Mathematics school curricula are organized into strands that classify mathematics as a strictly compartmentalized discipline with an over-emphasis on computation and formulas. This organization makes it almost impossible for students to see mathematics as a continuously growing scientific field that continually spreads into new fields and applications. Students are not positioned to see overarching concepts and relations, so mathematics appears to be a collection of fragmented pieces of factual knowledge.
Steen (1990) puts it somewhat differently: School mathematics picks very few strands (e.g., arithmetic, algebra, geometry) and arranges them horizontally to form a curriculum. First is arithmetic, then simple algebra, then geometry, then more algebra, and finally—as if it where the epitome of mathematical knowledge—calculus. This layer-cake approach to mathematics education effectively prevents informal development of intuition along the multiple roots of mathematics. Moreover, it reinforces the tendency to design each course primarily to meet the prerequisites of the next course, making the study of mathematics largely an exercise in delayed gratification.
“What is mathematics?” is not a simple question to answer. A person asked at random will most likely answer, “Mathematics is the study of Number.” Or, if you’re lucky, “Mathematics is the science of number.” And, as Devlin (1994) states in his very successful book, “Mathematics: The Science of Patterns,” the former is a huge misconception based on a description of mathematics that ceased to be accurate some 2,500 years ago. Present-day mathematics is a thriving, worldwide activity, it is an essential tool for many other domains like banking, engineering, manufacturing, medicine, social science, and physics. The explosion of mathematical activity that has taken place in the twentieth century has been dramatic. At the turn of the nineteenth century, mathematics could reasonably be regarded as consisting of about 12 distinct subjects: arithmetic, geometry, algebra, calculus, topology and so on. The similarity between this list and the present-day school curricula list is amazing.
A more reasonable figure for today, however, would be between 60 and 70 distinct subjects. Some subjects (e.g., algebra, topology) have split into various sub fields; others (e.g., complexity theory, dynamical systems theory) are completely new areas of study. In our list of principles, we mentioned content: Mathematics should be relevant, meaning that mathematics should be seen as the language that describes patterns—both patterns in nature and patterns invented by the human mind. Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, purely utilitarian or of little more than recreational interest. They can arise from the world around us, from depth of space and time, or from the inner workings of the human mind (Devlin, 1994). For this reason, we have not chosen traditional content strands as the major dimensions for describing content.
Instead we have chosen to organize the content of the relevant mathematics around “big ideas” or “themes.” The concept of big ideas is not new. In 1990, the Mathematical Sciences Education Board published On the Shoulders of Giants: New Approaches to Numeracy (Steen, 1990), a book that made a strong plea for educators to help students delve deeper to find the concepts that underlie all mathematics and thereby better understand the significance of these concepts in the world. To accomplish this, we need to explore ideas with deep roots in the mathematical sciences without concern for the limitations of present schools of curricula.
Many big ideas can be identified and described. In fact the domain of mathematics is so rich and varied that it would not be possible to identify an exhaustive list of big ideas. It is important for purposes of classroom assessment, however, for any selection of big ideas that is offered to represent a sufficient variety and depth to reveal the essentials of mathematics and their relations to the traditional strands. The following list of mathematical big ideas meets this requirement:
• Change and growth.
• Space and shape.
• Quantitative reasoning.
• Uncertainty.
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