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Implications for teacher professional development are discussed. Skip to Main Content. Search in: This Journal Anywhere. Advanced search. Submit an article Journal homepage. Alibali University of Wisconsin-Madison. Pages Published online: 05 Dec Original Articles. Article Metrics Views. Article metrics information Disclaimer for citing articles.

People also read Article. Mitchell J. Nathan et al. Mathematical Thinking and Learning Volume 9, - Issue 3. Published online: 5 Dec Nicole M. McNeil et al. Cognition and Instruction Volume 24, - Issue 3. Agriculturally based activities, such as 4H and Future Farmers of America, have for many years used the farm setting and students' interest in farming to teach a variety of skills. It takes only a little imagination to think of how to use the social, economic, and scientific bases of agriculture to motivate and illustrate skills and knowledge from all of the academic disciplines. Many schools are now using internships and projects based on local business activities as teaching tools.

One example among many is the integrated program offered by the Thomas Jefferson High School for Science and Technology in Virginia, linking biology, English, and technology through an environmental issues forum. Students work as partners with resource managers at the Mason Neck National Wildlife Refuge and the Mason Neck State Park to collect data and monitor the daily activities of various species that inhabit the region.

They search current literature to establish a hypothesis related to a real world problem, design an experiment to test their hypothesis, run the experiment, collect and analyze data, draw conclusions, and produce a written document that communicates the results of the experiment. The students are even responsible for determining what information and resources are needed and how to access them.

Student projects have included making plans for public education programs dealing with environmental matters, finding solutions to problems caused by encroaching land development, and making suggestions for how to handle the overabundance of deer in the region. These examples suggest the potential that a more integrated education could have for all students.

Thus continuing to maintain a sharp distinction between vocational and academic instruction in high school does not serve the interests of many of those students headed for four-year or two-year college or of those who expect to work after high school. Work-bound students will be better prepared for work if they have stronger academic skills, and a high-quality curriculum that integrates school-based learning into work and community applications is an effective way to teach academic skills for many students.


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Despite the many examples of innovative initiatives that suggest the potential for an integrated view, the legacy of the duality between vocational and academic education and the low status of work-related studies in high school continue to influence education and education reform. In general, programs that deviate from traditional college-prep organization and format are still viewed with suspicion by parents and teachers focused on four-year college.

Indeed, college admissions practices still very much favor the traditional approaches. Interdisciplinary courses, "applied" courses, internships, and other types of work experience that characterize the school-to-work strategy or programs that integrate academic and vocational education often do not fit well into college admissions requirements. The general principle should be to try to design standards that challenge rather than reinforce the distinction between vocational and academic instruction. Academic teachers of mathematics and those working to set academic standards need to continue to try to understand the use of mathematics in the workplace and in everyday life.

Such understandings would offer insights that could suggest reform of the traditional curriculum, but they would also provide a better foundation for teaching mathematics using realistic applications. The examples in this volume are particularly instructive because they suggest the importance of problem solving, logic, and imagination and show that these are all important parts of mathematical applications in realistic work settings.

But these are only a beginning. In order to develop this approach, it would be helpful if the NCTM standards writers worked closely with groups that are setting industry standards. The NCTM's Curriculum Standards for Grades include both core standards for all students and additional standards for "college-intending" students.

The argument presented in this essay suggests that the NCTM should dispense with the distinction between college intending and non-college intending students. Most of the additional standards, those intended only for the "college intending" students, provide background that is necessary or beneficial for the calculus sequence. A re-evaluation of the role of calculus in the high school curriculum may be appropriate, but calculus should not serve as a wedge to separate college-bound from non-college-bound students.

Clearly, some high school students will take calculus, although many college-bound students will not take calculus either in high school or in college. Thus in practice, calculus is not a characteristic that distinguishes between those who are or are not headed for college. Perhaps standards for a variety of options beyond the core might be offered. Mathematics standards should be set to encourage stronger skills for all students and to illustrate the power and usefulness of mathematics in many settings.

They should not be used to institutionalize dubious distinctions between groups of students. Bailey, T. School-to-work for the collegebound.

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Hoachlander, G. Organizing mathematics education around work. Steen Ed. Levy, F. Journal of Economic Literature , 30 , National Commission on Excellence in Education. A nation at risk: The imperative for educational reform. Washington, DC: Author. Vocational education has been shaped by federal legislation since the first vocational education act was passed in According to the current legislation, the Carl D. Perkins Vocational and Technical Education Act of , vocational students are those not headed for a baccalaureate degree, so they include both students expecting to work immediately after high school as well as those expecting to go to a community college.

Approaches to Algebra

VATEA also promoted a program, dubbed "tech-prep," that established formal articulations between secondary school and community college curricula. For an argument about how education may be organized around broad work themes can enhance learning in mathematics see Hoachlander The Goals Educate America Act, for example, established the National Skill Standards Board in to serve as a catalyst in the development of a voluntary national system of skills standards, assessments, and certifications for business and industry.

For decades our industrial society has been based on fossil fuels. In today's knowledge-based society, mathematics is the energy that drives the system.

Advances in Linear Algebra Research - Nova Science Publishers

New mathematical techniques and analyses and even conceptual frameworks are continually required in economics, in finance, in materials science, in physics, in biology, in medicine. Just as all scientific and health-service careers are mathematically based, so are many others. Interaction with computers has become a part of more and more jobs, and good analytical skills enhance computer use and troubleshooting.


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  • In addition, virtually all levels of management and many support positions in business and industry require some mathematical understanding, including an ability to read graphs and interpret other information presented visually, to use estimation effectively, and to apply mathematical reasoning. Education in mathematics and the ability to communicate its predictions is more important than ever for moving from low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton , had a section "Focus.

    But precisely what mathematics should students learn in school? Mathematicians and mathematics educators have been discussing this question for decades. This essay presents some thoughts about three areas of mathematics—estimation, trigonometry, and algebra—and then some thoughts about teaching and learning.

    Estimation is one of the harder skills for students to learn, even if they experience relatively little difficulty with other aspects of mathematics. Many students think of mathematics as a set of precise rules yielding exact answers and are uncomfortable with the idea of imprecise answers, especially when the degree of precision in the estimate depends on the context and is not itself given by a rule. Yet it is very important to be able to get an approximate sense of the size an answer should be, as a way to get a rough check on the accuracy of a calculation I've personally used it in stores to detect that I've been charged twice for the same item, as well as often in my own mathematical work , a feasibility estimate, or as an estimation for tips.

    Trigonometry plays a significant role in the sciences and can help us understand phenomena in everyday life. Often introduced as a study of triangle measurement, trigonometry may be used for surveying and for determining heights of trees, but its utility extends vastly beyond these triangular applications. Students can experience the power of mathematics by using sine and cosine to model periodic phenomena such as going around and around a circle, going in and out with tides, monitoring temperature or smog components changing on a hour cycle, or the cycling of predator-prey populations.

    No educator argues the importance of algebra for students aiming for mathematically-based careers because of the foundation it provides for the more specialized education they will need later. Yet, algebra is also important for those students who do not currently aspire to mathematics-based careers, in part because a lack of algebraic skills puts an upper bound on the types of careers to which a student can aspire.

    Former civil rights leader Robert Moses makes a good case for every student learning algebra, as a means of empowering students and providing goals, skills, and opportunities. The same idea was applied to learning calculus in the movie Stand and Deliver. How, then, can we help all students learn algebra? For me personally, the impetus to learn algebra was at least in part to learn methods of solution for puzzles. Suppose you have 39 jars on three shelves. There are twice as many jars on the second shelf as the first, and four more jars on the third shelf than on the second shelf.

    How many jars are there on each shelf? Such problems are not important by themselves, but if they show the students the power of an idea by enabling them to solve puzzles that they'd like to solve, then they have value.

    Mathematics (MATH) Courses

    We can't expect such problems to interest all students. How then can we reach more students? One of the common tools in business and industry for investigating mathematical issues is the spreadsheet, which is closely related to algebra. Writing a rule to combine the elements of certain cells to produce the quantity that goes into another cell is doing algebra, although the variables names are cell names rather than x or y.

    Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires. By exploring mathematics via tasks which come from workplace and everyday settings, and with the aid of common tools like spreadsheets, students are more likely to see the relevance of the mathematics and are more likely to learn it in ways that are personally meaningful than when it is presented abstractly and applied later only if time permits. Thus, this essay argues that workplace and everyday tasks should be used for teaching mathematics and, in particular, for teaching algebra.

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    It would be a mistake, however, to rely exclusively on such tasks, just as it would be a mistake to teach only spreadsheets in place of algebra. Communicating the results of an analysis is a fundamental part of any use of mathematics on a job.

    go site There is a growing emphasis in the workplace on group work and on the skills of communicating ideas to colleagues and clients. But communicating mathematical ideas is also a powerful tool for learning, for it requires the student to sharpen often fuzzy ideas. Some of the tasks in this volume can provide the kinds of opportunities I am talking about. Another problem, with clear connections to the real world, is the following, taken from the book entitled Consider a Spherical Cow: A Course in Environmental Problem Solving , by John Harte The question posed is: How does biomagnification of a trace substance occur?

    For example, how do pesticides accumulate in the food chain, becoming concentrated in predators such as condors?