"There are at least four fundamental purposes that the study of mathematics should attain. First, it should serve as a functional tool in solving our individual everyday problems. These questions: How much? How many? What form or shape? and Can you prove it? arise every day in the lives of every citizen.
In the second place, mathematics serves as a handmaiden for the explanation of the quantitative situations in other subjects, such as economics, physics, navigation, finance, biology, and even the arts. The mathematics used in these areas of practice is exactly the same mathematics and involves the same mathematical concepts and skills. It is only the things to which the mathematics is applied that are different, and this is immaterial if one really understands the mathematics.
In the third place, mathematics, when properly conceived, becomes a model for thinking, for developing scientific structure, for drawing conclusions, and for solving problems. Its postulational nature, that is, accepted relations axioms or postulates), undefined terms, definitions, theorems, and logic. aids ail other areas of knowledge to approach scientific This same structure aids us in problem solving methods in which we collect, organize, and analyze data, and deduce conclusions for future action. For example, one who understands the mathematical method can easily frame the problem, [...]
In the fourth place, mathematics is the best describer of the universe about us. In an age that has become statistical and scientific in much of its human endeavor, the need for people to understand these phenomena is not only a cultural necessity but to some extent a necessity for intelligent action." (Howard F Fehr, "Reorientation in Mathematics Education", Teachers Record 54, 1953)
"Every problem-solving effort must begin with creating a representation for the problem - a problem space in which the search for the solution can take place. Of course, for most of the problems we encounter in our daily personal or professional lives, we simply retrieve from memory a representation that we have already stored and used on previous occasions. Sometimes, we have to adapt the representation a bit to the new situation, but that is usually a rather simple matter." (Herbert A Simon, "The Sciences of the Artificial", 1968)
"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)
"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)
"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)
"Thinking involves reasoning about a situation, and to do that we must have some kind of dynamic "model" of the situation in our heads. Any changes we make to this mental model of the world should ideally mirror changes in the real world."
"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)
"Good numeric representation is a key to effective thinking that is not limited to understanding risks. Natural languages show the traces of various attempts at finding a proper representation of numbers. [...] The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)
"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)
“A conceptual model is a mental image of a system, its components, its interactions. It lays the foundation for more elaborate models, such as physical or numerical models. A conceptual model provides a framework in which to think about the workings of a system or about problem solving in general. An ensuing operational model can be no better than its underlying conceptualization.” (Henry N Pollack, “Uncertain Science … Uncertain World”, 2005)
"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)
"Mental models are formed over time through a deep enculturation process, so it follows that any attempt to align mental models must focus heavily on collective sense making. Alignment only happens through a process of socialisation; people working together, solving problems together, making sense of the world together." (Robina Chatham & Brian Sutton, "Changing the IT Leader’s Mindset", 2010)
"Mathematics does not merely describe the problem in an abstract way, it allows us to find a previously unknown 'solution' from the abstract description. It is surprising that the unknown can be transformed into the well known when we succeed in describing the problem mathematically." (Waro Iwane, "Mathematics in Our Company: What Does It Describe?", [in "What Mathematics Can Do for You"] 2013)
"Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it." (Sandip Banerjee, "Mathematical Modeling: Models, Analysis and Applications", 2014)
"Mental imagery is often useful in problem solving. Verbal descriptions of problems can become confusing, and a mental image can clear away excessive detail to bring out important aspects of the problem. Imagery is most useful with problems that hinge on some spatial relationship. However, if the problem requires an unusual solution, mental imagery alone can be misleading, since it is difficult to change one’s understanding of a mental image. In many cases, it helps to draw a concrete picture since a picture can be turned around, played with, and reinterpreted, yielding new solutions in a way that a mental image cannot." (James Schindler, "Followership", 2014)
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