The Nature and Role of Algebra in the K-14 Curriculum Proceedings of a National Symposium (1998) / Chapter Skim
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Pages 7-22
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From page 7... ...
Bass) Making Algebra Dynamic and Motivating: A National Challenge hi.
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From page 9... ...
1, 2, 3, 4, 5,.... Then, in order to solve equations such as 2x = 3, with integer coefficients, we are led to introduce fractions, i.e., the rational numbers: numbers of the form p/q with p and q (i O)
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For example, we know the familiar quadratic formula for the solutions of quadratic polynomial equations. Is there a similar formula, involving only rational operations and extraction of roots (square roots, cube roots, and so on)
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This gives us a natural intuitive access to and geometric sense of the real line as a scale of measurement, even before we have a developed notational system for designating individual real numbers. Our geometric intuition is a sound and coherent foundation on which to build a geometric discussion of the basic arithmetic operations with real numbers, as I shall try to illustrate.
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From page 12... ...
First of all, one has an intuitively accessible and authentic model of the real numbers that can be apprehended at a very early age Second, the model's geometric representation is completely intrinsic and not dependent on the structure of place-value or any other arbitrary notation. Third, the basic operations of +, -, and x have very natural aeon Metric interpretations, making some properties transparent when they are not at all in the combinatorial or counting model.
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From page 13... ...
This reconciliation is itself a very instructive undertaking and can be based on some of the fundamental properties of real numbers, which themselves perhaps deserve increased attention. Place-value flows from a basic result, sometimes called "Euclidean Division," which I believe deserves some emphasis, since, as I shall illustrate below, it has many other significant applications as well.
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From page 14... ...
Further, if a > 0 is not an integer, then we can apply the above to the integer part of a, and then apply a related but modified procedure to the fractional part of a to derive the place-value representation of a to the right of the "b-ecimal" point. While this is all a bit technical, I simply wish to emphasize that Euclidean Division contains the genesis of the place-value representation of all numbers, to an arbitrary integer base b > 1.
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From page 15... ...
SUMMARY I have argued that it is both natural and advantageous to give an early emphasis to the geometric real line model of the real numbers, in which basic arithmetic operations are interpreted geometrically and developed alongside the more algorithmic development rooted in base-10 place-value representation. I also have used Euclidean Division to illustrate several important algebraic phenomena.
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From page 17... ...
We are the survivors of that system and the sifting process it helped define. The very skills and knowledge that gave us our identifications as mathematicians and mathematics educators those knowledgeable about mathematics and its teaching were a result of our performance as we moved through "algebra." Now, one might say, "We have the Curriculum and Evaluation Standards for School Mathematics now, and, so, things are changing." But what do we see when we look at the National Assessment of Educational Progress (NAEP)
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From page 18... ...
We teaching professionals indeed most educated adults have never successfully refuted her claim, at least not with students and not on a regular basis. This symposium provides us with an opportunity to focus on what school algebra is, how it can be structured, and to note the roles that algebra plays in the lives of citizens.
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From page 19... ...
That means developing a program of study that enables students to meet four goals of algebra in the context of the school mathematics program. FOUR GOALS FOR SCHOOL ALGEBRA The following four goals for school algebra programs are to help students see and use algebra as a way of · representing quantity and relationships among quantities; · predicting what happens in quantitative settings; · controlling, where possible, the outcomes to quantitative processes; · extending the applications and establishing the validity of new relationships in the structure of algebra.
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As elementary-school children begin to build out from their study of patterns, they are developing the foundation of the next big idea linearity. Many children approach linearity recursively and then move to an understanding that approximates seeing linearity like a function of a real variable.
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A similar form of algebraic representation allows a can of Campbell_' s soup to be identified and accurately priced by the scanner in your local grocery store, as well as allows the grocer to make almost instant pricing changes. Bar codes are also used to identify individuals in security settings.
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From page 22... ...
The sequence of steps involves rich data, recognition of pattern, developing quantity within that pattern, representing that quantity through the use of vanable, developing a function-like expression to represent typical values within the pattern, using equations to study the pattern, and using the models developed to represent, manipulate, predict, and control with algebra. How do we mold these visions of algebra into a coherent program for students in grades K to 14?
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