Measuring tools
Measuring tools are instruments designed to quantify various physical properties, such as distance, area, temperature, mass, and time. Historically, humans have utilized body parts for measurements, but as needs expanded, more sophisticated tools emerged, including notched bones used for counting in ancient cultures. Today, measuring devices range from basic items like rulers and thermometers to advanced technologies such as laser measurement tools and planimeters, which calculate areas through mechanical tracing.
In various fields like astronomy, navigation, and engineering, measuring tools enable both direct and indirect measurements. For example, rulers provide direct length comparisons, while devices like interferometers utilize wave frequencies for complex calculations. The evolution of these instruments reflects advancements in mathematics and technology, leading to increased accuracy, efficiency, and portability. In educational settings, measuring tools continue to play a vital role, helping students explore mathematical concepts through hands-on activities and dynamic software. As measurement technology advances, it remains integral to both scientific inquiry and daily life.
Measuring tools
SUMMARY: Mathematicians have developed a number of tools to make accurate measurements.
Body parts, including the thumb, hand, and foot, have long been used to measure distance. Some of the oldest known mathematical measuring tools were notched bones, such as the Lochango and Ishango bones, which may have been designed for use in counting or multiplication. However, many concepts and objects in astronomy, navigation, surveying, optics, medicine, and other fields cannot be directly or accurately measured with body parts or tools like marked bones. Indirect measurements require advances in engineering and instrumentation, sometimes using sophisticated mathematical transformations. Twenty-first century digital measuring tools have also aided in both accuracy and efficiency, and laser technology has led to instruments that can measure longer ranges. As these technologies advance, measuring instruments have become even smaller, making transportability far easier as well.
Scientists, mathematicians, and inventors have created many ingenious tools to accurately quantify concepts such as distance, area, temperature, mass, and time. Measuring devices are used to collect data, create mathematical models, verify mathematical relationships, and make predictions. They also have widespread applications in everyday life, including household thermometers, rulers, and watches. Teachers bring measuring devices into the classroom in order to help their students learn mathematics. Some, such as rulers and compasses, form the basis for an object of mathematical study. Others, such as yardsticks, are used to discover or verify relationships. In the twenty-first century, measuring devices continue to be refined and improved for greater precision and accuracy, as well as to develop theories and to solve new problems.
Direct Comparison Tools
In many cases, it is possible to physically measure an object or event directly by making comparisons. For example, rulers and tape measures directly compare lengths of objects to standard units of length. Protractors directly measure angles, balance scales are used for weights, and measuring cups and graduated cylinders and pipettes are used for volumes. Hourglasses and water clocks compare known units of time, measured out by the device, to the time people try to measure. Many such measuring tools that use direct comparisons of units were developed relatively early in the history of humanity, with different versions built by many different cultures.
Indirect Measurement
Other measurements are indirect. While people can directly experience temperature and pressure and are sensitive to relatively small variations in them, the physical properties and the measurements of temperature and pressure are less directly observable and comparable than length or weight. Because of this fact, units of temperature and pressure, as well as tools for measuring them, were developed several thousand years later than units of length and weight.
Planimeters are tools for measuring area and provide an interesting example of relatively sophisticated use of mathematics in measurement. They use a mechanical arm that traces the perimeter of an object, while its other end moves along a straight axis. The principle of the device, designed through calculus, is that the distance the end of the arm traces on the axis is proportional to the area of the object. Units of area were used in ancient times, but area was always separated into rectangles or right triangles for direct comparison with units. Planimeters do not depend on such direct comparison.
Calculating Measures
While there are tools for direct comparison of lengths of certain magnitudes, it is impossible to use these tools for very large lengths, such as the distance to the moon, or very small lengths, such as wavelengths of different colors. For these cases, various computational tools are more appropriate. For example, large distances can be measured by sending radio, light, or other wave pulses to distant objects and measuring the return time. The distance is equal to the product of velocity and time. An interferometer is a tool for observing changes in wave frequencies when there is wave interference. Known frequencies can be used to compute wavelength, inversely proportional to them.
Antiquated Measuring Tools
Some measuring inventions are no longer in use because of advances in mathematics and technology that also leads to changes in educational emphases. For instance, the astrolabe is an ancient measuring device that was once very popular. In the tenth century, Abd al-Rahmân al-Sufî detailed the flexibility of the astrolabe with reportedly 1000 applications. In the twenty-first century, it has mostly been relegated to collections and astronomy history and education. The sextant, which replaced the astrolabe for navigation in the eighteenth century, has mostly been replaced by global positioning system (GPS) devices.
Measuring in the Classroom
In the twenty-first century, students in mathematics classrooms use a variety of tools and systems of measurements. Ruler and compass constructions were a focus of ancient investigations, and students in mathematics classroom continue to explore them using physical instruments and dynamic software programs. In the late eighteenth century, Dr. Buxton obtained a patent for printed graph paper. In the early nineteenth century, mathematicians such as E. H. Moore advocated that graph paper be used to help students in algebra, and it took on an increasingly important role in schools. Cartographers were using protractors to measure angles in the late sixteenth century. Mathematician Alexis Clairaut described protractors in his 1741 book Elements de géometrie, and protractors appeared in some geometry and trigonometry textbooks in the nineteenth century. However, they were not common in mathematics classrooms in the United States until the early twentieth century. Representations and measurements of geometric solids have been the focus in mathematics since antiquity. Teachers and mathematics departments in the nineteenth and twentieth century showcased models made of a variety of different materials, including wood and string. These physical models became rarer because of the software that can perform measurement calculations and present interactive three-dimensional models. However, young children continue to fill plastic geometry shapes with water or sand to measure volume.
Measurement systems are also explored in mathematics classrooms. Those that have high-enough accuracy and precision for the given purpose are called “valid.” Precision and accuracy are established using statistical calculations such as mean and standard deviation and statistical laws such as the central limit theorem. Accuracy and precision are expressed using significant figures of numbers, with the error margin being half of the last significant place value. For example, the weight of 3.0×104g means the last significant place value is thousands and the error margin is 1000g÷2=500g. On the other hand, 3.00×104g means the last significant place value is hundreds and the error margin is 100g÷2=50g, which is more precise.
Bibliography
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Kidwell, Peggy, Amy Ackerberg-Hastings, and David Roberts. Tools of American Mathematics Teaching, 1800–2000. Baltimore, MD: Johns Hopkins University Press, 2008.
“Measurement – A Timeline.” Science Learning Hub, 19 Aug. 2019, www.sciencelearn.org.nz/interactive‗timeline/13-measurement-a-timeline. Accessed 2 Oct. 2024.
Stephenson, Bruce, Marvin Bolt, and Anna Friedman. The Universe Unveiled: Instruments and Images Through History. Cambridge, England: Cambridge University Press, 2000.
Turner, Gerard. Scientific Instruments, 1500–1900: An Introduction. London: Philip Wilson Publishers, 1998.
“A Turning Point for Humanity: Redefining the World's Measurement System.” National Institute of Standards and Technology, 12 May 2018, www.nist.gov/si-redefinition/turning-point-humanity-redefining-worlds-measurement-system. Accessed 2 Oct. 2024.