The Nature and Role of Algebra in the K-14 Curriculum Proceedings of a National Symposium (1998) / Chapter Skim
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Presentations on Day Two
Presentations on Day Two
Pages 55-94
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The Role of Algebraic Structure in the Mathematics Curriculum of Gracles ~1-14 (G. Foley)
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This pattern of change, or joint variation, becomes the object of study as certain kinds of change produce recognizable tables, graphs, and symbolic expressions. At the middle-school level, these important families of predictable joint variation are linear, quadratic, and exponential functions.
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One can predict the distance traveled for any number of seconds, which is a very different pattern of joint variation from the first situation. Students can graph both functions on the same axes and see that the two lines cross.
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The three problems I have presented here would be appropriate for different stages of a student' s development of algebraic skill, but, nonetheless, all three serve to illustrate the centrality of variable and joint variation in understanding and using function to make sense of situations.
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GOALS AND APPROACHES From a functions and relations perspective, the continued study of algebra at the high-school level should enable all students to develop the ability to examine data or quantitative conditions; to choose appropriate algebraic models that fit patterns in the data or conditions; to write equations, inequalities, and other calculations to match important questions in the given situations; and to use a variety of strategies to answer the questions. Achievement of these goals would suggest that the study of algebra be rooted in the modeling of interesting data and phenomena in the physical, biological, and social sciences, in economics, and in students' daily lives.
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SUMMARY Organizing school algebra around functions and their use in mathematical modeling can provide a meaningful and broadly useful path to algebra for all students. Algebra as a language and means of representation is a natural by-product of this approach.
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extracting information from a situation and representing that information mathematically the process of "mathematizing"; and (b) interpreting and applying mathematical findings to have meaning within specific situations.
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An appropriate situation has the following characteristics: it is engaging for many middle school students; it can lead to significant mathematical explorations at an appropriate level of complexity; and it is manageable within the classroom. Next, let' s expand upon the mathematical representations for the middle-school algebra curriculum.
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Diagram 5 on the next page shows some categories of patterns students should understand; knowledge and techniques students will need to solve equations and inequalities; some tools for testing conjectures; and types of functions that students should become able to recognize and apply to understanding situations. All of these can be introduced in the middle-school curriculum, in many cases at an informal, context-based level, that forms a conceptual base for the more formal and abstract understandings that will develop in later grades.
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66 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Mathematical \ Findings Identify Patterns Additive ~Operations Square I I Inverse Operations Cubic Solve for I Unknowns , . _ Equivalence Signed Number I Multiplicative ~Operations Substitution Multi-variate ~Distributive Property I I Simplifying Step Test If/Then Conjectures Identify Functional Relationships Backtracking ~ Spreadsheets ~Direct Graphical Informal I I Inverse Solutions Proofs Linear Exponential Exponent ~Isolating Operations I Unknowns 1 Successive Approximation Finding Inequality Solution Sets Diagram 5 Quadratic
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Application of mathematical ideas is not the same as modeling, but the two are related. And modeling is not just "curve fitting," although that may, at times, be one part of the modeling process.
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, matrices, and good old arithmetic, to name a few. The process of moving from a set of assumptions to mathematical representations has long been a part of traditional school algebra through the dreaded "word problems." The modeling process extends this math-context interface in both directions.
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They provide a mental picture for repeated reference, not just for the investigation where they were used but for the class of problems about similar processes. Mathematical models of the dynamics of different drugs in the bloodstream and similar processes, such as elimination of pollution from lakes, can provide context for a range of algebraic concepts.
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· developing a habit of thinking mathematically (mathematical models of interest to students support this) · learning to communicate mathematics · developing an expectation of being able to make mathematical models so that they could answer questions and solve problems that arise in their own fields · learning mathematical concepts in depth, with an understanding of their place in the logic of mathematics and their value in practice If we do it well, modeling real-world problem situations has great value for the learning of mathematics.
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An issue for the teacher or curriculum developer becomes how to present graphing tasks that engage children and help them see the need to move toward more systematic representations. One way to do this is to move attention from the specific data to the overall graph shape.
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74 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Elizabeth's Graph of Changes in Population at Home 3:20 p.m. Elizabeth came home from school 3:35 p.m.
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Children will also move toward more systematic representations in order to compare data or to combine data into one graph or table. Labeling an axis with the measured heights of one plant 1 cm., 1.5 cm., 2.5 cm., and 4 cm.
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. Transparency 3a Elevator Graph: Third Grader's Invented Graph
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Net Change O I would keep going forward and forward and I would go out the side of the building. Transparency 3c Descriptions of Elevator Trips with Repeating Sets of Changes /
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78 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM We also can embed conventions into the task, asking students to collect data at regular time intervals, to interpret a systematic graph, and/or to make a graph or table on a provided template. When templates are provided marked at regular intervals, most fifth-graders are able to invent data for tables or sketch graphs that fit with stones such as, "Run about halfway, then go slower and slower until the end." (See Transparencies 4a, 4b, and 4c.)
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We believe they should be introduced in early middle school, as one type of regular function among others, such as cycles or waves or graphs that go faster and faster or slower and slower. The significance of a pattern being linear or not depends on our expectations.
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on the display. In the broken calculator work as with the graphing work, children are doing construction problems, finding for themselves patterns that allow them to make many problems of a similar type.
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· The net change is minus one. If you start on four and go minus one you get to three, so four is the answer.
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. First, I will try to clarify the term "algebra." ARITHMETIC, ALGEBRA, AND CALCULUS In the discussions about "algebra for all" there is confusion about what is actually meant by "algebra." Algebra is the term that is used for school algebra; a domain within the mathematics curriculum as it is taught at schools (elementary, middle, secondary)
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Algebra in the context of school algebra is a coherent integration of elements from the three domains: arithmetic, algebra, and calculus. ALGEBRA AT THE MIDDLE GRADES Over the past five years, the Freudenthal Institute has been involved in a curriculum development project in which a complete, new mathematics curriculum named MiC has been developed for American students between 10 and 14 years-of-age.
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The MiC algebra strand serves as an example of how this goal can be achieved.
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A careful reader of the NCTM Curriculum and Evaluation Standards (1989) will notice an overarching theme of "Mathematics as Reasoning" and will see that the document says highschool students should learn about matrices, abstraction and symbolism, finite graphs, sequences, recurrence relations, algorithms, and mathematical systems and their structural characteristics, and that, in addition, collegeintending students should gain facility with formal proof, algebraic transformations, operations on functions, linear programming, difference equations, the complex number system, elementary theorems of groups and fields, and the nature and purpose of axiomatic systems.
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Furthermore, in keeping with the NCTM curriculum standards for collegeintending students, high-school teachers need to be able to convey such understanding to their upper level students. This should be reinforced, amplified, and extended in lower division postsecondary mathematics courses, especially those in discrete mathematical structures and linear algebra.
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Having students work in groups and share representations makes them aware that different representations can be equivalent yet look quite different. This is a powerful experience in middle school and will pay huge benefits at the high-school level.
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To develop algebraic thinking, we need to include informal work with algebraic concepts in the middle school and not move too quickly to the abstract level. For example, being able to set up graphs or tables in various problem settings brings mathematical power and understanding to students.
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~? ~m Developer Diane Resek San Francisco State University San Francisco, California In this symposium, I am approaching the issue of teaching algebra from the perspective of a curriculum developer.
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Unfortunately, that kind of assessment system does not match the way students learn. At any level first grade, high school, or college students do not study a deep topic and suddenly "get it." Understanding comes gradually.
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Technology is changing how much technical skill students need to have in drawing graphs by hand, but it has not changed the fact that students need to understand how the zeros and symmetries of a function appear on a graph, where to expect asymptotes, and what sort of scale will show all the features of the graph. Numerical data, usually in a table, is for many students the least familiar representation.
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94 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Mastery of the language of algebra requires a two-pronged approach: 1. What does it mean?
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