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What Does Integration of Science and MathematicsReally Mean?

David M. Davison Kenneth W. MillerMontana State University-Billings

Dixie L. Metheny Montana State University-Billings Montana State University-Billings

In this era of curriculum reconstruction, considerable attention is being focused on curriculumintegration. The integration of science and mathematics continues to be interpreted in different ways. Inthis article, five different meanings of integration of science and mathematics-discipline specific,content specific, process, methodological and thematic-are investigated along with insturctionalimplications of these different approaches to integration.

Why Should Science and Mathematics BeIntegrated?

Plans to change schools fundamentally requirethat we face many harsh realities. First, schools resistchange with a remarkable resiliency. Efforts to re-structure mathematics and science curricula histori-cally seem to have had little effect on conventional usesof the textbook and methods of delivery. Second, allstudents, especially many low-income and minoritystudents, need continuity between schooling and therest of their lives. The inclusion of science in amathematics curriculum, and vice versa, is one way toprovide this continuity. The key thought behind thisprocess is to develop relevancy and applicability of thediscipline to the existing student experiences. Studentsmust see mathematics, as well as science, as relevantcomponents of their world. In other words, mathemat-ics should no longer be seen as a discipline studied andapplied for mathematics sake, but rather, because itwill help make sense out of some part of our world. The“doing” of mathematics and the “doing” of sciencecreates a new way for students to look at the world-a way that develops depth rather than breadth in amathematics curriculum.

What Does Integration Mean?

The expression “integration of science and math-ematics” is used in different ways throughout thescience and mathematics education community. Be-cause integration has been a commitment of the SchoolScience and Mathematics Association, teachers needto understand different ways in which the term integra-tion can be used and how they apply to the teaching ofscience and mathematics (Underhill, 1994). SchoolScience and Mathematics has taken the lead in present-ing teachers with models for integrating mathematicsand science (Berlin, 1991; Berlin&White, 1994). Two

questions seem to emerge from this discussion:

l To what extent can these integration effortsrepresent a bona fide integration of science and math-ematics?

To what extent has the integration of science andmathematics been merely cosmetic?

Answers to questions such as these are critical in thisclimate of significant curriculum reform. We becameinterested in exploring such questions as we sought toredesign preservice teacher education courses to inte-grate science and mathematics.

The Curriculum and Evaluation Standards forSchool Mathematics (National Council of Teachers ofMathematics [NCTM], 1989) are radically changingviews of school mathematics, and Project 2061: Sci-ence for all Americans (AAAS, 1989) and subsequentbenchmarks present blueprints for changing scienceeducation. Few educators would argue about the needfor an interwoven, cross-disciplinary curriculum, butto many, the nature of the integration in many interdis-ciplinary projects is not readily apparent. A mompervasive problem is that integration means differentthings to different educators. The purpose of thisarticle is to describe the method, type, and value ofintegration between and among the two disciplines andto discuss the meaning of such integration.

Many topics in mathematics and science aretouched upon at surface level, but few topics, histori-cally, are covered or developed in much depth. Con-tent coverage, rather than the provision of contextualunderstanding has been the valued mode in mathemat-ics and science teaching. For these reasons, scienceand mathematics can be integrated to make disciplinesrelevant and meaningful to the learner. Mathematics,when integrated with science, provides the opportunityfor students to apply the discipline to real situations,situations that are relevant to the student’s world and

School Science and Mathematics

presented from the student’s own perspective.Integration deals with the extent to which teachers

use examples, data, and information from a variety ofdisciplines and cultures to illustrate the key concepts,principles, generalizations, and theories in their sub-ject area or discipline (Banks, 1993). Understandingdifferent types of integration becomes necessary inorder to begin to understand the integration foundbetween and among science and mathematics. Fivetypes of science and mathematics integration (disci-pline specific, content, process methodological, andthematic) can be used in interdisciplinary curriculumdevelopment (Miller, Davison, & Metheny, 1993).

Discipline Specific IntegrationThis approach to integration involves an activity

that includes two or more different branches of math-ematics or science. For example, discipline specificintegration might include activities involving algebraand geometry in mathematics and activities infusingbiology, chemistry, and physics in science. The projecton Scope, Sequence, and Coordination of SecondarySchool Science (SS&C), initiated by the National Sci-ence Teachers’ Association (NSTA), is a major reformof science at the secondary level. SS&C recommendsthat all students study science every year for six yearsand advocates carefully sequenced, well-coonlinatedinstruction of all the sciences. As opposed to thetraditional “layer cake” curriculum in which science istaught in year-long discrete and compressed disci-plines, the NSTA project provides for spacing-thestudy of each of the sciences spread out over severalyears (NSTA, nd). Also, in mathematics, systemicefforts in a number of states integrate the variousbranches of mathematics.

Examples from the traditional mathematics cur-riculum that integrate various branches of mathematicsinvolvethestudy oftriangles. Consider the Pythagoreanrelation: The result is typically studied in a course ingeometry, and a common proof uses similar triangles,while other readily accessible proofs use algebra. Like-wise, proofs of compound angle formulas in trigonom-etry use analytical geometry.

In the science content areas, an example of adiscipline specific integration activity might involvethe study of an environmental issue. Models andstrategies for teaching controversial and relevant envi-ronmental issues affecting all the sciences target disci-pline specific integration. Effective strategies fordealing with controversial issues within the sciencedisciplines should consist of (a) the selection of anappropriate and relevant problem which is of local,

regional, and national concern, (b) a problem that ismultifaceted and influenced by scientific, environ-mental, and socioeconomic impacts on a decision, (c)the integration of geological, environmental, ecologi-cal, biological, and chemical databases specific to theissue, (d) strategies to structure the problem so that it issolved by the students, and (e) a problem that does nothave one right answer. This type of integration re-quires a problem where students reach an informeddecision based upon data analysis from all the disci-plines and their use of critical thinking and problemsolving skills. An example for discipline specificintegration might consist of one similar to the proposalof a high altitude hard rock mining site located inMontana in proximity to established wilderness areasand Yellowstone National Park. Students are requiredto gather their own data as well as data presented inpreliminary environmental impact statements. Geo-logical, environmental, biological, chemical, and so-cioeconomic databases affecting the problem are col-lected and used as the basis for making an informeddecision regarding the impact of this mine.

Students learn that branches of mathematics aswell as the branches of science are interrelated. Theconnections between sciences or fields of mathematicsare preeminent. However, there are times when thebranches of mathematics or science must be taughtseparately so that the students know the basic concepts,skills, and procedures.

Content Specific IntegrationContent specific integration involves choosing an

existing curriculum objective from mathematics andone from science. An activity is planned which willinvolve instruction in each of these objectives. It iscontent specific because it conforms to the previouslydeveloped curriculum, infusing the objectives fromeach discipline. In this type of integration, the chal-lenge to the teacher is to weave together the existingprograms in science and mathematics with objectivesfrom two separate and distinct curricular For example,suppose that the content objective for mathematics ismeasurement and the science content objective is thestudy of dinosaurs. Then, using masking tape on thegym floor, they create life-size dinosaurs.

Another example is the study of simple machinesin science and proportions in mathematics. Groups ofstudents are given meter sticks, a fulcrum, and variousmetric weights. After balancing the lever, the studentsare asked to determine the relationship between themasses of the weights and their distances from thefulcrum. In mathematics, the students are working

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228 What Does Integration Mean?

with proportions; in science, they are identifying therelationshipbetween work and a simple machine calleda lever. After the students have worked on the relation-ship, the teacher can help the students arrive at theformula for a lever and its proportional relationship.The teacher might choose to give the students theformula m

1 * d 1 = m2 * d

2 and let them check the formulafor their set of weights. Alternatively, the teacher canhelp the students derive the formula from the data theyhave collected.

The students explore the connections betweenmathematics and science and begin to see the relevancyof mathematics in the reality of science and vice versa.Note, not all mathematic and scientific concepts can be integrated. Basic mathematical and scientific conceptsand processes may need to be taught first, and some-times separately. Also, specific integration activitiesmay involve only surface level organization and devel-opment. For example, in a mathematics class, countingparamecia under a microscope generally does not con-stitute valid integration of science.

Process IntegrationAnother approach to integrating curriculum in

mathematics and science is through the use of real-lifeactivities in the classroom. By conducting experi-ments, collecting data, analyzing the data, and report-ing results, students experience the processes of sci-ence and perform the needed mathematics. A compari-son of science processes with the mathematics stan-

dards in Table 1 iilustrates this idea.The students practice formulating their own ques-

tions and finding the answers. These questions havemeaning to the students and are ones they have someinterest in answering. In small groups, they identify aproblem, decide how to collect data, collect it, and theninterpret these data. In doing this, the students count,measure, and compute. In addition, the students com-municate the data orally and graphically. What isimportant is that mathematical operations are per-formed for a purpose: to answer questions that are ofconcern to the students about the problem under inves-tigation and, generally about the real world.

For example, AIMS activities for Grades 5-9,Math + Science: A Solution (1987), contains manyappropriate illustrations. In the M&M’s (TM) activity,What’s in the bag?" (p. 2 1), the science processes ofidentifying and controlling variables, hypothesizing,interpreting, and predicting are integrated with themathematical skills of averaging, graphing, and esti-mating. General standards such as problem solving,communication, and reasoning are observed to repre-sent both disciplines. A more student-centered ap-proach to the M&M’s activity involves having eachgroup of students formulate a hypothesis about theM&M’s in their bag. One group might conjecture thatthere will be more than 53 M&M's in the bag. Anothergroup might claim that there will be more browns thanany other color, and yet another group may believe thatthere will be fewer greens than any other color. The

l Problem Solvingl Communicationl Reasoningl C o n n e c t i o n s

l Estimationl Number sense and numerationl Whole number operationsl Whole number computationl Geometry and spatial sensel Measurementl Statistics and probabilityl Fractions and decimals- Patterns and relationships

studentstesttheirownhypothesisandprepare a chart to demonstrate to therest of the class. At the same time,theycheckthehypotheses oftheothergroup. Each group reports the resultor testing Us own hypothesis and co-ordinates the findings of the othergroups with their hypothesis; a judg-ment about the truth or falsity of thehypothesis is then based on majorityfindings from the class. By beinginvolved in decision making in thismanner, the students are practicingthe processes of science and math-ematics.

Table 1. Process Integration.

l Observing. communicatingl using space relationshipsl Using time relationships. Classifyingl Using numbersl Measuringl Predicting

l Inferringl Controlling variables9 Interpreting datal Testing hypothesisl Defining operationallyl Experimentingl Imagining and creating

Methodological IntegrationThe integration of methodology

is rarely mentioned in current litera-ture. In effect, “good” science meth-odology is integrated in "good" math-ematics teaching. For example, math-

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ematics developed under constructivist theory usingscience discovery and inquiry teaching techniques andbuilding on prior knowledge characterizes anotherform of integration. The Standards of the NCTM statethat students should be able to “apply mathematicalthinking and modeling to solve problems that arise inother disciplines” (NCTM, 1989, p. 84).

But, the reverse might also be true. Should westrive to integrate mathematics with other disciplinesor other disciplines with mathematics? If we choosethe latter, the integration of science with mathematics,for example, it can be accomplished using sciencemethods as the medium of integration Theuseofwhatis commonly known as the learning cycle (Karplus,1967) as a method of teaching science can be directlyinfused with the development of teaching and learningmodels in mathematics. The learning cycle, as ateaching strategy, includes three main components:exploration, conceptual invention, and expansion ofthe idea. Historically, the most common learningmodel used in schools has been stimulus-response (S-R) where the student is rewarded for providing correctresponses. This model assumes that the learner willrespond if the stimulus is adequate. In contrast, thelearning cycle provides the learner the opportunity tobuild upon previous knowledge, leads the learner torespond to those experiences, and develops new expe-riential structures applied to relevant situations (Renner& Marek, 1988). In the learning cycle format, thechildren work through an exploration usingmanipulatives and familiar activities to develop theconcepts before symbols, procedures, and algorithmsare taught. The teacher discusses the material todevelop the conceptual understanding necessary, andeventually asks that the students apply what they havelearned.

The methodological approach to the integration ofscientific methods clearly focuses on “experimentalscience.” At present, a focus on integration of method-ologies means that students will investigate issues inboth science and mathematics using related strategiessuch as inquiry, discovery, and the learning cycle.

Thematic IntegrationThe thematic approach begins with a theme which

then becomes the medium with which all the disci-plines interact. McDonald and Czemiak (1994) de-scribe how the theme “Sharks” can be used to design anintegrated curriculum unit In another example, thetheme could be oil spills: in mathematics you would beworking withvolume, surface area, and cost of cleanup;in science, you would be working with density and

What Does Integration Mean?

environmental aspects of oil spills. However, thethematic unit goes beyond integration of mathematicsand science by including all other disciplines typicallyfound in elementary and middle schools. For example,this unit would include an investigation of the eco-nomic and social implications of oil spills. In thematicintegration, while science and mathematics integrationis important, integration of individual disciplines issubsumed under the integration implied by the inves-tigation of the thematic topic.

Conclusion

What does it really mean to integrate science andmathematics? Whether the integration of science andmathematics occurs within the disciplines or is infusedwith the disciplines, integration will provide for a morereality-based learning experience. As national, state,and local curriculum efforts continue, closer linksbetween science and mathematics will be explored,and thereby lead to more obviously integrated scienceand mathematics curricula.

We have presented five different meanings ofintegrationinthisarticle. Eachoftheseintetpretationsprovides a valid approach to integrating the disci-plines. We believe that the most potent approach tointegration is to focus, not on science and mathematicscontent, but on scientific processes. We have used thisstrategy as a guide in the redesign of a methods coursein science and mathematics. In an atmosphere ofcurriculum restructuring which focuses on connec-tions between the disciplines, the use of scientificprocesses as an emphasis is deemed appropriate Suchintegration between science and mathematics can serveas a model for process integration across the curricu-lum. Theteachereducationcommunityneedsto moreactively explore ways in which programs can be rede-signed to respond to this change.

References

American Association for the Advancement ofScience. (1989). Project 2061: Science for all Ameri-cans. Washington, DC: Author.

Banks, J. (1993). Multicultural education: Devel-opment, dimensions, andchallenges. Phi Delta Kappan75 (1) 22-28.

Berlin, D. F. (1991). A bibliography of integratedscience and mathematics teaching and learning litera-ture. School Science and Mathematics AssociationTopicsfor Teachers Series number 6. Columbus, OH:ERIC Clearinghouse for Science, Mathematics and

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