Natl Council of Teachers of Mathematics, Reston, VA, 2000. Soft Cover. First edition, VG+ lg. format, 402pp, color illustrated text is bright, clean and fresh-in NF cond.
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9 of 16 people found the following review helpful:
3.0 out of 5 stars
Naive,
This review is from: Principles and Standards for School Mathematics (Paperback)
This book has a reasonable basic outlook, which includes such things as emphasis on conceptual understanding, discovery and empathic discussion rather than drill and teacher-authority. But on the whole the book is very poorly written. For one thing, virtually everything it says is completely vacuous, as one can see by inverting the trivial statements that it makes. Consider for example the following passage:
"By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages." The inversion of which would be: "By learning problem solving in mathematics, students should acquire habits of negligence and apathy, and fear of unfamiliar situations. In everyday life and in the workplace, being a good problem solver is worthless." And so it goes throughout the book; truism upon truism upon truism, page after page after page. The miniscule fraction of the book that actually says something of substance is well-meaning but betrays a considerable naiveté, as I shall illustrate by some examples. ----Faith in preemptive learning. "Research indicates a variety of student difficulties with the concept of variable, so developing understanding of variable over the grades is important." "Research indicates that [calculus] is not an area that students typically understand with much depth, even after taking calculus. If ideas of change receive a more explicit focus from the early grades on, perhaps students will eventually enter calculus with a stronger basis for understanding the ideas at that level. In prekindergarten through grade 2, students can, at first, describe qualitative change ('I grew taller over the summer') and then quantitative change ('I grew two inches in the last year'). Using graphs and tables, students in grades 3-5 can begin to notice and describe change, such as the changing nature of the growth of a plant, 'It grows slowly, then grows faster, then slows down.' And as they look at sequences, they can distinguish between arithmetic growth (2, 5, 8, 11, 14, ...) and geometric growth (2, 4, 8, 16, ...). With a strong middle-grades focus on linearity, students should learn about the idea that slope represents the constant rate of change in linear functions and be ready to learn in high school about classes of functions that have nonconstant rates of change." "Cutting apart and rearranging the pieces of a shape may change the perimeter but will not affect the area. ... Such observations can offer glimpses of sophisticated mathematical concepts such as invariance under certain transformations." "An important measurement technique in high school is successive approximation, a precursor to calculus concepts." Cf. Whitehead, Aims of Education, p. 6: "Whatever interest attaches to your subject-matter must be evoked here and now; whatever powers you are strengthening in the pupil, must be exercised here and now; whatever possibilities of mental life your teaching should impart, must be exhibited here and now. That is the golden rule of education, and a very difficult rule to follow." ----Formalism considered an end in itself. "Ultimately, a mathematical proof is a formal way of expressing particular kinds of reasoning and justification." "By the end of secondary school, students should be able to understand and produce mathematical proofs---arguments consisting of logically rigorous deductions of conclusions from hypotheses---and should appreciate the value of such arguments." "Students can explore motions such as slides, flips, and turns by using mirrors, paper folding, and tracing. Later, their knowledge about transformations should become more formal and systematic. ... The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers." Cf. Whiteside, Patterns of Mathematical Thought in the Later Seventeenth Century, p. 184: "a 17th century mathematical proof ... is a psychologically satisfying sketch and no more. ... it has a directness and immediacy---even a warmth and guilelessness---which is very often lacking in the cool surgical precision of its modern equivalent." ----Discovery learning used to "explore" boring technicalities. "Using ... dynamic geometry software, students can engage actively with geometric ideas ... [and] make and explore conjectures about geometry ... For example, using objects or dynamic geometric software to experiment with a variety of rectangles, students in grades 3-5 should be able to conjecture that rectangles always have congruent diagonals that bisect each other." ----Life sucked out of problems for which discovery learning would actually be fun. "From a ship on the sea at night, the captain can see three lighthouses and can measure the angles between them. If the captain knows the position of the lighthouses from a map, can the captain determine the position of the ship?" "When the problem is translated into a mathematical representation, the ship and the lighthouses become points in the plane, and the problem can be solved without knowing that it is about ships. ... As soon as the problem is represented in some form, the classic solution methods for that mathematical form may be used to solve the problem." ----Content generally unquestioned, i.e., traditional topics and motivations taken for granted even when they are nonsense. "[Students should] understand complex numbers as solutions to quadratic equations that do not have real solutions. ... Students' understanding of the mathematical development of number systems--from whole numbers to integers to rational numbers and then on to real and complex numbers--should be a basis for their work in finding solutions for certain types of equations. Students should understand the progression and the kinds of equations that can and cannot be solved in each system." ----Somewhat derogatory view of intuitive geometry. "Geometry is a natural place for the development of students' reasoning and justification skills ... The Geometry Standard includes a strong focus on the development of careful reasoning and proof, using definitions and established facts." Thus the book commends such things as "an algebraic justification of a visual argument for the Pythagorean theorem," where the original argument was in no need of such "justification."
0 of 3 people found the following review helpful:
4.0 out of 5 stars
Needed Book,
Amazon Verified Purchase(What's this?)
This review is from: Principles and Standards for School Mathematics (Paperback)
I bought this book for my wife because she needed it for a class she was taking. She got a B in the class. so I guess the book worked well enough.
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