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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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TEACHING IN CONTEXT: ENHANCING THEPROCESSES OF TEACHING AND LEARNING INCOMM UNITY COLLEGE M ATHEMATICS

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

Teaching in context can be deüned 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 de�ned as teaching a mathematical idea orprocess by using a problem, situation, or data to enhance the teachingand learning process. An alternative de�nition 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 ‘‘What’sthis 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: 383–390, 2001Copyright Ó 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 students�rst 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. Many mathematicseducators 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

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universities and 58% in two-year colleges (Loftsgaarden, Rung, &Watkins, 1997, p. 7). Textbooks for developmental mathematics rarelydiffer signi�cantly in pedagogy from those used in kindergartenthrough twelfth grade. These books typically both re�ect and in�uencethe 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 author’s 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 suf�cient interest in teaching in contextthey will provide the resources for student success through meaningfulmathematics.

SUM MARY 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 y’s divided by the difference of thex’s.’ ’ 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 de�nition 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 form and 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 between‘‘increasing’’ 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 speci�c 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 Dt

on the graph and then explain how they arrived at their decision.

DECIDING ON A M ODEL

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

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data set linear in a strict mathematical sense? Would a physicist thinkthe data set is linear? If it isn’t 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 L3

respectively. 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 signi�cance 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 ProtectionAgency’s 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 tore�ect 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 would need orcould use your mathematical model to help solve a problem? If yes,what kind of professional? How would they use it?

(17) What page(s) of the text did you use as a reference for working onthis project? Did you use any other reference work or resourceperson? If yes, what or who?

(18) Give a detailed description of the thought process you used todevelop your mathematical model.

(19) Using the model from # 1 above, how much garbage does oneperson in the U.S. create in one year? The Japanese creategarbage at half the rate in the U.S. How much garbage wouldone person in Japan create in one year?

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(20) Research the problem with garbage (often referred to a solidwaste). Are there problems �nding space for garbage? Whatcan be learned about military and industrial waste leeching intothe soil?

(21) How could the amount of garbage we create be reduced? Whoshould be responsible for solutions to the problem?

A SECOND OPTION

Additional investigations and questions are provided below foralternate ways to teach average rate of change in context at thebeginning or intermediate algebra level.

(1) A taxi company charges $0.45 per É/Ì mile, explain the meaning of‘‘$0.45 per É/Ì mile.’’

(2) A road sign near Naha City in Okinawa simply reads 5% grade.What is the meaning of this?

(3) What is the meaning of ‘‘60 miles per hour’’?(4) A hospital patient’s IV is set to release 20 drips per minute. What

is the meaning of ‘‘20 drips per minute’’?(5) A speed-reader can read 500 words per minute. What is the

meaning of ‘‘500 words per minute’’?(6) My real estate property is taxed $54 per $1,000 in assessed value.

What is the meaning of ‘‘$54 per $1,000’’?(7) The economy is growing at a rate of 5% per year. Explain ‘‘5% per

year.’’(8) I take three vitamin pills per day. What is the meaning of ‘ ‘three

pills per day’ ’?(9) On a typical commercial airline �ight, the plane ascends about

1,200 feet per minute on its way to a cruising altitude. Explainwhat ‘‘1,200 feet per minute’ ’ means.

(10) The author of a textbook may earn $1.50 for every $10 in originalsales of the book. What does ‘‘$1.50 for every $10’’ mean?

(11) The roof on my house rises 12 feet for every 12 feet of horizontalwidth of the house. Explain ‘‘rises 12 feet for every 12 feet ofhorizontal width.’’

(12) Describe one common feature found in all questions above.(13) List three more examples of questions like those above.

REFERENCES

American Mathematical Association of Two-Year Colleges. (1995). Crossroads in

Mathematics: Standards for Introductory College Mathematics Before Calculus.Memphis, TN: Author.

Teaching in Context 389

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Forman, S. L. & Steen, L. A. (2000). Beyond eighth grade: Functional mathematics for lifeand work. In Learning Mathematics for a New Century: NCTM 2000 Yearbook. Reston,VA: National Council of Teachers of Mathematics.

Loftsgaarden, D.O., Rung, D.C., & Watkins, A.E. (1997). Statistical Abstract of

Undergraduate Programs in the Mathematical Sciences in the United States, Fall

1995 CBMS Survey. Washington, DC: Mathematical Association of America.Mathematical Sciences Education Board of the National Research Council. (1993).

Measuring W hat Counts—A Policy Brief. Washington, DC: National Academy Press.Secretary’s Commission on Achieving Necessary Skills (SCANS). (1991). What work

requires of schools: A SCANS report for America 2000 (Executive summary).Washington, DC: U.S. Department of Labor

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