crew finds out that half of the building is contaminated with mold and must be closed down.
Expectations change or increase. Group members must innovate more efficient or effective ways of doing something. Example—a firehouse crew finds out that the district they are responsible for is being expanded.
Something goes wrong and expectations change or increase. Group members must fix/stop and become more efficient/effective. Example—the firehouse crew has to close half the building and must start responding to more calls due to the expanding district.
The problem existed from the beginning. Group members must go back to the origins of the situation and walk through and analyze the steps again to decide what can be done differently. Example—a firehouse crew has consistently had to work with minimal resources in terms of building space and firefighting tools.

In each of the cases, the need for a decision may be more or less urgent depending on how badly something is going wrong, how high the expectations have been raised, or the degree to which people are fed up with a broken system. Decisions must be made in situations ranging from crisis level to mundane.

Personality Influences on Decision Making

A long-studied typology of value orientations that affect decision making consists of the following types of decision maker: the economic, the aesthetic, the theoretical, the social, the political, and the religious (Spranger, 1928).

The economic decision maker makes decisions based on what is practical and useful.
The aesthetic decision maker makes decisions based on form and harmony, desiring a solution that is elegant and in sync with the surroundings.
The theoretical decision maker wants to discover the truth through rationality.
The social decision maker emphasizes the personal impact of a decision and sympathizes with those who may be affected by it.
The political decision maker is interested in power and influence and views people and/or property as divided into groups that have different value.
The religious decision maker seeks to identify with a larger purpose, works to unify others under that goal, and commits to a viewpoint, often denying one side and being dedicated to the other.

In the United States, economic, political, and theoretical decision making tend to be more prevalent decision-making orientations, which likely corresponds to the individualistic cultural orientation with its emphasis on competition and efficiency. But situational context, as we discussed before, can also influence our decision making.

Personality affects decision making. For example, “economic” decision makers decide based on what is practical and useful.

One Way Stock – Tough Decisions Ahead – CC BY-ND 2.0.

The personalities of group members, especially leaders and other active members, affect the climate of the group. Group member personalities can be categorized based on where they fall on a continuum anchored by the following descriptors: dominant/submissive, friendly/unfriendly, and instrumental/emotional (Cragan & Wright, 1999). The more group members there are in any extreme of these categories, the more likely that the group climate will also shift to resemble those characteristics.

Dominant versus submissive. Group members that are more dominant act more independently and directly, initiate conversations, take up more space, make more direct eye contact, seek leadership positions, and take control over decision-making processes. More submissive members are reserved, contribute to the group only when asked to, avoid eye contact, and leave their personal needs and thoughts unvoiced or give into the suggestions of others.
Friendly versus unfriendly. Group members on the friendly side of the continuum find a balance between talking and listening, don’t try to win at the expense of other group members, are flexible but not weak, and value democratic decision making. Unfriendly group members are disagreeable, indifferent, withdrawn, and selfish, which leads them to either not invest in decision making or direct it in their own interest rather than in the interest of the group.
Instrumental versus emotional. Instrumental group members are emotionally neutral, objective, analytical, task-oriented, and committed followers, which leads them to work hard and contribute to the group’s decision making as long as it is orderly and follows agreed-on rules. Emotional group members are creative, playful, independent, unpredictable, and expressive, which leads them to make rash decisions, resist group norms or decision-making structures, and switch often from relational to task focus.

Cultural Context and Decision Making

Just like neighborhoods, schools, and countries, small groups vary in terms of their degree of similarity and difference. Demographic changes in the United States and increases in technology that can bring different people together make it more likely that we will be interacting in more and more heterogeneous groups (Allen, 2011). Some small groups are more homogenous, meaning the members are more similar, and some are more heterogeneous, meaning the members are more different. Diversity and difference within groups has advantages and disadvantages. In terms of advantages, research finds that, in general, groups that are culturally heterogeneous have better overall performance than more homogenous groups (Haslett & Ruebush, 1999). Additionally, when group members have time to get to know each other and competently communicate across their differences, the advantages of diversity include better decision making due to different perspectives (Thomas, 1999). Unfortunately, groups often operate under time constraints and other pressures that make the possibility for intercultural dialogue and understanding difficult. The main disadvantage of heterogeneous groups is the possibility for conflict, but given that all groups experience conflict, this isn’t solely due to the presence of diversity. We will now look more specifically at how some of the cultural value orientations we’ve learned about already in this book can play out in groups with international diversity and how domestic diversity in terms of demographics can also influence group decision making.

International Diversity in Group Interactions

Cultural value orientations such as individualism/collectivism, power distance, and high-/low-context communication styles all manifest on a continuum of communication behaviors and can influence group decision making. Group members from individualistic cultures are more likely to value task-oriented, efficient, and direct communication. This could manifest in behaviors such as dividing up tasks into individual projects before collaboration begins and then openly debating ideas during discussion and decision making. Additionally, people from cultures that value individualism are more likely to openly express dissent from a decision, essentially expressing their disagreement with the group. Group members from collectivistic cultures are more likely to value relationships over the task at hand. Because of this, they also tend to value conformity and face-saving (often indirect) communication. This could manifest in behaviors such as establishing norms that include periods of socializing to build relationships before task-oriented communication like negotiations begin or norms that limit public disagreement in favor of more indirect communication that doesn’t challenge the face of other group members or the group’s leader. In a group composed of people from a collectivistic culture, each member would likely play harmonizing roles, looking for signs of conflict and resolving them before they become public.

Power distance can also affect group interactions. Some cultures rank higher on power-distance scales, meaning they value hierarchy, make decisions based on status, and believe that people have a set place in society that is fairly unchangeable. Group members from high-power-distance cultures would likely appreciate a strong designated leader who exhibits a more directive leadership style and prefer groups in which members have clear and assigned roles. In a group that is homogenous in terms of having a high-power-distance orientation, members with higher status would be able to openly provide information, and those with lower status may not provide information unless a higher status member explicitly seeks it from them. Low-power-distance cultures do not place as much value and meaning on status and believe that all group members can participate in decision making. Group members from low-power-distance cultures would likely freely speak their mind during a group meeting and prefer a participative leadership style.

How much meaning is conveyed through the context surrounding verbal communication can also affect group communication. Some cultures have a high-context communication style in which much of the meaning in an interaction is conveyed through context such as nonverbal cues and silence. Group members from high-context cultures may avoid saying something directly, assuming that other group members will understand the intended meaning even if the message is indirect. So if someone disagrees with a proposed course of action, he or she may say, “Let’s discuss this tomorrow,” and mean, “I don’t think we should do this.” Such indirect communication is also a face-saving strategy that is common in collectivistic cultures. Other cultures have a low-context communication style that places more importance on the meaning conveyed through words than through context or nonverbal cues. Group members from low-context cultures often say what they mean and mean what they say. For example, if someone doesn’t like an idea, they might say, “I think we should consider more options. This one doesn’t seem like the best we can do.”

In any of these cases, an individual from one culture operating in a group with people of a different cultural orientation could adapt to the expectations of the host culture, especially if that person possesses a high degree of intercultural communication competence (ICC). Additionally, people with high ICC can also adapt to a group member with a different cultural orientation than the host culture. Even though these cultural orientations connect to values that affect our communication in fairly consistent ways, individuals may exhibit different communication behaviors depending on their own individual communication style and the situation.

Domestic Diversity and Group Communication

While it is becoming more likely that we will interact in small groups with international diversity, we are guaranteed to interact in groups that are diverse in terms of the cultural identities found within a single country or the subcultures found within a larger cultural group.

Gender stereotypes sometimes influence the roles that people play within a group. For example, the stereotype that women are more nurturing than men may lead group members (both male and female) to expect that women will play the role of supporters or harmonizers within the group. Since women have primarily performed secretarial work since the 1900s, it may also be expected that women will play the role of recorder. In both of these cases, stereotypical notions of gender place women in roles that are typically not as valued in group communication. The opposite is true for men. In terms of leadership, despite notable exceptions, research shows that men fill an overwhelmingly disproportionate amount of leadership positions. We are socialized to see certain behaviors by men as indicative of leadership abilities, even though they may not be. For example, men are often perceived to contribute more to a group because they tend to speak first when asked a question or to fill a silence and are perceived to talk more about task-related matters than relationally oriented matters. Both of these tendencies create a perception that men are more engaged with the task. Men are also socialized to be more competitive and self-congratulatory, meaning that their communication may be seen as dedicated and their behaviors seen as powerful, and that when their work isn’t noticed they will be more likely to make it known to the group rather than take silent credit. Even though we know that the relational elements of a group are crucial for success, even in high-performance teams, that work is not as valued in our society as the task-related work.

Despite the fact that some communication patterns and behaviors related to our typical (and stereotypical) gender socialization affect how we interact in and form perceptions of others in groups, the differences in group communication that used to be attributed to gender in early group communication research seem to be diminishing. This is likely due to the changing organizational cultures from which much group work emerges, which have now had more than sixty years to adjust to women in the workplace. It is also due to a more nuanced understanding of gender-based research, which doesn’t take a stereotypical view from the beginning as many of the early male researchers did. Now, instead of biological sex being assumed as a factor that creates inherent communication differences, group communication scholars see that men and women both exhibit a range of behaviors that are more or less feminine or masculine. It is these gendered behaviors, and not a person’s gender, that seem to have more of an influence on perceptions of group communication. Interestingly, group interactions are still masculinist in that male and female group members prefer a more masculine communication style for task leaders and that both males and females in this role are more likely to adapt to a more masculine communication style. Conversely, men who take on social-emotional leadership behaviors adopt a more feminine communication style. In short, it seems that although masculine communication traits are more often associated with high status positions in groups, both men and women adapt to this expectation and are evaluated similarly (Haslett & Ruebush, 1999).

Other demographic categories are also influential in group communication and decision making. In general, group members have an easier time communicating when they are more similar than different in terms of race and age. This ease of communication can make group work more efficient, but the homogeneity may sacrifice some creativity. As we learned earlier, groups that are diverse (e.g., they have members of different races and generations) benefit from the diversity of perspectives in terms of the quality of decision making and creativity of output.

In terms of age, for the first time since industrialization began, it is common to have three generations of people (and sometimes four) working side by side in an organizational setting. Although four generations often worked together in early factories, they were segregated based on their age group, and a hierarchy existed with older workers at the top and younger workers at the bottom. Today, however, generations interact regularly, and it is not uncommon for an older person to have a leader or supervisor who is younger than him or her (Allen, 2011). The current generations in the US workplace and consequently in work-based groups include the following:

The Silent Generation. Born between 1925 and 1942, currently in their midsixties to mideighties, this is the smallest generation in the workforce right now, as many have retired or left for other reasons. This generation includes people who were born during the Great Depression or the early part of World War II, many of whom later fought in the Korean War (Clarke, 1970).
The Baby Boomers. Born between 1946 and 1964, currently in their late forties to midsixties, this is the largest generation in the workforce right now. Baby boomers are the most populous generation born in US history, and they are working longer than previous generations, which means they will remain the predominant force in organizations for ten to twenty more years.
Generation X. Born between 1965 and 1981, currently in their early thirties to midforties, this generation was the first to see technology like cell phones and the Internet make its way into classrooms and our daily lives. Compared to previous generations, “Gen-Xers” are more diverse in terms of race, religious beliefs, and sexual orientation and also have a greater appreciation for and understanding of diversity.
Generation Y. Born between 1982 and 2000, “Millennials” as they are also called are currently in their late teens up to about thirty years old. This generation is not as likely to remember a time without technology such as computers and cell phones. They are just starting to enter into the workforce and have been greatly affected by the economic crisis of the late 2000s, experiencing significantly high unemployment rates.

The benefits and challenges that come with diversity of group members are important to consider. Since we will all work in diverse groups, we should be prepared to address potential challenges in order to reap the benefits. Diverse groups may be wise to coordinate social interactions outside of group time in order to find common ground that can help facilitate interaction and increase group cohesion. We should be sensitive but not let sensitivity create fear of “doing something wrong” that then prevents us from having meaningful interactions. Reviewing Chapter 8 “Culture and Communication” will give you useful knowledge to help you navigate both international and domestic diversity and increase your communication competence in small groups and elsewhere.

Key Takeaways

Every problem has common components: an undesirable situation, a desired situation, and obstacles between the undesirable and desirable situations. Every problem also has a set of characteristics that vary among problems, including task difficulty, number of possible solutions, group member interest in the problem, group familiarity with the problem, and the need for solution acceptance.

The group problem-solving process has five steps:

Define the problem by creating a problem statement that summarizes it.
Analyze the problem and create a problem question that can guide solution generation.
Generate possible solutions. Possible solutions should be offered and listed without stopping to evaluate each one.
Evaluate the solutions based on their credibility, completeness, and worth. Groups should also assess the potential effects of the narrowed list of solutions.
Implement and assess the solution. Aside from enacting the solution, groups should determine how they will know the solution is working or not.
Before a group makes a decision, it should brainstorm possible solutions. Group communication scholars suggest that groups (1) do a warm-up brainstorming session; (2) do an actual brainstorming session in which ideas are not evaluated, wild ideas are encouraged, quantity not quality of ideas is the goal, and new combinations of ideas are encouraged; (3) eliminate duplicate ideas; and (4) clarify, organize, and evaluate ideas. In order to guide the idea-generation process and invite equal participation from group members, the group may also elect to use the nominal group technique.
Common decision-making techniques include majority rule, minority rule, and consensus rule. With majority rule, only a majority, usually one-half plus one, must agree before a decision is made. With minority rule, a designated authority or expert has final say over a decision, and the input of group members may or may not be invited or considered. With consensus rule, all members of the group must agree on the same decision.

Several factors influence the decision-making process:

Situational factors include the degree of freedom a group has to make its own decisions, the level of uncertainty facing the group and its task, the size of the group, the group’s access to information, and the origin and urgency of the problem.
Personality influences on decision making include a person’s value orientation (economic, aesthetic, theoretical, political, or religious), and personality traits (dominant/submissive, friendly/unfriendly, and instrumental/emotional).
Cultural influences on decision making include the heterogeneity or homogeneity of the group makeup; cultural values and characteristics such as individualism/collectivism, power distance, and high-/low-context communication styles; and gender and age differences.
Scenario 1. Task difficulty is high, number of possible solutions is high, group interest in problem is high, group familiarity with problem is low, and need for solution acceptance is high.
Scenario 2. Task difficulty is low, number of possible solutions is low, group interest in problem is low, group familiarity with problem is high, and need for solution acceptance is low.
Scenario 1: Academic. A professor asks his or her class to decide whether the final exam should be an in-class or take-home exam.
Scenario 2: Professional. A group of coworkers must decide which person from their department to nominate for a company-wide award.
Scenario 3: Personal. A family needs to decide how to divide the belongings and estate of a deceased family member who did not leave a will.
Scenario 4: Civic. A local branch of a political party needs to decide what five key issues it wants to include in the national party’s platform.
Group communication researchers have found that heterogeneous groups (composed of diverse members) have advantages over homogenous (more similar) groups. Discuss a group situation you have been in where diversity enhanced your and/or the group’s experience.

Adams, K., and Gloria G. Galanes, Communicating in Groups: Applications and Skills , 7th ed. (Boston, MA: McGraw-Hill, 2009), 220–21.

Allen, B. J., Difference Matters: Communicating Social Identity , 2nd ed. (Long Grove, IL: Waveland, 2011), 5.

Bormann, E. G., and Nancy C. Bormann, Effective Small Group Communication , 4th ed. (Santa Rosa, CA: Burgess CA, 1988), 112–13.

Clarke, G., “The Silent Generation Revisited,” Time, June 29, 1970, 46.

Cragan, J. F., and David W. Wright, Communication in Small Group Discussions: An Integrated Approach , 3rd ed. (St. Paul, MN: West Publishing, 1991), 77–78.

de Bono, E., Six Thinking Hats (Boston, MA: Little, Brown, 1985).

Delbecq, A. L., and Andrew H. Ven de Ven, “A Group Process Model for Problem Identification and Program Planning,” The Journal of Applied Behavioral Science 7, no. 4 (1971): 466–92.

Haslett, B. B., and Jenn Ruebush, “What Differences Do Individual Differences in Groups Make?: The Effects of Individuals, Culture, and Group Composition,” in The Handbook of Group Communication Theory and Research , ed. Lawrence R. Frey (Thousand Oaks, CA: Sage, 1999), 133.

Napier, R. W., and Matti K. Gershenfeld, Groups: Theory and Experience , 7th ed. (Boston, MA: Houghton Mifflin, 2004), 292.

Osborn, A. F., Applied Imagination (New York: Charles Scribner’s Sons, 1959).

Spranger, E., Types of Men (New York: Steckert, 1928).

Stanton, C., “How to Deliver Group Presentations: The Unified Team Approach,” Six Minutes Speaking and Presentation Skills , November 3, 2009, accessed August 28, 2012, http://sixminutes.dlugan.com/group-presentations-unified-team-approach .

Thomas, D. C., “Cultural Diversity and Work Group Effectiveness: An Experimental Study,” Journal of Cross-Cultural Psychology 30, no. 2 (1999): 242–63.

Communication in the Real World Copyright © 2016 by University of Minnesota is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

Identify the Problem
Define the Problem
Form a Strategy
Organize Information
Allocate Resources
Monitor Progress
Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

Asking questions about the problem
Breaking the problem down into smaller pieces
Looking at the problem from different perspectives
Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

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You can become a better problem solving by:

Practicing brainstorming and coming up with multiple potential solutions to problems
Being open-minded and considering all possible options before making a decision
Breaking down problems into smaller, more manageable pieces
Asking for help when needed
Researching different problem-solving techniques and trying out new ones
Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors. The Psychology of Problem Solving . Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving . Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

40 problem-solving techniques and processes

All teams and organizations encounter challenges. Approaching those challenges without a structured problem solving process can end up making things worse.

Proven problem solving techniques such as those outlined below can guide your group through a process of identifying problems and challenges , ideating on possible solutions , and then evaluating and implementing the most suitable .

In this post, you'll find problem-solving tools you can use to develop effective solutions. You'll also find some tips for facilitating the problem solving process and solving complex problems.

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What is problem solving?

Problem solving is a process of finding and implementing a solution to a challenge or obstacle. In most contexts, this means going through a problem solving process that begins with identifying the issue, exploring its root causes, ideating and refining possible solutions before implementing and measuring the impact of that solution.

For simple or small problems, it can be tempting to skip straight to implementing what you believe is the right solution. The danger with this approach is that without exploring the true causes of the issue, it might just occur again or your chosen solution may cause other issues.

Particularly in the world of work, good problem solving means using data to back up each step of the process, bringing in new perspectives and effectively measuring the impact of your solution.

Effective problem solving can help ensure that your team or organization is well positioned to overcome challenges, be resilient to change and create innovation. In my experience, problem solving is a combination of skillset, mindset and process, and it’s especially vital for leaders to cultivate this skill.

A group of people looking at a poster with notes on it

What is the seven step problem solving process?

A problem solving process is a step-by-step framework from going from discovering a problem all the way through to implementing a solution.

With practice, this framework can become intuitive, and innovative companies tend to have a consistent and ongoing ability to discover and tackle challenges when they come up.

You might see everything from a four step problem solving process through to seven steps. While all these processes cover roughly the same ground, I’ve found a seven step problem solving process is helpful for making all key steps legible.

We’ll outline that process here and then follow with techniques you can use to explore and work on that step of the problem solving process with a group.

The seven-step problem solving process is:

1. Problem identification

The first stage of any problem solving process is to identify the problem(s) you need to solve. This often looks like using group discussions and activities to help a group surface and effectively articulate the challenges they’re facing and wish to resolve.

Be sure to align with your team on the exact definition and nature of the problem you’re solving. An effective process is one where everyone is pulling in the same direction – ensure clarity and alignment now to help avoid misunderstandings later.

2. Problem analysis and refinement

The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem . Choosing the right problem to solve means you are on the right path to creating the right solution.

At this stage, you may look deeper at the problem you identified to try and discover the root cause at the level of people or process. You may also spend some time sourcing data, consulting relevant parties and creating and refining a problem statement.

Problem refinement means adjusting scope or focus of the problem you will be aiming to solve based on what comes up during your analysis. As you analyze data sources, you might discover that the root cause means you need to adjust your problem statement. Alternatively, you might find that your original problem statement is too big to be meaningful approached within your current project.

Remember that the goal of any problem refinement is to help set the stage for effective solution development and deployment. Set the right focus and get buy-in from your team here and you’ll be well positioned to move forward with confidence.

3. Solution generation

Once your group has nailed down the particulars of the problem you wish to solve, you want to encourage a free flow of ideas connecting to solving that problem. This can take the form of problem solving games that encourage creative thinking or techniquess designed to produce working prototypes of possible solutions.

The key to ensuring the success of this stage of the problem solving process is to encourage quick, creative thinking and create an open space where all ideas are considered. The best solutions can often come from unlikely places and by using problem solving techniques that celebrate invention, you might come up with solution gold.

4. Solution development

No solution is perfect right out of the gate. It’s important to discuss and develop the solutions your group has come up with over the course of following the previous problem solving steps in order to arrive at the best possible solution. Problem solving games used in this stage involve lots of critical thinking, measuring potential effort and impact, and looking at possible solutions analytically.

During this stage, you will often ask your team to iterate and improve upon your front-running solutions and develop them further. Remember that problem solving strategies always benefit from a multitude of voices and opinions, and not to let ego get involved when it comes to choosing which solutions to develop and take further.

Finding the best solution is the goal of all problem solving workshops and here is the place to ensure that your solution is well thought out, sufficiently robust and fit for purpose.

5. Decision making and planning

Nearly there! Once you’ve got a set of possible, you’ll need to make a decision on which to implement. This can be a consensus-based group decision or it might be for a leader or major stakeholder to decide. You’ll find a set of effective decision making methods below.

Once your group has reached consensus and selected a solution, there are some additional actions that also need to be decided upon. You’ll want to work on allocating ownership of the project, figure out who will do what, how the success of the solution will be measured and decide the next course of action.

Set clear accountabilities, actions, timeframes, and follow-ups for your chosen solution. Make these decisions and set clear next-steps in the problem solving workshop so that everyone is aligned and you can move forward effectively as a group.

Ensuring that you plan for the roll-out of a solution is one of the most important problem solving steps. Without adequate planning or oversight, it can prove impossible to measure success or iterate further if the problem was not solved.

6. Solution implementation

This is what we were waiting for! All problem solving processes have the end goal of implementing an effective and impactful solution that your group has confidence in.

Project management and communication skills are key here – your solution may need to adjust when out in the wild or you might discover new challenges along the way. For some solutions, you might also implement a test with a small group and monitor results before rolling it out to an entire company.

You should have a clear owner for your solution who will oversee the plans you made together and help ensure they’re put into place. This person will often coordinate the implementation team and set-up processes to measure the efficacy of your solution too.

7. Solution evaluation

So you and your team developed a great solution to a problem and have a gut feeling it’s been solved. Work done, right? Wrong. All problem solving strategies benefit from evaluation, consideration, and feedback.

You might find that the solution does not work for everyone, might create new problems, or is potentially so successful that you will want to roll it out to larger teams or as part of other initiatives.

None of that is possible without taking the time to evaluate the success of the solution you developed in your problem solving model and adjust if necessary.

Remember that the problem solving process is often iterative and it can be common to not solve complex issues on the first try. Even when this is the case, you and your team will have generated learning that will be important for future problem solving workshops or in other parts of the organization.

It’s also worth underlining how important record keeping is throughout the problem solving process. If a solution didn’t work, you need to have the data and records to see why that was the case. If you go back to the drawing board, notes from the previous workshop can help save time.

What does an effective problem solving process look like?

Every effective problem solving process begins with an agenda . In our experience, a well-structured problem solving workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

The format of a workshop ensures that you can get buy-in from your group, encourage free-thinking and solution exploration before making a decision on what to implement following the session.

This Design Sprint 2.0 template is an effective problem solving process from top agency AJ&Smart. It’s a great format for the entire problem solving process, with four-days of workshops designed to surface issues, explore solutions and even test a solution.

Check it for an example of how you might structure and run a problem solving process and feel free to copy and adjust it your needs!

For a shorter process you can run in a single afternoon, this remote problem solving agenda will guide you effectively in just a couple of hours.

Whatever the length of your workshop, by using SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Complete problem-solving methods

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

The Six Thinking Hats #creative thinking #meeting facilitation #problem solving #issue resolution #idea generation #conflict resolution The Six Thinking Hats are used by individuals and groups to separate out conflicting styles of thinking. They enable and encourage a group of people to think constructively together in exploring and implementing change, rather than using argument to fight over who is right and who is wrong.

Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly It doesn’t matter where you work and what your job role is, if you work with other people together as a team, you will always encounter the same challenges: Unclear goals and miscommunication that cause busy work and overtime Unstructured meetings that leave attendants tired, confused and without clear outcomes. Frustration builds up because internal challenges to productivity are not addressed Sudden changes in priorities lead to a loss of focus and momentum Muddled compromise takes the place of clear decision- making, leaving everybody to come up with their own interpretation. In short, a lack of structure leads to a waste of time and effort, projects that drag on for too long and frustrated, burnt out teams. AJ&Smart has worked with some of the most innovative, productive companies in the world. What sets their teams apart from others is not better tools, bigger talent or more beautiful offices. The secret sauce to becoming a more productive, more creative and happier team is simple: Replace all open discussion or brainstorming with a structured process that leads to more ideas, clearer decisions and better outcomes. When a good process provides guardrails and a clear path to follow, it becomes easier to come up with ideas, make decisions and solve problems. This is why AJ&Smart created Lightning Decision Jam (LDJ). It’s a simple and short, but powerful group exercise that can be run either in-person, in the same room, or remotely with distributed teams.

Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

The 5 Whys

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.

World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.

Problem Tree

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.

SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!

Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!

LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

Journalists

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for brainstorming solutions

Now you have the context and background of the problem you are trying to solving, now comes the time to start ideating and thinking about how you’ll solve the issue.

Here, you’ll want to encourage creative, free thinking and speed. Get as many ideas out as possible and explore different perspectives so you have the raw material for the next step.

Looking at a problem from a new angle can be one of the most effective ways of creating an effective solution. TRIZ is a problem-solving tool that asks the group to consider what they must not do in order to solve a challenge.

By reversing the discussion, new topics and taboo subjects often emerge, allowing the group to think more deeply and create ideas that confront the status quo in a safe and meaningful way. If you’re working on a problem that you’ve tried to solve before, TRIZ is a great problem-solving method to help your team get unblocked.

Making Space with TRIZ #issue analysis #liberating structures #issue resolution You can clear space for innovation by helping a group let go of what it knows (but rarely admits) limits its success and by inviting creative destruction. TRIZ makes it possible to challenge sacred cows safely and encourages heretical thinking. The question “What must we stop doing to make progress on our deepest purpose?” induces seriously fun yet very courageous conversations. Since laughter often erupts, issues that are otherwise taboo get a chance to be aired and confronted. With creative destruction come opportunities for renewal as local action and innovation rush in to fill the vacuum. Whoosh!

Mindspin

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.

MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

Idea and Concept Development

Brainstorming without structure can quickly become chaotic or frustrating. In a problem-solving context, having an ideation framework to follow can help ensure your team is both creative and disciplined.

In this method, you’ll find an idea generation process that encourages your group to brainstorm effectively before developing their ideas and begin clustering them together. By using concepts such as Yes and…, more is more and postponing judgement, you can create the ideal conditions for brainstorming with ease.

Idea & Concept Development #hyperisland #innovation #idea generation Ideation and Concept Development is a process for groups to work creatively and collaboratively to generate creative ideas. It’s a general approach that can be adapted and customized to suit many different scenarios. It includes basic principles for idea generation and several steps for groups to work with. It also includes steps for idea selection and development.

Problem-solving techniques for developing and refining solutions

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to develop and refine your ideas in order to bring them closer to a solution that actually solves the problem.

Use these problem-solving techniques when you want to help your team think through their ideas and refine them as part of your problem solving process.

Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.

Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

Ensuring that everyone in a group is able to contribute to a discussion is vital during any problem solving process. Not only does this ensure all bases are covered, but its then easier to get buy-in and accountability when people have been able to contribute to the process.

1-2-4-All is a tried and tested facilitation technique where participants are asked to first brainstorm on a topic on their own. Next, they discuss and share ideas in a pair before moving into a small group. Those groups are then asked to present the best idea from their discussion to the rest of the team.

This method can be used in many different contexts effectively, though I find it particularly shines in the idea development stage of the process. Giving each participant time to concretize their ideas and develop them in progressively larger groups can create a great space for both innovation and psychological safety.

1-2-4-All #idea generation #liberating structures #issue analysis With this facilitation technique you can immediately include everyone regardless of how large the group is. You can generate better ideas and more of them faster than ever before. You can tap the know-how and imagination that is distributed widely in places not known in advance. Open, generative conversation unfolds. Ideas and solutions are sifted in rapid fashion. Most importantly, participants own the ideas, so follow-up and implementation is simplified. No buy-in strategies needed! Simple and elegant!

15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

Problem-solving techniques for making decisions and planning

After your group is happy with the possible solutions you’ve developed, now comes the time to choose which to implement. There’s more than one way to make a decision and the best option is often dependant on the needs and set-up of your group.

Sometimes, it’s the case that you’ll want to vote as a group on what is likely to be the most impactful solution. Other times, it might be down to a decision maker or major stakeholder to make the final decision. Whatever your process, here’s some techniques you can use to help you make a decision during your problem solving process.

How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.

How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.

Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

Straddling the gap between decision making and planning, MoSCoW is a simple and effective method that allows a group team to easily prioritize a set of possible options.

Use this method in a problem solving process by collecting and summarizing all your possible solutions and then categorize them into 4 sections: “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”.

This method is particularly useful when its less about choosing one possible solution and more about prioritorizing which to do first and which may not fit in the scope of your project. In my experience, complex challenges often require multiple small fixes, and this method can be a great way to move from a pile of things you’d all like to do to a structured plan.

MoSCoW #define intentions #create #design #action #remote-friendly MoSCoW is a method that allows the team to prioritize the different features that they will work on. Features are then categorized into “Must have”, “Should have”, “Could have”, or “Would like but won‘t get”. To be used at the beginning of a timeslot (for example during Sprint planning) and when planning is needed.

When it comes to managing the rollout of a solution, clarity and accountability are key factors in ensuring the success of the project. The RAACI chart is a simple but effective model for setting roles and responsibilities as part of a planning session.

Start by listing each person involved in the project and put them into the following groups in order to make it clear who is responsible for what during the rollout of your solution.

Responsibility (Which person and/or team will be taking action?)
Authority (At what “point” must the responsible person check in before going further?)
Accountability (Who must the responsible person check in with?)
Consultation (Who must be consulted by the responsible person before decisions are made?)
Information (Who must be informed of decisions, once made?)

Ensure this information is easily accessible and use it to inform who does what and who is looped into discussions and kept up to date.

RAACI #roles and responsibility #teamwork #project management Clarifying roles and responsibilities, levels of autonomy/latitude in decision making, and levels of engagement among diverse stakeholders.

Problem-solving warm-up activities

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process. Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

Doodling Together

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.

Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.

Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!

Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.

Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Closing activities for a problem-solving process

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.

One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

Who What When Matrix

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.

Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.

Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Tips for effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.

Try different approaches

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Create psychologically safe spaces for discussion

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner.

It can be tough for people to stand up and contribute if the problems or challenges are emotive or personal in nature. Try and create a psychologically safe space for these kinds of discussions and where possible, create regular opportunities for challenges to be brought up organically.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

Save time and effort creating an effective problem solving process

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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The Top 10 Characteristics of Problem Solvers

September 24th, 2017

Have you ever noticed that some people seem to be natural born problem solvers? Look closer, and you’ll discover that problem solving is more a skill than a gift. Effective problem solvers share ten common characteristics.

1. They have an “attitude”!

Simply expressed, effective problem solvers invariably see problems as opportunities, a chance to learn something new, to grow, to succeed where others have failed, or to prove that “it can be done”. Underlying these attitudes is a deeply held conviction that, with adequate preparation, the right answer will come.

2. They re-define the problem.

Problem solving is a primary consulting skill. Seasoned consultants know that, very often, the initial definition of the problem (by the client) is incorrect or incomplete. They learn to discount statements such as, “Obviously, the problem is that …” and follow their own leadings, but…

3. They have a system.

Perhaps the most common model is the old consulting acronym: DACR/S in which the letters stand for Describe, Analyze, Conclude, and Recommend/Solve. As with many formulas, its usefulness stems from the step-by-step approach it represents. Effective problem solvers take the steps in order and apply them literally. For example, in describing the problem (the first step), they strenuously avoid making premature judgments or ruling out possibilities. In analyzing the information, they are careful that their own prejudices do not interfere. In developing conclusions, they are aware of the need to test them thoroughly. Finally, most astute problem solvers recognize that there is almost always more than one solution, so they develop several alternatives from which to choose.

4. They avoid the experience trap.

The world is becoming increasingly non-linear. Things happen in pairs, triads, and groups and often don’t follow traditional lines from past to present and cause to effect. In such an environment, where synchronicity and simultaneity rather than linearity prevails, past experience must be taken with a grain of salt. Seasoned problem solvers know the pitfalls of relying on what worked in the past as a guide to what will work in the future. They learn to expect the unexpected, illogical, and non-linear.

5. They consider every position as though it were their own.

For effective problem solvers, standing in the other person’s shoes is more than a cute saying. It’s a fundamental way of looking at the problem from every perspective. This ability to shift perspectives quickly and easily is a key characteristic of effective problem solvers. As one especially capable consultant put it, “I take the other fellow’s position, and then I expand upon it until I understand it better than he does”.

6. They recognize conflict as often a prerequisite to solution.

When the stakes are high in a problem situation, the parties are often reluctant to show their hands and cautious about giving away too much. In such instances, managed conflict can be an effective tool for flushing out the real facts of a situation.

7. They listen to their intuition.

Somewhere during the latter stages of the fact-finding (description) process, effective problem solvers experience what can best be called, “inklings”-gut-level feelings about the situation. When this happens, they listen, hypothesize, test and re-test. They realize that, while intuition may be partially innate, effective intuition is overwhelmingly a developed faculty-and they work to develop it!

8. They invariably go beyond “solving the problem”.

On a time scale, just solving the problem at hand brings you to the present, to a point you might call, ground-zero. Truly effective problem solvers push further. They go beyond simply solving the problem to discover the underlying opportunities that often lie concealed within the intricacies of the situation. Implicit in this approach is the premise that every problem is an opportunity in disguise.

9. They seek permanent solutions.

Permanent, as opposed to band-aid solutions, has two characteristics: (1) they address all aspects of the problem, and (2) they are win/win in that they offer acceptable benefits to all parties involved. Symptomatic problem solving, like bad surgery or dentistry, leaves part of the decay untouched, with the result that, over time, it festers and erupts. Just for the record, a permanent solution is one that STAYS solved and doesn’t come back to bite you.

10. They gain agreement and commitment from the parties involved.

It’s easy, in the heady rush of finding “the answer” to a problem, to fail to gain agreement and commitment on the part of everyone involved. For effective problem solvers, just “going along” via tacit agreement isn’t enough. There must be explicit statements from all parties that they concur and are willing to commit to the solution. Agreement and concurrence really constitute a third characteristic of the “permanent” solution discussed above, but they are so often ignored that it is important that they be viewed separately.

A penalty-type method for solving inverse optimal value problem in second-order conic programming

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Information & Contributors

Bibliometrics & citations, view options, recommendations, a penalty function method for solving inverse optimal value problem.

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn-Tucker optimality condition of ...

The inverse optimal value problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear ...

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC^1) convex ...

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Inverse optimal value problem
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The Ultimate Guide To The Case Method: Definition And Characteristics

16 Jun 2024
Benk1 topictrek

What is the case method and what are its characteristics?

The case method is a teaching method in which students learn by analyzing real-world business cases. This method is often used in business schools to teach students about business strategy, management, and other business topics. The case method is based on the idea that students can learn more effectively by studying real-world examples of business problems than by simply reading textbooks or listening to lectures.

The case method typically involves students reading a case study, which is a detailed description of a real-world business problem. Students then analyze the case study and develop their own solutions to the problem. The students then present their solutions to the class and discuss them with the professor and other students. The case method is a highly interactive teaching method that allows students to learn from each other as well as from the professor.

The case method has a number of benefits over traditional teaching methods. First, the case method allows students to learn by doing. By analyzing real-world business problems, students can develop their problem-solving skills and critical thinking skills. Second, the case method helps students to develop their communication skills. By presenting their solutions to the class, students can learn how to communicate their ideas effectively. Third, the case method helps students to develop their teamwork skills. By working together to solve business problems, students can learn how to work effectively in a team environment.

The case method is a valuable teaching method that can help students to learn about business in a more engaging and effective way.

The Case Method

Interactive: The case method is a highly interactive teaching method that allows students to learn from each other as well as from the professor.
Engaging: The case method is a more engaging way to learn about business than traditional teaching methods.
Practical: The case method helps students to develop practical skills that they can use in their careers.
Collaborative: The case method encourages students to work together to solve business problems.
Critical Thinking: The case method helps students to develop their critical thinking skills.
Problem Solving: The case method helps students to develop their problem-solving skills.
Communication: The case method helps students to develop their communication skills.

The case method is a valuable teaching method that can help students to learn about business in a more engaging and effective way. By analyzing real-world business problems, students can develop the skills they need to be successful in their careers.

Interactive

As a component of "mtodo del caso que es y sus caractersticas", the interactive nature of the case method plays a significant role in fostering a dynamic and collaborative learning environment. Unlike traditional lectures or textbook-based learning, the case method places students at the center of the learning process, encouraging them to actively engage with the material and contribute their perspectives. Through class discussions, group work, and presentations, students have the opportunity to share their insights, challenge each other's ideas, and learn from diverse viewpoints. This interactive approach not only enhances the comprehension of complex business concepts but also promotes critical thinking, communication skills, and teamwork.

Real-life examples of the case method's effectiveness can be found in various business schools worldwide. For instance, Harvard Business School, a pioneer in case-based learning, has consistently ranked among the top business schools globally. Its graduates often attribute their success to the case method's ability to develop their analytical, problem-solving, and leadership skills.

Understanding the connection between the interactive nature of the case method and "mtodo del caso que es y sus caractersticas" is crucial for educators seeking to implement this teaching approach effectively. By creating a highly interactive environment, educators can foster a deeper understanding of business principles, enhance student engagement, and prepare students to tackle real-world business challenges.

Within the framework of "mtodo del caso que es y sus caractersticas", the engaging nature of the case method holds significant importance. Unlike traditional teaching methods that often rely on passive learning through lectures and textbooks, the case method actively involves students in the learning process, fostering a more immersive and engaging experience.

The case method presents students with real-world business scenarios, making the learning process relatable and applicable to practical situations. By analyzing and discussing actual business challenges, students develop a deeper understanding of the complexities and nuances involved in real-life decision-making.

The case method promotes interactive discussions among students, encouraging them to share their perspectives, challenge ideas, and engage in collaborative problem-solving. This dynamic exchange of thoughts fosters critical thinking, enhances communication skills, and develops a sense of community within the classroom.

The case method provides students with hands-on learning opportunities, allowing them to apply their knowledge and skills to practical business situations. Through case studies, students actively participate in analyzing data, developing recommendations, and presenting their solutions, which reinforces their understanding and prepares them for real-world business challenges.

The case method exposes students to diverse perspectives and approaches to business problems. By examining multiple case studies from different industries and contexts, students develop a well-rounded understanding of business practices and strategies, enabling them to adapt and innovate in a dynamic business environment.

The engaging nature of the case method aligns seamlessly with the principles of "mtodo del caso que es y sus caractersticas" by fostering active learning, promoting collaboration, and providing practical, real-world experiences. Through this engaging approach, the case method equips students with the skills, knowledge, and confidence to navigate the complexities of the business world.

Within the framework of "mtodo del caso que es y sus caractersticas", the practical nature of the case method plays a vital role in equipping students with applicable skills for their future careers. By delving into real-world business scenarios, students gain hands-on experience in analyzing, evaluating, and solving complex business problems.

The case method fosters analytical thinking and problem-solving abilities. Students learn to break down complex business issues into manageable components, identify key factors, and develop logical solutions, enhancing their analytical prowess.

Through case studies, students grapple with real-world dilemmas, requiring them to make informed decisions based on incomplete information. This process cultivates their decision-making capabilities and prepares them for the challenges of the business world.

The case method emphasizes effective communication, both written and verbal. Students present their analyses and recommendations, engage in discussions, and defend their viewpoints, honing their communication skills essential for career success.

Many case studies involve teamwork, fostering collaboration and interpersonal skills. Students learn to work effectively in teams, contribute diverse perspectives, and negotiate solutions, preparing them for the collaborative nature of the workplace.

The practical orientation of the case method aligns with the principles of "mtodo del caso que es y sus caractersticas" by providing students with opportunities to develop and refine skills that are highly sought after by employers. By equipping students with practical abilities, the case method empowers them to make meaningful contributions to organizations and navigate the complexities of the business world.

Collaborative

Within the framework of "mtodo del caso que es y sus caractersticas", the collaborative nature of the case method holds significant importance in developing students' teamwork and interpersonal skills, preparing them for the collaborative nature of the business world.

The case method often involves group work, where students collaborate to analyze case studies, discuss alternative solutions, and present their recommendations. This collaborative approach mirrors real-world business environments, where teamwork is essential for success.

Collaborative case study analysis fosters effective communication and conflict resolution skills. Students learn to articulate their ideas clearly, actively listen to others' perspectives, and engage in constructive discussions, even when faced with differing viewpoints.

Group work in the case method encourages students to work with individuals from diverse backgrounds and perspectives. This exposure to diversity prepares students to work effectively in multicultural teams and appreciate the value of different viewpoints.

The collaborative aspect of the case method aligns with the principles of "mtodo del caso que es y sus caractersticas" by providing students with opportunities to develop essential skills for success in collaborative business environments. By fostering teamwork, communication, and diversity, the case method equips students to navigate the complexities of the modern business world effectively.

Critical Thinking

Within the framework of "mtodo del caso que es y sus caractersticas", the focus on critical thinking occupies a central position. The case method fosters the development of critical thinking skills, enabling students to analyze complex business scenarios, evaluate evidence, and make sound judgments.

Critical thinking serves as a cornerstone for effective decision-making in the business world. By engaging with real-world case studies, students learn to identify key issues, assess the credibility of information, and develop logical arguments. These skills are essential for navigating the complexities of modern business environments, where data and information are abundant, and the ability to discern truth from falsehood is crucial.

The practical significance of critical thinking in the case method cannot be overstated. Students who develop strong critical thinking skills are better equipped to:

Identify and solve problems effectively
Make informed decisions based on evidence
Communicate their ideas clearly and persuasively
Adapt to changing business landscapes

In conclusion, the connection between critical thinking and "mtodo del caso que es y sus caractersticas" is inseparable. The case method provides a fertile ground for the development of critical thinking skills, which are essential for success in the business world and beyond.

Problem Solving

Within the framework of "mtodo del caso que es y sus caractersticas", the focus on problem-solving skills holds significant importance, as it equips students with the ability to analyze complex business scenarios, identify root causes, and develop innovative solutions.

The case method fosters analytical thinking and the ability to break down problems into manageable components. Students learn to identify key factors, evaluate evidence, and develop logical solutions, enhancing their analytical prowess.

The case method encourages students to think creatively and generate innovative solutions to complex problems. By examining multiple perspectives and challenging assumptions, students develop their ability to find unconventional approaches and drive positive change.

The connection between problem-solving skills and "mtodo del caso que es y sus caractersticas" is evident in the way the case method provides students with opportunities to develop and refine these skills through real-world business scenarios. By equipping students with problem-solving abilities, the case method empowers them to make meaningful contributions to organizations and navigate the complexities of the business world.

Communication

Within the framework of "mtodo del caso que es y sus caractersticas", the focus on communication skills holds utmost importance. The case method fosters the development of effective communication abilities, both written and verbal, which are essential for success in the business world.

Communication serves as a cornerstone for effective teamwork, persuasive presentations, and the ability to convey complex ideas clearly and persuasively. Through case studies, students engage in discussions, present their analyses, and defend their viewpoints, honing their communication skills in a practical and interactive setting.

The practical significance of communication skills in the case method cannot be overstated. Students who develop strong communication skills are better equipped to:

Articulate their ideas effectively in both written and verbal formats
Present their findings and recommendations with clarity and confidence
Engage in constructive discussions and debates, considering multiple perspectives
Build rapport with clients, colleagues, and stakeholders

In conclusion, the connection between communication skills and "mtodo del caso que es y sus caractersticas" is inseparable. The case method provides a fertile ground for the development of effective communication abilities, which are essential for success in the business world and beyond.

Frequently Asked Questions on "Mtodo del Caso

This section addresses common questions and misconceptions surrounding the case method, providing clear and concise answers to enhance understanding.

Question 1: What is the case method?

The case method is a teaching approach where students analyze real-world business scenarios to develop problem-solving and critical thinking skills. It involves studying case studies, engaging in discussions, and presenting solutions.

Question 2: What are the benefits of the case method?

The case method offers numerous benefits, including enhanced analytical skills, improved communication abilities, fostered teamwork and collaboration, and a deeper understanding of business practices.

Question 3: Is the case method suitable for all subjects?

While commonly used in business education, the case method can be adapted to various disciplines, including law, medicine, and public policy, where real-world problem-solving is essential.

Question 4: How does the case method promote critical thinking?

By analyzing case studies, students learn to identify key issues, evaluate evidence, and develop logical arguments. This process challenges assumptions and encourages them to think critically and creatively.

Question 5: What role does collaboration play in the case method?

The case method often involves group work, where students collaborate to analyze cases, share insights, and develop joint solutions. This fosters teamwork, communication, and the ability to work effectively in diverse teams.

Question 6: How can the case method be effectively implemented?

Successful implementation of the case method requires careful planning, relevant case study selection, and a supportive learning environment that encourages active participation and critical thinking.

In conclusion, the case method provides a dynamic and engaging learning experience that equips students with essential skills for success in various fields. Its emphasis on real-world problem-solving, critical thinking, and collaboration makes it a valuable teaching approach that continues to gain recognition and adoption.

Moving forward, we will explore the historical context and evolution of the case method, as well as its impact on business education and beyond.

Conclusion on "Mtodo del Caso

The case method has established itself as a valuable teaching approach, particularly in business education, due to its focus on real-world problem-solving, critical thinking, and collaboration. By analyzing case studies, students develop analytical, communication, and teamwork skills that are highly sought after in the professional world.

As the business landscape continues to evolve rapidly, the case method remains relevant and adaptable. Its emphasis on experiential learning and practical application prepares students to navigate complex and dynamic business environments effectively. Furthermore, the case method fosters a mindset of continuous learning and encourages students to embrace challenges as opportunities for growth.

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PDF 2 First-Order Equations: Method of Characteristics
Consider the following problem (ut +aux = 0 ujΓ = ` (2.6) where Γ is a curve in the xt-plane. This is an example of a Cauchy problem. More generally, for an (n¡1)-dimensional manifold Γ in Rn, ﬁnding a function u which satisﬁes (F(~x;u;Du) = 0 ujΓ = ` is a ﬁrst-order Cauchy problem. Remark. An (n ¡ 1)-dimensional manifold in Rn is a ...
Method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic and parabolic partial differential equation.The method is to reduce a partial differential equation to a family of ordinary differential equations along which the ...
1.3: Quasilinear Equations
The characteristic equations are. dτ = dt 1 = dx c = du 0. and the parametric equations are given by. dx dτ = c, du dτ = 0. These equations imply that. u = const. = c1. x = ct + const. = ct + c2. As before, we can write c1 as an arbitrary function of c2.
PDF 2. Method of Characteristics
2. Method of Characteristics In this section we explore the method of characteristics when applied to linear and nonlinear equations of order one and above. 2.1. Method of characteristics for ﬁrst order quasilinear equations. 2.1.1. Introduction to the method. A ﬁrst order quasilinear equation in 2D is of the form a(x,y,u) u x + b(x,y,u) u
PDF The Method of Characteristics
deﬁnes a surface above the x−t plane. In the Method of Characteristics, one tries to ﬁnd a relationship between x and t such that along the curve x(t), the equation for u simpliﬁes and can be solved. Often, u(x,t) is a constant along x(t). However, x(t) is only one trajectory that winds through x−t space.
PDF The Method of Characteristics
The Method of Characteristics. Recall that the first order linear wave equation. ut + cux = 0;u(x; 0) = f(x)is constant in the direction (1; c) in the (t; x)-plane, and is therefore constant o. lines of the form x ct = x0. To determine the value of u at (x; t), we go backward along these lines until we get to t = 0, and then determine the value of.
Method of Characteristics
The method of characteristics is a method that can be used to solve the initial value problem (IVP) for general first order PDEs. Consider the first order linear PDE. (1) in two variables along with the initial condition . The goal of the method of characteristics, when applied to this equation, is to change coordinates from ( x, t) to a new ...
PDF 9 The Method of Characteristics
f,g and h above) are known as the Cauchy data for the pde, and solving the pde subject to these conditions is said to be a Cauchy problem. According to Hadamard, the Cauchy problem is well-posed if - A solution to the Cauchy problem exists - The solution is unique - The solution depends continuously on the auxiliary data.
PDF The method of characteristics
the initial value: u = f(x0). Moreover, the characteristics are t. e straight lines x = x0 + ut. Thus, we may immediately integrate the rst ODE to obtain the. mplicit solution u = f(x ut). For some forms of the initial condition function, f(x), we may be able to algebraically solve this relatio.
PDF The Method of Characteristics
applying this method: Solving the system of characteristic ODEs may be diﬃcult (or impossible), especially if there is coupling between the equations. Passing from the parametric to the explicit form of the solution (i.e. solving for a and s in terms of x and y) may be diﬃcult (or impossible), especially is the expressions for x and y are ...
Method of Characteristics
The Method of Characteristics. The method of characteristics is a method which can be used to solve the initial value problem (IVP) for general first order (only contain first order partial derivatives) PDEs. Consider the first order linear PDE. (1) in two variables along with the initial condition .
PDF Method of characteristics: a special case
Example 4. We use the method of characteristics to show that the problem ux +uy = 1; u(x;x) = 1 has no solutions. In this case, the characteristic equations are x0 = y0 = u0 = 1 and so x = s+x0; y = s+y0; u = s+u0: Since the line y = x is one of the characteristic curves, it is better to avoid it and impose some other initial condition.
Method of Characteristics
First, the method of characteristics is used to solve first order linear PDEs. Next, I apply the method to a first order nonlinear problem, an example of a conservation law, and I discuss why the method may break down for nonlinear problems. I examine difficulties that appear in the nonlinear case, and I introduce the mathematical resolutions ...
Method of characteristics
The systems that we will encounter here will be solvable by elimination, that is by solving consecutively two ordinary differential equations. Example 1. Let a \in [0,+\infty). a ∈ [ 0, + ∞). Find the solution of the initial value problem \begin {align*} \frac {dx} {dt} & = 1 \\ \frac {dy} {dt} & = -x \\ \end {align*} d x d t d y d t = 1 ...
PDF THIRTEEN PROBLEM-SOLVING MODELS
The Six-Step method provides a focused procedure for the problem solving (PS) group. It ensures consistency, as everyone understands the approach to be used. By using data, it helps eliminate bias and preconceptions, leading to greater objectivity. It helps to remove divisions and encourages collaborative working.
Problem-Solving Strategies and Obstacles
Problem-solving is a vital skill for coping with various challenges in life. This webpage explains the different strategies and obstacles that can affect how you solve problems, and offers tips on how to improve your problem-solving skills. Learn how to identify, analyze, and overcome problems with Verywell Mind.
Critical Thinking and Problem-Solving
Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by ...
14.3 Problem Solving and Decision Making in Groups
Step 2: Analyze the Problem. During this step a group should analyze the problem and the group's relationship to the problem. Whereas the first step involved exploring the "what" related to the problem, this step focuses on the "why.". At this stage, group members can discuss the potential causes of the difficulty.
What is Problem Solving? Steps, Process & Techniques
Finding a suitable solution for issues can be accomplished by following the basic four-step problem-solving process and methodology outlined below. Step. Characteristics. 1. Define the problem. Differentiate fact from opinion. Specify underlying causes. Consult each faction involved for information. State the problem specifically.
The Problem-Solving Process
Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue. The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything ...
40 problem-solving techniques and processes
An effective process is one where everyone is pulling in the same direction - ensure clarity and alignment now to help avoid misunderstandings later. 2. Problem analysis and refinement. The process of problem analysis means ensuring that the problem you are seeking to solve is the right problem.
The Top 10 Characteristics of Problem Solvers
Effective problem solvers share ten common characteristics. 1. They have an "attitude"! Simply expressed, effective problem solvers invariably see problems as opportunities, a chance to learn something new, to grow, to succeed where others have failed, or to prove that "it can be done". Underlying these attitudes is a deeply held ...
7 Problem-Solving Skills That Can Help You Be a More ...
Although problem-solving is a skill in its own right, a subset of seven skills can help make the process of problem-solving easier. These include analysis, communication, emotional intelligence, resilience, creativity, adaptability, and teamwork. 1. Analysis. As a manager, you'll solve each problem by assessing the situation first.
A penalty-type method for solving inverse optimal value problem in
This inverse problem can be reformulated as a minimization problem with some second-order cone complementarity constraints. To tackle these bilinear constraints, we apply a penalty-type method and show that the associated penalty term is exact, which avoids the hurdles of penalty-type methods in the update strategy of the penalty parameter.
The Ultimate Guide To The Case Method: Definition And Characteristics
The Case Method. The case method is a teaching method in which students learn by analyzing real-world business cases. This method is often used in business schools to teach students about business strategy, management, and other business topics. The case method is based on the idea that students can learn more effectively by studying real-world ...