Monday, February 8, 2010

Mathematically Unsound "Discovering" Reviews

You can find links to all 25 pages on W. Stephen Wilson's home page HERE.


A few basic goals of high school mathematics will be looked at closely in the top programs chosen for high school by the state of Washington. Our concern will be with the mathematical development and coherence of the programs and not with issues of pedagogy.


Wilson On Discovering Algebra and Discovering Advanced Algebra:

Summary: In addition to the failure to deal with basic foundational issues associated with symmetry, the material is skewed toward the study of graphs rather than the study of quadratic functions. The low status afforded the functions and the algebra of the functions is disturbing. While the two textbooks have a nice collection of problems, including max/min problems, they have seldom done the mathematics to justify the solutions they find. Algebra tends to be better than Advanced Algebra in its development of the mathematics. However, logic breaks down at many points in the presentation. We have issues with the lack of development of symmetry, and going through three decimal place approximations using a calculator to end up with a precise function makes no sense. In this way and others, graphing calculators are used to undermine the structure of the mathematics. The logic of finite differences is not presented. Discovering relies on its claim to have shown that all quadratics are transformations of the function x-squared but it is a spurious claim. These problems detract from this program and keep it from being consistent, coherent, and mathematically sound.


Wilson on Discovering Geometry:

Summary: The text consists of 690 pages of inductive geometry followed by a short attempt to do rigorous deductive geometry. Unfortunately, the rigorous attempt depends on vague and “discovered” definitions scattered throughout the first 690 pages. This is a highly unsatisfactory geometry text.

Wilson on Holt Algebra:
Summary: Algebra 2 is much better than Algebra 1. In Algebra 1 formulas are given to you and the unjustified “second differences” is used. One could think of this as an empirical introduction, but it is perhaps better to avoid such an approach. The attempt to deal with symmetry is begun in Algebra 2 by actually showing that x2 is symmetrical around the y-axis. Transformations that take this parent function to the general quadratic in vertex form are quite explicit. Then Algebra 2 shows how to go between the general standard and vertex forms of quadratics. This is the best attempt at doing symmetry of the four programs. Much more detail would be better even here. The vertex and the line of symmetry area actually calculated. Although there are many well-conceived problems, there are few that require the students to produce quadratic functions or equations rather than solve those given in the problem.

Wilson on Holt Geometry:
Summary: The main criticism of this program is the use of redundant postulates. The mathematics is all in order. This is a sound, coherent presentation of the triangle sum theorem.



0.4 Criteria and Method
The task was to examine the mathematical soundness of the programs in relation to the aforementioned five standards. Pedagogy, per se, was not considered. I used the following criteria for mathematical soundness:
1. Mathematical justification
• Are central theorems stated and proved?
• Are solution methods to problems, conditions, and relations justified?
• Does the program develop norms for mathematical justification, where students gradually learn that empirical observations do not constitute justifications, though they can be a source for forming conjectures?
2. Symbolism and structure
• Does the program emphasize algebraic manipulations and reasoning in general terms?
• Is there an explicit attempt to help students organize what they have learned into a coherent logical structure?
• Does the program attend to crucial elements of deductive reasoning, such as “existence” and “uniqueness,” “necessary condition” and “sufficient condition,” and the distinction among “definition,” “theorem,” and “postulate?”
3. Language
• Is the language used clear and accurate?

Harel on Discovering:

2.4 Summary
The text does not justify fundamental theorems on linear and quadratic functions. In different places and in different contexts these theorems are demonstrated empirically. A common approach throughout the text is to present the problems and material through non-holistic problems, which mask the big ideas intended for students to learn. Consistently the text generalizes from empirical observations without attention to mathematical structure and justifications. There is nothing wrong with beginning with particular cases to understand something and make a conjecture about it. In many cases it is advantageous to do so and sometimes even necessary. However, students need to learn the difference between a conjecture generated from particular cases and an assertion that has been proved deductively. Unfortunately, the demarcation line between empirical reasoning and deductive reasoning is very vague in this program.
The approach the program applies to geometry is similar to that applies to algebra. It, too, amounts to empirical observations of geometric facts; it has little or nothing to do with deductive geometry. There is definitely a need for intuitive treatment of geometry in any textbook, especially one intended for high-school students. But the experimental geometry presented in the first 700 pages of the book is not utilized to develop geometry as deductive system. Most, if not all, assertions appear in the form of conjectures and most of the conjectures are not proved. It is difficult, if not impossible, to systematically differentiate which of the conjectures are postulates and which are theorems. It is difficult to learn from this text what a mathematical definition is or to distinguish between a necessary condition and sufficient condition.

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