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## Gaining Insights into Children's Geometric Knowledge

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## The Duodecimal System

In view of the fact that there seems to be a growing tendency toward the use of decimal notation in almost all lines of numerical computation, let us examine the possibility of a scheme which might be more useful than our present one. Our present method of representing numbers is based on 10. Probably this is due to the fact that we have two hands of five digits each, and that either consciously or unconsciously we use these digits to help us in making numerical calculations. Furthermore, since the psychologists tell us that some sort of muscular reaction is associated with almost every thought process which we perform, there are people who feel that we are merely a race of finger counters and that it would be folly ever to try to adopt any base other than 10. We agree that it would be premature seriously to recommend the adoption of any other base at the present time. But there is ample evidence that the world has long realized that 10 is not always the most convenient base. The dozen has been found useful in many ways; and though the foot may be a rather arbitrary unit of length, the fact that it is divided into 12 equal parts is more likely due to the convenience of 12 than mere accident.

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## The Use of Mathematical History in Teaching

For several centuries subject matter was considered the only thing of importance in a course in mathematics. Then came a demand for the teaching of the applications of mathematics. Consequently there was written into our textbooks various types of applied problems. In the old order of things it was considered sufficient for the pupil to prove as an exercise-theorem that if the diagonals of a parallelogram are equal the figure is a rectangle. Now we suggest in connection with this exercise that a boy may test his accuracy in laying out a rectangular tennis court by measuring the diagonals. In more recent years we have also come to the realization that there is educational value in knowing the history of the subject; consequently we tell the pupil how and when the human race first discovered and proved this important fact about parallelograms with equal diagonals-or any other important and useful part of mathematics. So, in the last quarter of a century, we find the history of mathematics creeping into our high school textbooks. Let us consider then what value may be found in the history of mathematics, either in high school or college teaching, and how we may make use of mathematical history in teaching.

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## Some Difficult Problems

If, As The writers of our year book would have us believe, it is certain that the schools of our day do not know how to teach arithmetic successfully; if our present day civilization is one which depends on the universal recognition of the importance of the idea of precision, since precision is the soul of science and commerce; if number is an ever guiding principle of life; if the number system has changed the life of men and is a mode of thinking that cultivates a general idea of regularity, arrangement and order in all thinking; if the number system, formulas of Algebra and Geometry have helped the race to organize and arrange the world in which we live; if mathematics is linked up with a large number of the branches of human knowledge; if every educated man or woman should know what mathematics means, what its greatest uses are, and something of its soul; if it enters into the making of a good citizen because of its value as a mental discipline; if the contact with absolute truth, the style of reasoning, the habit of rigorous thinking, the love for beauty develops transferable power for independent investigation and gives a keener insight into life; if these should emerge in the mind of the student, the conception of an ordered, lawful universe, a universe in which the reign of law is absolute; then these facts make the teaching of mathematics a problem that challenges our skill and study, and makes our work top the pinnacle of life service.

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## Teacher Care and Students’ Sense of Connectedness in the Urban Mathematics Classroom

Mathematics education has long adopted a colorblind perspective, where issues of race, racism, or culture have not been perceived to shape an individual’s educational access or ability to learn or perform well ( Martin, 2009 ). This perspective

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## Problems to Ponder

hung such that the flat, unpainted surface aligns with the wall, what percentage of the total surface area of the two halves is touching the wall? 26 Shania is part of the PlayItForward team cycling in a race to raise money for youth sports in her town

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## Chapter 6: Next Steps for Catalyzing Change

race, class, language, gender, and ­ability status. Actions for Teachers Educators Teacher educators, including coaches, professional development facilitators, and university faculty, engage in important work to support and improve the mathematics

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## GPS: Spring Frogs

./sec., respectively. If the length of the track is 40 feet, when will the green frog, brown frog, and red frog meet at the starting point for the first time after they start the race? Show as many methods as you can to solve this problem. Does the answer depend on the

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## Tiering Instruction for Seventh-Grade Students

enter a distance and time for each of two cars and run the race (see figure 1 ). Fig. 1 NewRace is a GeoGebra app that can be accessed at https://www.geogebra.org/m/vabtrttr . On days 9 and 10, students worked on tasks in which they were to make the

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## Now: We Can Do Hard Things

.edu , is an assistant professor of Mathematics Education Leadership and Secondary Education at George Mason University in Fairfax, Virginia. Her research focuses on understanding how race impacts mathematics teacher education and enhancing advanced