TEACHING IN CONTEXT: ENHANCING THE PROCESSES OF TEACHING AND LEARNING IN COMMUNITY COLLEGE MATHEMATICS

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  • This article was downloaded by: [Colorado College]On: 18 November 2014, At: 18:53Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

    Community CollegeJournal of Research andPracticePublication details, including instructionsfor authors and subscription information:http://www.tandfonline.com/loi/ucjc20

    TEACHING IN CONTEXT:ENHANCING THEPROCESSES OF TEACHINGAND LEARNING INCOMMUNITY COLLEGEMATHEMATICSEdward D. LaughbaumPublished online: 15 Dec 2010.

    To cite this article: Edward D. Laughbaum (2001) TEACHING IN CONTEXT:ENHANCING THE PROCESSES OF TEACHING AND LEARNING IN COMMUNITYCOLLEGE MATHEMATICS, Community College Journal of Research andPractice, 25:5-6, 383-390, DOI: 10.1080/106689201750192238

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    http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions

  • TEACHING IN CONTEXT: ENHANCING THEPROCESSES OF TEACHING AND LEARNING INCOMMUNITY COLLEGE MATHEMATICS

    Edward D. LaughbaumThe Ohio State University, Department of Mathematics, Columbus,Ohio, USA

    Teaching in context can be dened as teaching a mathematical idea or process by

    using a problem, situation, or data to enhance the teaching and learning process.

    The same problem or situation may be used many times, at different mathematical

    levels to teach different objectives. A common misconception exists that

    assigning/teaching applications is teaching in context. W hile both use problems,

    the difference is in timing, in purpose, and in student outcome. In this article,

    one problem situation is explored thoroughly at different levels of understanding

    and other ideas are suggested for classroom explorations.

    Teaching in context can be dened as teaching a mathematical idea orprocess by using a problem, situation, or data to enhance the teachingand learning process. An alternative denition might be as follows:using a problem, situation, or data to motivate a mathematical topic.A common misconception exists that using applications is teachingin context. While both use problems, the difference is in timing, inpurpose, and in student outcome. Applications are assigned aftermathematics is taught and they are used to apply the mathematicsalready learned. When used in this way, problems and situationsdo not enhance the initial learning process nor help in the conceptualdevelopment of the mathematics.

    Applications may be used to answer the student question Whatsthis stuff good for? Applications allow teachers to validate whatwe have taught. Teaching in context also uses problems, situations,or data, but these devices are used to introduce a math topic forthe purpose of helping students understand the mathematics beingpresented or to create a motivating experience for the mathematics

    Address correspondence to Edward D. Laughbaum, The Ohio State University,Department of Mathematics, 231 West 18th Avenue, Columbus, OH 43210, USA. E-mail:elaughba@ math.ohio-state.edu

    383

    Community College Journal of Research and Practice, 25: 383390, 2001Copyright 2001 Taylor & Francis1066-8926/01 $12.00 + .00

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  • that follows. A problem or situation may be used many times tointroduce different mathematical concept or to motivate learning atdifferent levels each time with a different teaching objective. Onesituation may have many inherent teachable features that can be usedto aid in student understanding of mathematics. Using situations,problems, or data that directly relate to the lives of students engagesthem. Once engaged they are better able to make sense of themathematics and to master conceptual understandings.

    The advantages of introducing and motivating mathematics byteaching in context is supported in many of the documents andreports that have guided mathematics education reform for the lastdecade.

    We believe, after examining the ndings of cognitive science, that themost effective way of learning skills is in context, placing learningobjectives within a real environment rather than insisting that studentsrst learn in the abstract what they will be expected to apply. (SCANS,1991, Executive Summary).

    Basic skills, general principles, algorithms, and problem solvingstrategies should be introduced to students in the context of real,understandable problem-solving situations. . . . The problems used shouldbe relevant to the needs and interests of the students in class. Suchproblems provide a context as well as a purpose for learning. . . .(American Mathematical Association of Two-Year Colleges, 1995, p. 4).

    Students learn important mathematics when they use mathematics inrelevant contexts (Mathematical Sciences Education Board, 1993, p. 6).

    For most students, interesting contexts make rigorous learning possible.(Forman & Steen, 2000, p. 140).

    Contextual instruction asks students engage problems rst and thenmathematical formality, . . . (Forman & Steen, 2000 , p. 138).

    Many two-year college students in the United States and elsewhere,who are taking pre-college (developmental) courses are studyingmaterial they covered previously in high school or other collegecourses and did not learn. Or they may have learned it beforeand did not retain it due to a lack of understanding. Manymathematicseducators believe, as evidenced by the quotes above, that studentslearn best when they learn in context. Yet, few textbooks at any levelpresent mathematics in context. A great many college students inthe United States are taking developmental courses (15% in

    384 E. D. Laughbaum

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  • universities and 58% in two-year colleges (Loftsgaarden, Rung, &Watkins, 1997, p. 7). Textbooks for developmental mathematics rarelydiffer signicantly in pedagogy from those used in kindergartenthrough twelfth grade. These books typically both reect and inuencethe way teachers at all levels teach.

    The material that follows provides examples of how teaching incontext can be used in introductory classes, particularly those atthe developmental level. One example is presented with the relatedpedagogical discussion. The authors premise is that if faculty seethe power of teaching in context, it will become a priority in theirprofessional teaching lives. These teachers can then nd problems,situations, and data. Availability of materials can be accomplishedby posting their lessons on their home pages. Furthermore,professional associations might act as a repository of this information.If textbook publishers note sufcient interest in teaching in contextthey will provide the resources for student success through meaningfulmathematics.

    SUMMARY EXAMPLE

    The objective of the following example is to teach the concepts of increasing and rate of change to algebra students. It also servesto help algebra students calculate rates of change and understand thatit is more than the difference of the ys divided by the difference of thexs. This example would help students at another level choose amongdifferent mathematical models. At another level, the example could beused to help students decide if a linear function is a good model of thegiven data. And if so, what would be the symbolic form of the model?They can demonstrate their understanding of the model through theirapproach to the questions provided. For this example we use thefollowing data:

    Year (t) 1960 1970 1980 1988 1993

    Garbage (g) generated in pounds perperson per day

    2.7 3.2 3.6 4.0 4.1

    THE CONCEPT OF CHANGE

    As the independent variable of a function changes in value from low tohigh, the dependent variable may increase or decrease in one of manyways, remain constant, or oscillate. Students at the introductory level

    Teaching in Context 385

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  • may not have previously used a function in symbolic form or functionnotation. For this and other reasons it is inappropriate to begin withthe formal denition of an increasing function. Yet the concept issimple and easily understandable by beginning-level students whenapproached in the context of the above data. A student who cannotlook at a function in symbolic formand determine whether the functionis increasing can answer the question, Is the amount of garbagecreated per person per day increasing as time passes? Beginningalgebra students can look at the numeric information in the contextualdata and know that the amount of garbage generated per person peryear is increasing. A general understanding is only obvious whenplaced in an appropriate context. Students can look at the dataand see that the values in g are increasing as the values in t increase.They can graph the data and make the connection betweenincreasing and the graph rising to the right. They can then makethe connection to the slope (rate of change) of the linear functionin symbolic form.

    CALCULATING THE SLOPE

    Teaching in context can be enhanced by asking leading questionsbefore the context is provided. For example, the instructor can askif we are generating (creating) garbage at a constant rate year afteryear. If so, at what rate? Or, is the rate rising or falling as time passes?At what rate is it rising? If the rate is increasing, is this a problem forsociety? Why? The instructor proceeds to asking specic questions.What is the change (increase) in garbage generation in the 10-yearperiod from 1960 to 1970? What might the yearly average changebe during this period? How is it calculated? The same questions shouldbe asked about each of the periods of 1970 to 1980, 1980 to 1988, and1988 to 1993. Does the fact that these periods are not of equal lengthimpact our study? Students can plot the data and show Dg and Dton the graph and then explain how they arrived at their decision.

    DECIDING ON A MODEL

    Students need to decide if a linear model is appropriate. The rst test isobservation: Does it look linear? (Figure 1). What other method canbe used to determine if the data t a linear function? Consider thecalculation of DL2/DL1 in the third column of the calculator display(Figure 2). What information does it provide about linearity? Is the

    386 E. D. Laughbaum

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  • data set linear in a strict mathematical sense? Would a physicist thinkthe data set is linear? If it isnt linear, is it quadratic? Exponential?Rational? Can the data be modeled by any elementary function?

    Suppose we nd the means of the numbers in L1, L2, and L3respectively. What information is provided by the means? Plot the pairof numbers (mean of L1, mean of L2) along with the data. Whatadditional information does this provide? If we assume the data tobe linear, do we have enough information to nd a model? What isthe signicance of the mean of L3? Can we nd a model using the pointwith the coordinates (mean of L1, mean of L2) and the mean of L3?Students can ll in the following table using the idea of mean rateof change (0.04) and the calculated mean point (1978, 3.52). This willlead to the symbolic form of the model.

    Year 1978 1979 1980 1981 1982 1983 1984 t

    Garbage 3.52

    FURTHER EXPLORATIONS WITH THE MODEL

    A contextual problem like the one provided allows for the explorationof mathematics and its connection to other disciplines. Students can beencouraged to consider the rate of growth of the amount of garbage andits implications for the environment, the economy, politics, and health.Some of the questions following are strictly mathematical; othersmight best be answered after students discuss the issue in othercourses or do research on the Web.

    FIGURE 1 FIGURE 2

    Teaching in Context 387

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  • (1) What is your nal mathematical model for garbage generated perperson per day in the United States?

    (2) Should the domain of your model be restricted to numbers from1960 to 1993? Explain your response.

    (3) Should your model have a zero? Explain your response.(4) Describe a situation under which your model of garbage

    generation will not apply.(5) What are the limitations to your model?(6) What do you think is the biggest reason for the increasing amount

    of garbage we are creating? Explain your thinking.(7) Using your mathematical model, how much garbage per person

    per day was generated in 1945? The Environmental ProtectionAgencys model predicts 4.4 pounds per person per day in 2000.What does your model predict?

    (8) Is your answer above for 1945 realistic? Explain.(9) What rate of change did you use in your nal model, if you used a

    linear function?(10) In terms of garbage creation, what is your concrete interpretation

    of the abstract mathematical concept of increasing?(11) How much garbage is generated at a zero of your mathematical

    model?(12) In your model of this data what would negative values of t and g

    represent, if that is possible?(13) If the rate at which we generate garbage were greater than what it

    is, what function parameter in your model can you change toreect this?

    (14) If there were 2.8 108 people in the U.S. in 1995, how many poundsof garbage were generated in the entire year? How big is this pile?Will it ll the Super Dome? Would it ll Lake Erie?

    (15) If one point on the graph of your model has coordinates (1995, 4.3),explain what these numbers mean.

    (16) Do you think anyone working at a professional job woul...

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