Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 23
Presentations on Day One Transforming Algebra from an Engine of Inequity to an Engine of Mathematical Power by "AIgebrafying" the K-12 Curriculum If. Kaput) Developing a Coherent and Focused K-12 Algebra Curriculum (E. Phillips) Enhancing Algebraic Reasoning with Technology (G. Akst) Algebra for Everyone? With or Without Technology? (M. Norman) How Might Technology Enhance Algebraic Reasoning? (R. Zbiek) What Do We Know about K-14 Students' Learning of Algebra? hi. Confrey) Algebra: What All Students Can Learn (S. Williams and D. Molina) Improving K-14 Algebra Instruction: A Discussion of Teachers' Responsibilities and Students' Opportunities (B. Moore-Harris) 23
OCR for page 24
OCR for page 25
Transformin,g Al,ge bra flora an Engine of Inequity to an Engine of Mathematical Power by "Al~eb~afyin~" the 1(-12 CnrricnIn~ ,James,J. Kaput University of Massachusetts Dartmouth, Massachusetts In this brief note I assume, first, that, just as algebra has acted as a constricted gateway to significant mathematics and all that follows from mastery of that mathematics, algebra reform is the gateway to K-12 mathematics reform for the next century; and, second, that by acknowledging the several different aspects of algebra and their roots in younger children's mathematical activity, a deeply reformed algebra is not only possible but very achievable within our current capacity for change. The key to algebra reform is to integrate algebraic reasoning across all grades and all topics to "algebrafy" school mathematics. This integration solves three major problems: 1. It opens curricular space for the 21st century mathematics desperately needed at the secondary level, space currently locked up by the 19th century high-school curriculum now in place across the nation; 2. It adds a new level of coherence, depth, and power to school mathematics, both as a curriculum and as a habit of mind; and 3. It eliminates the most pernicious curricular element of today's school mathematics late, abrupt, isolated, and superficial high-school algebra courses. An early-start, strands approach to algebra also fits well with an inclusive, big-idea strands oriented approach to the curriculum at large, which contrasts with the layer cake-filter structure that delays and ultimately denies access to powerful ideas for all students but a few. An algebrafied K-12 curriculum supports democratization of access to powerful ideas. Our discussions of algebra must be as honest and clear as possible. Towards clarity, we need to distinguish "Algebra, the Institution" from "Algebra, the Web of Knowledge and Skill" that we want students to develop in school, so that criticisms of the former are not heard as statements about the latter. "Algebra, the Institution" is a peculiarly American enterprise embodying the standard courses, textbooks, tests, remediation industry, and associated economic arrangements, as well as the supporting intellectual and social infrastructure of course and workplace prerequisites, cultural expectations relating success in algebra to intellectual ability and academic promise, special interests, relations between levels of schooling, and so on. Exhortation for and legislation of "Algebra for All" tacitly assume the viability and legitimacy of this institution. But this algebra is the disease for which it purports to be the cure! It alienates even the nominally successful students from genuine mathematical experience, prevents real reform, and acts as an engine of inequity for an egregiously high number of students, especially those who are the less advantaged of our society. Our challenge is to create an implementable alternative to this inimical institution, to transform an engine of inequity to an engine of mathematical power. One first step is to get a bit clearer about what we mean by "Algebra, the Web of Knowledge and Skill" so that we can see what it might offer as a goal to work towards and how it differs 25
OCR for page 26
26 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM from the intellectual content of the courses that are part of institutional algebra. We can then ask how we might achieve this algebra of knowledge and skill. FIVE FORMS OF ALGEBRAIC REASONING In my view, algebraic reasoning is a complex composite of five interrelated forms of reasoning. The first two underlie all the others (kernels), the next two constitute topic strands, and the last reflects algebra as a web of languages. All five richly interact conceptually as well as in activity to understand this algebra is to make connections. All five can and should be started early. Algebra as Generalizing and Formalizing Patterns and Constraints, especially, but not exclusively, Algebra as Generalized Arithmetic Reasoning and Algebra as Generalized Quantitative Reasoning; 2. Algebra as Syntactically-Guided Manipulation of Formalisms; 3. Algebra as the Study of Structures and Systems Abstracted from Computations and Relations; 4. Algebra as the Study of Functions, Relations, and Joint Variation; and 5. Algebra as a Cluster of (a) Modeling and (b) Phenomena-Controlling Languages. As kernels, forms 1. and 2. underlie all the others listed above, with the reasoning in 1. based both inside and outside of mathematics and the reasoning in 2. done in conjunction with 1. It is difficult to point to mathematical activity that does not involve generalizing and formalizing in a central way. It is one of the features of thinking that makes it mathematical. Also, the actions one performs with formalisms identified as 2. typically are not generalizing and formalizing per se but, rather, typically occur as the direct or indirect result of prior formalizing. The formalisms also may be of many different types, not merely variables over sets of familiar numbers (or transcendentals over some field). It is also possible for the manipulation to yield general patterns and structures at another level of generalizing and formalizing, which is the essence of 3., the structural form of algebraic reasoning. In order to use or communicate generalizations, one needs languages in which to express them, which leads to 5., and 5., in turn, permeates 1. through 4. While 3. is a school mathematics topic strand found today mainly at the advanced levels, it is also an important domain of mathematics in its own right abstract algebra. On the other hand, topic strand 4. functions is more a school mathematics domain that lives in the world of mathematics more as a general purpose conceptual tool than as a branch of mathematics. Traditional school algebra focuses on form 2. above at the expense of all the others. And while calls for a functions approach to algebra were ignored for almost a century, some of our contemporaries tend to say that form 4. is what algebra is all about. Although functions are an extremely powerful organizer of mathematical activity across topics and grade levels, this also is true of the other forms of algebra listed, which is exactly why algebra can play the key role across K-12 mathematics, as I and others have suggested. This wider view of algebra emphasizes its deep but varied connections with all of mathematics. ALGEBRA BEFORE ACNE: THE ROLE OF TEACHERS The language aspect of algebra supports both early and integrated algebra: early because students require repeated use of a language over an extended amount of time to become fluent in its use; integrated because in order to learn a language, students need to use it to express something significant to them, such as the quantitative relationships arising inside mathematics (for example, that occur in arithmetic and geometry) and outside mathe- matics (when we use it to model our world). As several speakers will show during this symposium, appropriate instructional materials can "seed" each aspect of algebra that I have listed through relatively ordinary, elementary mathematical activity. One key is that teachers need to be able to identify and nurture these roots of algebraic reasoning in forms that appear very different from what is deemed "algebra" under the auspices of "Algebra. the Institution." For example. generalization ~7 1 ~7 ' 1 ' ~7 initially can be expressed using ordinary language, intonation, and gesture rather than formal symbolism. This will require teacher development focused on student thinking rather than traditional formalisms. Another key is that these beginnings need to go somewhere mathematically, both in terms of growth in notational competence and in terms of the significance of the big ideas that these notations are used to express. This will require careful designed classroom materials to help guide the way.
OCR for page 27
Developing a Coherent and Focused 1(-12 Algebra C?~ic?~?~m Elizabeth Phillips Michigan State University East Lansing, Michigan Successful completion of algebra is seen as a "gateway" to future success in scientific and other postsecondary endeavors. Getting through the algebra gate is so important that it has given rise to other gates, such as state- mandated algebra tests and a variety of pre-algebra courses that consist of a combination of arithmetic drills and simple equation solving. The recent Third International Mathematics and Science Study and National Assessment of Educational Progress data (U.S. National Research Center, 1996; Mullis et al., 1991) confirm that the U.S. is falling short in the goal of improving the algebraic understanding of all students. One conclusion of the TIMSS curriculum analysis is that the American mathematics curriculum is a "mile wide and an inch deep." Such reports suggest a lack of coherence and focus in the K-12 mathematics curricula across the country, with little time devoted at any grade level to developing deeper understandings of fewer important ideas (Schmidt et. al., 1996~. Nowhere is this lack of focus and cohesion more noticeable than in the area of algebra. Continued demonstra- tion of students' lack of basic skills despite two decades of a skill-and-drill curriculum should convince us to look for a new way of bringing students and algebra together. In the search for new ways of teaching and learning algebra, we must seek answers to the following questions: How do we build a coherent algebra curriculum across the grades that develops understanding of important algebra concepts and skills and students' quantitative reasoning abilities? How do we assess these skills and understandings? The paper, "A Framework for Constructing a Vision of Algebra: A Discussion Document" (National Council of Teachers of Mathematics [NCTMi, 1997), suggests that algebraic concepts are best learned first within a context or problem. While a contextual setting can become a vehicle for understanding and remembering these concepts, there is an intrinsic difficulty in building a coherent curriculum around a range of concrete contexts in which the student and teacher may not see any common patterns. The NCTM algebra paper offers a way to connect these contexts by using organizing themes: Functions and Relations, Modeling, Structure, and Language and Representations. "It is the ability to generalize, extend, apply, and connect ideas throughout many different situations and across several grades that enables students to make sense of mathematics; thinking of it as a set of unifying ideas rather than a set of disjointed, unrelated problems." "Algebra becomes manifest through exploration of the compelling ideas within the contextual settings, while the themes bring logic and organization to the discipline." (NCTM, 1997) Several of the recent National Science Foundation-funded curriculum projects have organized their curriculum around interesting problems. If mathematical concepts are developed from problem situations or contexts, then the variables in the situation and relationship between them become real to the student. The problem context can provide the scaffolding needed to reason about the quantities in the situation and how they change in relationship to 27
OCR for page 28
28 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM each other. Variables and patterns are central to a problem-centered curriculum; that is, what are the variables, and how are they related? In the "Connected Mathematics Project" (CMP) (Lappan et al., 1995), one of the NSF-funded middle-school curriculum projects, we have found that, over time, through such experiences, students develop a rich language, moving freely among the different forms of representations, pictures, words, tables, graphs, and symbols. They come to understand what each form of representation can tell them about a situation, and they learn which representation best captures their reasoning about patterns they observe between related variables. They understand what kinds of questions or information are more easily explored through a particular representation of the data or a combination of representations. The understanding needed to move freely among different representations takes time to develop and hence is an important part of the curriculum throughout. A problem-based curriculum, such as CMP, puts quantitative reasoning in the forefront and thereby provides the base on which students can investigate patterns of regularity among rates of change between the variables. It is the concept of "rate of change" that helps students identify, represent, and reason about families of functions, such as linear, quadratic, and exponential. Early on in the study of algebra, symbols are used to represent these situations, along with other representations. However, symbol manipulation, per se, is not the focus modeling situations and understanding the functions used to build such models are the focus. The contexts help students intuitively use symbols to represent their ideas about the relevant variables and their relationships. Eventually, students work with symbols, free of context. They do so in investigating the general characteristics of a family of functions, including solving for specific values of examples that belong to the family. Additionally, they investigate the syntax of the language of symbols as they look more closely at ways to represent problem situations symbolically, particularly those that give rise to different but equivalent expressions (Phillips and Lappan, 1997~. The CMP curriculum suggests that algebraic reasoning can be developed by exploring a variety of contextual settings connected by organizing themes: the problems bring meaning to the themes. More importantly, the students in the CMP curriculum are developing "habits of mind." As they move through the grades, the following questions become an integral part of their algebraic reasoning: "What are the variables? What are some patterns that relate these variables? How are these patterns captured? Can you predict from these representations? How are the variables changing in relation to each other? How is an increase in the independent variable related to a change in the dependent variable? Where is the dependent value changing the most? The least? How can this change be seen in a table? Detected in a story? Observed in a graph? Read from symbolic representations? Where does the graph cross the x-axis and the y-axis, and what is the significance of each of these intersections? Where does the dependent value reach its greatest value and its least value, and what is the significance of each? What effects does a change in parameter have on the situation?" (CMP) (Lappan et al., 1995) Taken altogether, the intuitive uses of algebraic thinking in a problem-based curriculum plus the specific development of the concepts of variable and relationships among variables constitute a powerful, sense-making approach to algebra in the K- 12 curriculum for all students. However, these practices are a radical departure from the traditional algebra curriculum that has focused almost exclusively on manipulating expressions and solving symbolic equations. These new curricula put new demands on teachers: to teach mathematics through rich problems requires the teacher to have a deeper understanding and a broader view of mathematics and a deeper knowledge of a pedagogy based on inquiry. Teachers for the K through 12th grades are key players in the improvement of student learning of algebra. Any effort to reform the algebra curriculum or methods of teaching must help them to see the potential for developing mathematical power for their students and help them communicate this mathematics power to parents. The keys are student growth and performance. QUESTIONS FOR DISCUSSION What do you see as the issues or big questions concerning the development of · a coherent algebra curriculum across the grades K-12; · criteria or lenses for selecting curriculum, appropriate pedagogy, and assessment; and · long-term support for teachers and Standards-based algebra? Also, where do we go from here?
OCR for page 29
PRESENTATIONS ONDAY ONE 29 REFERENCES Lappan, G., Fey, J., Friel, S., Fitzgerald, W., and Phillips, E. (1995.) The Connected Mathematics Project. Palo Alto, CA: Dale Seymour Publications. Mullis, I., Dossey, J., Owen, E., and Phillips, G. (1991.) The State of Mathematics Achievement: NAEP's 1990 Assessment of the Nation and the Trial Assessment of the States. Washington, DC: National Center for Educational Statistics. National Council of Teachers of Mathematics Algebra Working Group. (1997.) "A Framework for Constructing a a Vision of Algebra: A Discussion Document." Final report to the Board of Directors. East Lansing: Michigan State University. Phillips, E., and Lappan, G. (1998.) "Algebra: The First Gate," in Mathematics in the Middle, edited by Larry Leutzinger. Reston, VA: NCTM. Schmidt, W., et al. (1996.) Characterizing Pedagogical Flow: An Investigation of Mathematics and Science Teaching in Six Countries (pp. 49~. The Netherlands: Kluwer Academic Publishers. U.S. National Research Center. (1996.) "TIMSS" (Report No. 7) (pp. 9~. Washington, DC: National Center for Education Statistics.
OCR for page 30
OCR for page 31
Enhancin,g Al,geb~aic Reasoning with Technolo,gy Geoffrey Akst Borough of Manhattan Community College/CUNY New York, New York We tell our students that mathematical questions should be well defined. In mathematics education, however the important questions are usually ambiguous. Consider, for example, the question of whether technology can be used to enhance algebraic reasoning. Our response must depend on which technology we have in mind and on what meaning we attach to the term "algebraic reasoning." To start with the latter point, suppose that we construe algebraic reasoning narrowly: suppose we say it is deducing consequences from a symbolic relationship. Solving an equation, a kind of proof, is a case in point. Technology even the humble four-function calculator can help to foster this kind of reasoning by relieving the student of computational distractions and complexities, allowing him or her to focus attention on the structure of the algebraic argument. And a graphing or symbol-manipulating calculator can be used, at a minimum, in checking the validity of the conclusions drawn, step by step. Alternatively, suppose that we broaden our conception of algebraic reasoning to include a kind of intuition. For a student with this "algebra sense," symbol manipulation is a meaningful activity, and unreasonable answers are apparent. In this interpretation, algebraic reasoning includes the ability, through pattern recognition, to conjecture, even to have confidence in, symbolic generalizations. Surely technology has a role to play in developing this algebra sense. A student can use a computer algebra system to run through many examples of polynomial multiplication and develop the ability to predict the degree of a product or the number of terms the product contains. Finally, the case can be made that algebraic reasoning implies the ability to employ symbolic techniques in attacking word problems. Even anti-calculator educators must admit that the more realistic and less restrictive applications which calculators and computers allow beginning students to tackle can motivate interest and, hopefully, increase time-on-task. No matter how it is interpreted, algebraic reasoning can benefit from technology, even when we put technolog- ical tools aside. High-tech instructional delivery systems of increasing sophistication now present students with motivating lessons. CD-ROM-based computer programs and the World Wide Web are increasingly becoming attractive environments in which to learn about all subjects, including algebraic reasoning. 31
OCR for page 32
OCR for page 33
Al,geb~a for Everyone' With or Without Technolo,gy? Milton 0. Norman Granby High School Norfolk, Virginia Americans have placed an increasing number of unusual burdens on their educational system. Schools are expected to teach basic skills in reading, writing, mathematics, and reasoning. They also are supposed to introduce students to a complex cultural tradition in the liberal arts and the sciences, a tradition that now includes the entire world, not simply Western civilization. They further are charged with developing individual skills relating to such nonacademic areas as vocation, leisure-time activity, and citizenship, as well as providing such specialized programs as driver education, sex education, drug education, and parent education. As social institutions, schools are expected to be, simultaneously, agents of social stability and social change. By the 1970s, apprehensions grew that American schools were not accomplishing all these objectives: indeed, the objectives themselves were questioned as possibly unreasonable. The back-to-basics movement at the end of the 1970s was an attempt by educators to narrow and sharpen their ambitions. Other efforts began in the 1980s to restore interest in academic quality and excellence. A Nation at Risk, a 1983 report issued by a bipartisan federal commission, emphasized the need to upgrade American education at all levels. Technology is seen as a solution to upgrading mathematics education in general and to making algebra accessible to all students in particular. In simplest terms, the concern of the student of mathematics is "why" and of technology, "how." The relationship between the two is actually much more complex, however; some technologies are mathematics and science intensive, whereas the manufacture of such items as cigarettes or furniture depends much less on mathematics and science. Education must harness the particular skills of technologies to introduce in a more effective way concepts that were previously reserved for a selected group of students. Technology within the classroom has become a large issue, however, because of the many inequities in the United States. Some schools are without graphing calculators; others have full Internet access. This type of issue cannot be addressed solely by documents that propose the use of appropriate methods of teaching higher level skills. Rather, they must be addressed by adequate funding for equipment and essential training. Classroom teachers do not have the innate ability to grasp all the new technology introduced every day. How should we teach algebra? Over half of the objectives in the current "Standards of Learning" within the state of Virginia suggest the use of the graphing calculator and/or computers to reinforce algebraic concepts. The argument for the graphing calculator is its accessibility to each student and its mobility. The handheld computer/ calculator is capable of complex Computer Algebra Systems (CAS). The technologies appropriate for mathematics depend on an individual teacher's training, experience, and the information disseminated by forums such as this one. We must be cognizant of the differences in experience, philosophies, and political climate within individual states and school districts. These are all major contributors to how we, as teachers, will prepare students to participate in the new age. 33
OCR for page 38
38 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Research on symbol manipulation predominantly has consisted of a documentation of error patterns. To know common errors is useful to teachers. They can try to flag the problem areas, draw students' attention to the correct procedures, and provide controlled practice to encourage progressive automaticity. Unfortunately, the data show that these "fixes" often fail over short periods of time, and the errors resurface. Improvement is incremental and modest. The loss of student participation, approximately 50% per year, far exceeds the modest instructional gains. Other research has been devoted to designing a functions-based, context-based approach. A functions-based approach means that, instead of first teaching a whole layer of algebra as consisting of methods for solving equations and working with unknowns, one uses from the beginning the idea of relationships between or among quantities or sets as the basis for algebraic thinking. The context-based part of the approach means that the algebraic activities emerge from problems and tasks that relate frequently but not necessarily exclusively to referents from everyday activity. Researchers have offered a multitude of arguments for this, including the ease of identifying situations to which students can relate; the value of motivating math with a need to explain, predict, and model and thus actively to involve students in useful mathematics; the value of position, time, and rate to make the idea of variation accessible, to anticipate calculus, and to lessen dependence on symbol strings and to vary representations with graphs and tables, for example; and how making algebraic manipulation skills instrumentally rather than intrinsically valuable and using new technologies to carry out the algorithms of traditional algebra allows students to focus on using problem solving, communication, and learning technologies to engage in rich modeling-based activities. Focusing on functions could have some limitations. These could include a tendency to eliminate the study of the simple, elegant structures inherent in finite algebras, a lack of attention paid to proof and theorem-building, and inadequate time devoted to developing fluency in algebraic manipulation skills. Just as not all of the benefits listed can be attained by any presentation of a functional approach, neither will all of the potential limitations apply. Teachers or curriculum writers can choose what to emphasize and de-emphasize. In my research on functions-based approaches in 7th grade, 11th grade, and in university-level precalculus courses, I have found that, within contextual situations, an effective and knowledgeable instructor can carefully and intentionally incorporate and highlight the issues of focusing on structure, proof, and fluency in discussions and assignments. When this is done, student reasoning is strong and well-grounded. For instance, in a two-year study involving an eight-week introduction to algebra to an entire 7th grade class using functions, computers, and motion detectors, one teacher commented, "In the old, standard way of doing algebra, everybody has to do itin this strict formal, and there's no creativity or freedom of self-expression, no individuality to it. So you lose a lot of kids because they don't want to get boxed in or they want to be able to do it their own way. So they try and create it their own way, and it falls apart on them, and there' s nothing for them to fall back on. There's no support, you know, 'I think this should happen' and if that doesn't happen, boom, you're off the cliff and that's the end." In contrast, after using contextual problems and a functional approach, the same teacher commented, "When they get to solving an equation, their explanations are not, 'I need to subtract because it cancels multiplying.' It's, 'I need to subtract, because I need to get this hiker back to where he started.' They're using situations that they've seen along the way to explain things that they used not to have explanations for beyond, '...that cancels that out.' [Students can] believe their mathematical reasoning is right rather than just thinking it's right because 'I followed the right steps.' " Research on a functional approach with contextual problems has shown that students tend to prefer this approach. They find the use of "real-world" problems to be highly motivating, and they show evidence of deeper thought at younger grades. Instead of focusing exclusively on the manipulation of symbols, they can move from problem context to problem context, often using these as the comparative structure rather than the symbolic algebraic form. The challenge becomes how to order the context so as to be assured that the students are gaining progressive conceptual development and also that their use of such powerful tools as graphing calculators and computer software is accompanied by a flexible and deep level of understanding. At the same time, our research has shown that teachers find it challenging to move to a functional and contextual approach. Many of them were successful in traditional settings and lack experience in problem solving. Furthermore, unless a functional or contextual approach is aligned with methods of assessment, teachers tend to minimize these activities and to steer towards easily accessible and predictable skill development. Their use of new technologies is all too often limited to simple inductive inferences from a few examples or to a display of the
OCR for page 39
PRESENTATIONS ONDAY ONE 39 graphical image and identification of its critical points (vertices, intercepts, symmetries, etch. For teachers to use new technologies to explore new content opportunities or as evervdav or Practical tools takes extensive Professional development. THE DEEP USE AND COORDINATION OF MULTIPLE REPRESENTATIONS Pick up a reform document and you will see a call for the use of multiple representations. However, in most treatments, we still see the obvious preference for the symbolic equation over any other type of representation. This occurs in part due to technologies that are heavily symbolically driven or limited in screen space. In textbooks, lip service is given to the rule of three (graphs, tables, and equations), but the internal logic of working with each en, ~ `, ~ , , ~ ,, representation is given short shrift. In contrast, researchers have documented the potential of students to work actively and extensively in a variety of representations. This includes both the use and the invention of the representation. In the use of data tables, functional approaches often lead to the merging of statistical ideas with algebraic approaches. Similarly, in graph- related activities, tools such as "Function Probe" and even geometric sketchpads can lead to an integration of the visual, analytic, and symbolic. In our research, we have documented the essential role of the table in the development of functional reasoning and notation. Once the symbolism was available, the use of the table declined, but today, spread sheet capabilities could help us think about how to use tables once again. Graphically, we have shown that there are multiple approaches to the topic of transformations and that the "best" method depends on one's preference for affix) notation versus a more symmetrical x-y form of notation. In other work, researchers have documented the potential of introducing graph shape as a means to gain facility in the use of notation. Graph shape tends to highlight issues of rate of change and, with careful and early introduction, issues of accumulation. For example, students can view an exponential function as a bar graph and see the area under the curve as the sum of the previous bars. Hearing again from a classroom teacher, we report, "We looked at graphs qualitatively [in a college physics class], and it didn't make sense to me then. And it made more sense to me looking at it the [way] we did [in this project]. Think about letting kids look at graphs without having specific data, that was new. Just look at general forms of graphs rather than having particular pieces of data. That wasn't something I'd ever thought of doing." STUDENT METHODS AND INTERACTIONS TO GENERATE MATHEMATICAL INVENTION AND DISCUSSION _ _ communicate and make connections Critical to understanding a reform approach to algebra is considering how it fits into the larger framework for reform, including creating a need for the ideas and subsequently developing formal description as a means to As documented repeatedly, there are verbal challenges when the formal meanings are merged with informal ones. The solution of obliterating or erasing the informal meanings seldom works. More effective seems to be uncovering the rational path that was taken to evolve towards the formal definition. Also critical is a thorough understanding of the possibility of alternative definition. Learning to reach a consensual decision for clear communication is important. Reform approaches to algebra reach deep into our assumptions about learning. They challenge us to question our beliefs about who can learn and seek to eradicate harmful stereotyping. The use of rich contexts and tools and interactive and collaborative groups leads us to question the validity of theories of learning that only value individual progress and ignore group dynamics. Research on these contexts reminds us also to be wary of the idea that if all students participate in groups, all leave equally prepared. Finally, research in the use of new technologies and project-based learning allows us to question many peoples' assumptions about the required developmental sequences. It appears that when students operate in complex settings with powerful tools, the sequences of learning vary, and the insights that are gained are more a function of the tasks and tools than any rigid developmental sequence. Paramount among these are assumptions about the description of concrete versus abstract thinking.
OCR for page 40
40 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM SYSTEMIC AND COMMUNITY CHANGE TO SUPPORT HIGHER ALGEBRA SUCCESS Success in algebra is not simply a matter of getting individual students to develop psychologically or cognitively. The course acts within the community of learners as a signal of students' academic acuity and high career aspirations and expectations. Some students are provided extensive help on it at home, while other lack those resources and must either "get it" at school or cope with failure or exclusion. Since algebra is a gateway to advanced math course taking, its importance in the curriculum exceeds many other uniformly accessible courses or electives. For this reason, participation and success in it has been cast by some researchers as an issue of civil rights. It is at this critical juncture that our decisions about content, sequence, pace, curricular approach, use of technology, forms of interaction, types of help, and feedback to students carry out a political agenda that has ramifications way down the line in terms of participation and success. Research has shown that community and parental involvement can be critical to students' persistence and success. It also has shown that reform is difficult for teachers on their own and that the systems can respond to change with institutionalized racism, classism, and sexism. Even as females succeed at equal rates, their less favorable attitudes about the domain and the tendency of their male peers to sex-stereotype make this an important territory for intervention and counseling beyond discussions of cognitive approaches. In conclusion, research on reform in algebra suggests that progress will depend on a systemic response. Teachers must be deeply and consistently involved, must guide the process, and must be held accountable for equitable outcomes. We must introduce early the basic ideas so all children are well prepared for the transition and use a variety of materials and technologies, address the deep content needs of teachers in implementing reform, involve the larger communities, and constantly monitor our progress toward the important outcome of high achievement in algebra as a equal opportunity for all students.
OCR for page 41
Al,~ebra: What All Students Can [earn Susan E. Williams Department of Curriculum and Instruction The University of Houston Houston, Texas David Molina The Charles A. Dana Center The University of Texas at Austin Austin, Texas It is common for people to believe that they never use algebra after their formal course work and, hence, to believe that algebra is not important. The truth is, they may not use algebra in the way they experienced it in school, but algebra is prevalent in their lives. Algebra has been used as a vehicle for developing important skills necessary for further work in mathematics, but in education, the power of algebra its application in everyday situations has seldom been acknowledged. WHY IS ALGEBRA IMPORTANT? Traditionally, algebra has been approached as a generalized structure of our numeration system. Students have been asked to solve problems that involve rote computations in sterile or contrived settings. According to the Board of Directors of the National Council of Teachers of Mathematics (NCTM Board, 1993), the algebra of the past advances only a narrow range of by-hand skills for transforming, simplifying, and solving equations most divorced from any natural context. The algebraic understandings cultivated in this way are far removed from those needed in a technological world by both employment-bound and college-bound students. The algebra that most people use on a regular basis consists primarily of generalized and specific examples of function relationships that illustrate the systematic dependence of one quantity on another. This perspective of algebra is different from the perspective evident in traditional school algebra, but it is equally important. Whether one realizes it or not, determining gas mileage, predicting the amount of food to prepare for a party, and figuring the costs of renting videos are all examples of daily situations steeped in algebra and requiring a certain level of algebraic understanding. So is algebra for everyone? The answer depends in part on how one perceives algebra. Not every student needs proficiency in symbol manipulation skills. By choice or circumstance, many students will never reach the levels of mathematics study where they will use these skills. However, every student needs to understand how quantities depend on one another, how a change in one quantity affects the other, and how to make decisions based on these relationships. Every student is algebraically educable, although not every student needs to know how to simplify rational expressions or to derive the quadratic formula. Every student is capable of (a) learning about the use of symbols, (b) using patterns to look for generalizations, and (c) understanding the use of dependent, systematic relationships to model situations and make predictions. 41
OCR for page 42
42 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM WHY IS IT IMPORTANT FOR STUDENTS TO BE SUCCESSFUL IN ALGEBRA? The need for knowing algebra may not be as obvious as the need for knowing how to read, write, and do arithmetic, but the lack of such knowledge blocks the entrance to more advanced academic and vocational studies and often limits job advancement opportunities. Successful completion of high-school algebra appears to open the doors of the mind as well as the doors of opportunity. A recent study by the College Board (Hawkins, 1993) revealed that more than 70% of the students who took one or more years of algebra attended college within four years of high-school graduation. High-school students with one year or more of algebra were more than twice as likely to attend college as students who did not take algebra. The College Board also reported that, in general, majority students who complete high school algebra attend college at a much higher rate than underrepresented minority students. For example, at the national level in 1993, 58% of all white students as compared to only 47% of all African American students attended college. But successful completion of high-school algebra and/or geometry virtually eliminated the gap. For students who completed high-school geometry, 83% of the white students compared to 80% of the African American students enrolled in college. This description of the College Board's perspective of "Algebra as the Gatekeeper" raises concerns not only for educators but for the nation as a whole. The business community has expressed its concern by demanding that high- school graduates be given access to the entire mathematics curriculum (in particular, algebra, geometry, and calculus concepts). Important aspects of the curriculum have been withheld from students who have not mastered the basic skills of mathematics. Students today need more than basic skills; they need technical skills (including algebraic reasoning) to obtain and retain jobs. In the past, technical skills have separated the "cans" from the "cannots," but in the future, these skills will create an even greater division in our work force. Those with technical skills will be more in demand and, as a result, will earn higher wages. The "cans versus cannots" will be replaced with the "haves versus have note." Students of all races have been guaranteed equal education, but the students who most often are tracked into low-level mathematics courses are minorities and/or those of low socio-economic status. Data from the National Assessment of Educational Progress (NAEP) indicates that less than 50% of the students in urban schools take any mathematics beyond one year of algebra, and 20% never study algebra at all (Mullis et al., 1991~. This disturbing situation has led Bob Moses, a former mathematics educator and now noted civil rights activist, to identify access to algebra as an issue for a new civil rights movement (Jester, 1993; Michelmore, 1995~. HOW DO WE TEACH "ALGEBRA FOR ALL"? Whether the goal is access or increased mathematics achievement, finding the "Right Algebra for All" is a goal that we need to embrace (Chambers, 1994~. We know from experience that the current school approach to algebra is too abstract and an unmitigated disaster for most students (Steen, 1992~. This rigorous approach has served a purpose one of identifying groups of students who have proven worthy of continued study in the sciences. This may not have been the intended purpose; nonetheless, it has and continues to separate and "track" our students. If we are to develop an algebra that can provide all students with tools that will allow them to advance in both academic and employment settings, then administrators, curriculum leaders, and teachers are needed to help guard against "Algebra for All" as being interpreted to mean that every student should be placed in what one might call a traditional first-year algebra class. This approach will simply allow many students to experience failure at a higher level, according to Jack Price, NCTM Past President (NCTM Board, 1993~. If "Algebra for All" is to become a reality, it is vital that algebra' s connection to the world and its usefulness be evident. Students must be allowed to recognize the role and place of algebra. We need to use the information that we now have about how students learn best and apply that knowledge to our teaching. Students need to be involved in active learning situations, with new information being connected to the information they already possess. Algebra students need to be provided with opportunities to experiment and collect data, to analyze that data, and to draw reasonable conclusions based on the findings. Algebra needs to focus on "ways to solve a problem" rather than "the way to solve a problem" so as to equip students with multiple solution strategies. A major obstacle in creating such courses stems from the fact that teachers have never been taught how to teach in a way that engages a diverse student population in a variety of interesting and important learning
OCR for page 43
PRESENTATIONS ONDAY ONE 43 activities structured to develop the inherent critical thinking and problem-solving abilities of all students (Seeley, 1993). If administrators pressure teachers to make algebra accessible to all without allowing for planning and development time for teachers, the outcome will lead at best to mediocre results and has the potential to subject students to even worse situations than they are already experiencing. Teachers who are convinced that the goal is not attainable or who are uncertain of a viable approach may either (a) subject students to the rigors of traditional algebra and confirm their beliefs that many students just can't do algebra or (b) water down the content of their traditional course so that all students can succeed. Neither massive failure nor grade inflation are the desired outcomes of the algebra reform movement. To make "Algebra for All" a reality means focusing on fewer topics organized around central ideas and preserving mathematical integrity and rigor through the depth of study rather than the breadth of study. If we face reality, our students today have different needs for their high-school mathematics courses, and the courses must now serve different purposes. According to Cathy Seeley of the Texas State Systemic Initiative, it is not that we must lower the level of what we teach so that all kids can do it nor is it that we must simply raise our expectations and do the same thing for all students that we have done for some. Rather, we must shift the content and approach of what we do in order to come closer to serving some purpose in our students' lives. Given a school with teachers eager to meet the challenge of creating an "Algebra for All" and armed with a vision for implementation, the probability of sustained change is dim without the support and encouragement of the school's campus leaders. History has proven over and over again that strong campus leaders are the heart of successful programs. We ask campus leaders to join mathematics educators in rethinking the purpose of algebra, in verbalizing the importance of algebra for every student, and in supporting efforts to create such a program. Counting on You: Actions Supporting Mathematics Teaching Standards (National Academy Press, 1991), a document created by the National Research Council' s Mathematical Sciences Education Board, provides an action plan for school boards and school administrators to revitalize mathematics education by the year 2000. The seven key action items in the document are as follows: 1. Establish mathematics goals and programs in your schools that are consistent with national (and state) standards for curriculum and evaluation. 2. Provide sufficient resources (equipment, time, budget) to support an instructional program meeting the standards. 3. Discontinue use of standardized tests that are misaligned with national (state and local) standards for curriculum. 4. Provide time for your teachers: time to plan and evaluate their own teaching, consult with colleagues about teaching, and confer with supervisors about teaching effectiveness. 5. Give your teachers freedom to exercise their ability, judgment, and authority and involve them in decision making. 6. Institute a comprehensive inservice program consistent with national (and state) standards and involve teachers in its development and planning. 7. Evaluate teachers by using information gathered from various sources the teacher's goals and plans, students' accomplishments, repeated classroom observation and national professional standards. CLOSING REMARKS Mathematics teachers need to understand algebra very differently than we did a few years ago. Success in algebra in the past has been judged by mastery of a set of manipulative skills presented in isolation. In contrast, the algebra of today is expected to eliminate artificial and meaningless exercises and to de-emphasize symbolic manipulation while at the same time giving students opportunities for exploration and for conjecturing and discovering. Not only must we understand algebra differently, we must be prepared to illuminate others. The algebra reform movement put into motion by the NCTM Curriculum and Evaluation Standards for School Mathematics (1989) will not succeed if teachers are not prepared to rethink (a) what is algebra, (b) how do we teach algebra, and (c) who can be successful in algebra. It is actions such as those listed in this paper that will provide teachers with the initiative and courage to experiment with new instructional strategies and materials, to collaborate in their planning, and to recognize ways to attain the goals of the school and the district.
OCR for page 44
44 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM REFERENCES Chambers, D.L. (1994.) "The Right Algebra for All." Educational Leadership, 51~6), 85-86. Hawkins, B.D. (1993.) "Math The Great Equalizer: Equity 2000 and QUASAR, Improving Minority Standing in Gatekeeper Courses." Black Issues in Higher Education, 10~6), 38-41. Jetter, A. (February 21, 1993.) "Mississippi Learning." The New York Times Magazine; 28-32; 50-51, 64, 72. Michelmore, P. (1995.) "Bob Moses's Crusade." Reader's Digest, 875~3), 107-111. Mullis, I.V.S., Dossey, J.A., Owen, E.H., and Phillips, G.W. (1991.) The State of Mathematics Achievement: NAEP's 1990 Assessment of the Nation and the Trial Assessment of the States. Washington, DC: National Center for Education Statistics. National Council of Teachers of Mathematics. (1989.) Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics Board. (1993.) "Board Approves Statement on Algebra." NCTM News Bulletin, 30~6), 1, 3, 6. National Research Council, Mathematical Sciences Education Board. (1991.) Counting on You: Actions Supporting Mathematics Teaching Standards. Washington, DC: NationalAcademy Press. Persell, C., and Cookson, P. (1982.) "The Effect of Principals in Action" in The Elective Principal: A Research Summary. Reston, VA: National Association of Secondary School Principals. Seeley, C. (1993.) "Increasing Access or Ensuring Failure? Policy Makers Throw a Hammer into the Wall." Algebra for the Twenty-first Century: Proceedings of the August 1992 Conference (pp.43-45~. Reston, VA: National Council of Teachers of Mathematics. Silver, E.A. (1995.) "Rethinking 'Algebra for All.' " Educational Leadership, 50~6), 30-33. Steen, L. (1992.) "Does Everybody Need to Study Algebra?" Basic Education, 37~4), 9-13.
OCR for page 45
Improving K-14 Al,gelora Instruction: A Discussion of Teachers' Responsibilities and Students' Opportunities Beatrice Moore-Harris Houston Independent School District Houston, Texas We cannot have a discussion about how K-14 students learn without discussing how teachers teach. The improvement of students' learning of mathematics algebra in particular involves changes in student behavior as well as teacher behavior. To sustain changes in behavior, students and teachers must be able to reflect on new concepts, skills, and strategies as they practice them. Let us first explore teacher behaviors. In order to develop teacher behaviors that facilitate students' ability to think algebraically, planning must be provided throughout the school year to debrief; to discuss successes and failures; to develop, review and revise lessons plans; and to continue professional development follow-up activities that address areas of student and teacher needs as identified by teachers. Professional development opportunities that address instructional alignment, appropriate use of data (including test-item analysis), and equitable as well as appropriate grading practices are non-negotiable components. Additionally, sessions on technology, both the calculator and computer, critical thinking, reading and mathematics, appropriate use of manipulatives, curriculum alignment, developmentally appropriate instructional practices, learning styles, and cultural diversity as they relate to and support algebra instruction should be included in all professional development activities. Among the complexities and interactions of variables studied, many explanations and much confusion exist for the lack of student success in and preparation for algebra. There is clear agreement on one point, however; disparity begins to appear early in students' school careers (Camerlengo, 1993~. As we begin to focus on solutions and strategies that address improving student achievement in algebra, it is important that we discuss how we may be able to provide a rich program in pre-algebra experiences that is taught in nontraditional ways. These pre-algebra experiences should increase the relevance of mathematics to the lives of the students by making it engaging and inclusive of cultural contexts that are born of student' s environment, experiences, and interests. Furthermore, the use of activities that students find enjoyable, such as games, stories, dramatic play, and music, are critical instructional elements that incorporate what we have learned about learning styles and that bring mathematics to life. Unfortunately, many students are trapped in a system that does not allow them to develop critical thinking abilities or to access a curriculum intended for preparation for algebra because the emphasis of instruction is on mastery of basic facts. The remainder of this paper will provide some brief suggestions about how we could attempt to redesign America's K-14 classrooms to give all students opportunities and experiences that allow them to change their behavior as learners and that specifically allow them to internalize concepts and skills that facilitate algebraic thinking. My intention is to provide a springboard for discussing current realities and the development of solutions to issues that are paramount to all who are in positions of responsibility district curriculum supervisors, university-level instructors, public or private school building administrators, students, teachers, and parents. Vygotsky (1978) developed the concept of the "Zone of Proximal Development" (ZPD) in which he made 45
OCR for page 46
46 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM distinctions among the skills that a child demonstrates solidly on her own, those that she can demonstrate only with help, and those that she cannot demonstrate at all. It is the middle range (ZPD) that provides the most potential for growth. Often those who have just recently conquered a concept can best speak to the changes in understanding that have occurred. This middle zone can extend further up and further down than one may expect. On the upper end, some children can grapple with more complex ideas than they are generally given credit for. On the lower end, they often have not developed a solid understanding of even thoroughly explored concepts and, in new situations, need to remind themselves of what both we and they know they know. In schools with tracking, students in "lower" ability groups are not exposed to content that teachers perceive as too difficult for them. As a result, when students have developmental spurts, they may lose the opportunity to move ahead quickly because they are not being exposed to sufficiently challenging content. Children in these groups also do not have access to participation in sophisticated discussion about topics: thus, they lose that positive modeling. The goal is not to have students reach closure on a particular concept; rather, the goal is for important questions to remain open so that students' understanding can evolve. Using students' prior knowledge is a critical component of a successful mathematics program. The knowledge that students bring to a new experience will greatly shape their understanding of new concepts. Given a problem- solving task, students must bring to that task what they already know in order to develop solutions that are personally meaningful to them. Students may be asked to think of possibilities within a given situation and to create a word problem that involves real world contexts from their environment and experiences. This creates an opportunity for the students to explore language in a mathematical context by using language skills to discuss and create a world problem. By encouraging students to explain their thinking, we help them to solidify understanding. As they make deeper connections, develop new insights, and answer other students' questions, they further convince themselves of the truth of their understanding. Another technique is to remove the question from word problems. This forces the student to focus on the stated relationships in the problem, predict a conclusion, and then create a question that leads to a solution. As mentioned earlier, games are of interest to most students and provide rich contexts in which students can explore mathematical ideas or practice mathematical skills (Bright, Harvey, and Wheeler, 1985~. Many adults who are successful in mathematics had experiences as children where recreational time was spent engaged in games that provided practice in the acquisition of logical and analytical thinking abilities. Today's students do not come to school with these experiences because their recreational time is spent on activities that, in many cases, do not provide the same type of thinking. Thus, we need to provide recreational experiences in K-14 classrooms. Through the incorporation of additional experiences that are steeped in realistic and relevant contexts, the mathematics in a program becomes more accessible, and a more fertile foundation that promotes algebraic thinking is built. Concepts that are crucial to success in algebra, such as integers, fractions, the meaning of a variable, and ratio and proportion, are easily introduced, reinforced, reviewed, and maintained via a game format. In addition, students need to learn to move from one representation to another using the language of mathematics. For example, students may use a manipulative to set up a problem, draw a picture to provide the teacher with evidence of what was done, use written or oral language to explain the process used and to summarize results, and, finally, use symbolic representation to state mathematically the problem and solution. To transition successfully from one representation to another depends on the concept involved and the students' experiences in mathematics, so the teacher must carefully address the matter in a way that provides adequate time for students of varied levels of understanding to make the transition. As we look at the structure and organization of K-14 algebra instruction, we can consider the following example of how a typical class period is conducted during when the focus is to develop algebraic concepts, skills, and thinking processes. Imagine a classroom where students enter the room, are presented with a problem, and begin to work in small groups to solve it. The teacher monitors discussions, uses questioning strategies to assist students with their thinking, and organizes a discussion that allows students to share ideas and to summarize their findings or strategies. There is dialogue in which ideas are expressed in students' everyday language as well as appropriate mathematical language written and verbal. Of course, there are times where direct instruction or review of concepts and skills will be appropriate and required. At all times, however, instructional decisions are made on the basis of what is best for meeting the needs of the students and careful analysis of student performance. In conclusion, we must reflect once more on the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics. In particular, we should revisit the NCTM's five goals for all students:
OCR for page 47
PRESENTATIONS ONDAY ONE · Learn to value mathematics · Become confident in your ability to do math · Become mathematical problem solvers · Learn to communicate mathematically · Learn to reason mathematically 47 As we continue the discussion of how to best provide K-14 algebrainstruction for all students and how students best learn, these five goals are important to the discussion. REFERENCES Bright, G.W., Harvey, J.G., and Wheeler, M.M. (1985.) "Learning and Mathematical Games." Journal for Research in Mathematics Education Research Monograph Series, No. 1. Reston, VA: National Council of Teachers of Mathematics. Camerlengo, V. (1993.) "Mathematics Specialist-Teacher Program: An Intervention Strategy for All (Reaching All Students with Mathematics)." Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (1989.) Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author. Nelson, B.S., Silver, E.A., Smith, M.S. (1995.) "The QUASAR Project: Equity Concerns Meet Mathematics Education Reform in the Middle School." New Directions for Equity in Mathematics Education. New York: Cambridge University Press. Vygotsky, L.S. (1978.) Mind in Society: The Development of Higher Psychological Processes. Edited by Michael Cole, Vera John-Steiner, Sylvia Scribner, and Ellen Souberman. Cambridge, MA: Harvard University Press.
OCR for page 48
Representative terms from entire chapter: