Technology and Mathematics Standards: An Integrated Approach

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2004

Technology and Mathematics Standards: An Integrated Approach Technology and Mathematics Standards: An Integrated Approach

Chris Merrill Illinois State University

Mark Comerford Illinois State University

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Recommended Citation Recommended Citation Merrill, C., & Comerford, M. (2004). Technology and mathematics standards: An integrated approach. The Technology Teacher, 64(2), 8-12.

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TECHNOLOGY AND MATHEMATICS STANDARDS:AN INTEGRATED APPROACH

Chris Merrill

Mark Comerford

T

IntroductionThe use of standards-based teachingand learning has been gainingsignificant attention in the educationworld. State and national associationsnow base their specific subject areaor discipline solely on standards, i.e.,International Technology EducationAssociation (ITEA), National Council ofTeachers of Mathematics (NCTM),National Science Education Asso-ciation (NSEA). Moreover, at thepublic school level, state boards ofeducation are holding school districtsaccountable for teaching standards-based curricula. Standards-basedinstruction is not an educational fad,but a reality for public schools todayand for the future (Reeves, 2002). inaddition to a standards-basededucation initiative, integration ofdisciplines, especially withintechnology education, has gainedattention throughout the years (Brusic,1991; Childress, 1996; La Porte &Sanders, 1995; Loepp, 1999; Merrill,2002).

Technology andMathematicsIf you were to ask middle or highschool students to definemathematics, they would probablytell you course titles or functions ofmathematics that they have com-pleted; i.e., ratios, proportions,algebra, geometry. However, a moreopen-ended and broader definition ormeaning of mathematics is the studyof patterns. It is with the latterdefinition in mind that the authorscreated a standards-based, integratedtechnology and mathematics lessonusing the design and construction ofstair systems as the hands-on

A more open-ended and broader definition

or meaning of mathematics is the study of

patterns.

activity—the catalyst to bridge theoryand practice through an authentic,meaningful standards-based approach.Stair design offers teachers andstudents alike the opportunity to drawupon integrated learning whileapplying a standards-based approach.It should be noted that, while stairdesign may be a traditional type ofactivity, the authors consciouslylooked at developing this lesson andactivity with a focus on Standards forTechnological Literacy: Content for theStudy of Technology (STL), using abackward design approach.

Standard 3 in STL states that,"Students will develop an under-standing of the relationships amongtechnologies and the connectionsbetween technology and other fieldsof study" (ITEA, 2000, 2002, p. 44).Likewise, Principles and Standards forSchool Mathematics lists twostandards that are appropriate for thislesson: Connections and Measure-ment. One of the subparts of theconnections standard reads that"Students will recognize and applymathematics in contexts outside ofmathematics" (NCTM, 2000).

Curriculum DesignTo develop this lesson and activity,the backward design process, aspresented by Wiggins and McTighe(1998), was utilized for eleventh andtwelfth grade students. The backwarddesign process is a three stageprocess that teachers use to developcurriculum. More specifically, to startthis process, teachers start by asking

themselves: What is worthy andrequiring of understanding? To answerthis question, one must consider local,state, and national standards.Standards are the driving force behindtoday's education and therefore needto be addressed and taken seriously. Ifthe answer from this first question isnot based on the standards, it isprobably not worthy of teaching andlearning (Reeves, 2002; Wiggins &McTighe, 1998).

The big picture of the backwarddesign process is for teachers toteach for enduring understanding; forstudents to see the connections madebetween subject areas—to develop acognitive bank of knowledge that islearned and internalized, notmemorized. The backward designprocess draws upon what thestudents currently know and are ableto do with cognitive and proceduralconstructs. The authors believe that,by using stair design and constructionas the construct, students can makethe connection for enduring under-standing between what is currentlybeing taught in mathematics and thehands-on approach of technologyeducation.

The first question (What is worthy andrequiring of understanding?) istranslated into the first stage of thisprocess—identifying desired results.The desired results, which are whatthe students should know and he ableto do at the conclusion of the lessonor unit, were identified as:

October 2004 • THE TECHNOLOGY TEACHER

1. Use mathematical formulas andfunctions such as slope,Pythagorean Theorem, addition,subtraction, multiplication, anddivision to design a new stairway.

2. Design two different stairwaysand draw them to prescribedscales.

3. Use tools to construct twodifferent stair designs (with treadsand risers) out of cardboard at halfscale, based on a given total riseand total run.

4. Search various architecturalmagazines and the World WideWeb to identify various stairdesigns to assess their purposesfor a given space.

5. Describe the historical influence ofstair designs by writing a one-page paper on the history andapplication of stair designs.

The acceptable evidence (secondstage), based on the mathematics andtechnology education standards thatwere identified in stage one, consistsof the sketches (layout) of the stairdesign, mathematical formula usage,building/stair construction (finishedstringers), pictorial display of work,and the written paper. The acceptableevidence is what the teacher willaccept to show that, "yes, thestudents did understand, learn, andapply the content/constructs," andhere is the evidence or proof. Theacceptable evidence stage should bethought of as a continuum. Thiscontinuum takes into account differentlevels of cognitive, affective, andpsychomotor abilities. In fact, it maytake weeks for the students to exhibitthe "evidence" needed to prove to theteacher that they understand thematerial being presented.

The third stage of this curriculumdesign process is to plan the learningexperiences and instruction. Theauthors created a "tear out" lessonplan iLesson Plan Part 1) that could beutilized for this stage of the curriculumprocess. Lesson Plan Part 2 is asample activity that could also beimplemented for this stage in thedesign process, and is intended for thestudent. Figure 1 is an elevation viewof a simple stair stringer that could behanded out to students. Figures 2 and3 are examples of completed models

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Figure 1. Stair elevation handout for students.

Figure 2. Sample student-completed stair stringer.

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THE TECHNOLOGY TEACHER • October 2004 9

TFigure 3. Sample student-completed stair stringer.

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that define the stair stringers, treaddepth, riser height, and landing (if any)all constructed to a specific scale.

ConclusionIntegrating technology with otherdisciplines does not have to be aforce-fit. The use of mathematicswhen designing stairs is appropriateand necessary. Technology educationteachers basing their curriculum onstandards and benchmarks will readilysee the advantages of using multiple

T disciplines for students to developenduring understanding. By integratinga relatively simple technology edu-cation activity with other disciplines,students will begin to see the"connections or linchpins" thatconnect different fields of learning.

ReferencesBrusic, S. A. (1991). Determining effects on

fifth-grade students' achievement andcuriosity vwhen a technology educationactivity is integrated with a unit in

science. Dissertation AbstractsInternational, 52, 3204A.

Childress, V. W. (1996). Does integratingtechnology, science, and mathematicsimprove technological problem solving?A quasi experiment. Journal ofTechnology Education. 8{^). 16-26.

International Technology EducationAssociation. (2000, 2002). Standardsfor technological literacy: Content forthe study of technology. Reston, VA:Author,

LaPorte, J. E., & Sanders, M. E. (1995).Integrating technology, science, andmathematics education. In G. E. Martin(Ed.) Foundations for technologyeducation (pp. 179-219). Peoria. IL:Glencoe/McGraw Hill.

Loepp, F. (1999). Models of curriculumintegration. The Journal of TechnologyStudies, 25{2). 2]-2^.

Merrill, C. (2002). Integrated learning:Zoetropes in the classroom. TheTechnology Teacher, 6/(5), 7-12.

National Council of Teachers ofMathematics. (2000). Principles andstandards for school mathematics.Reston, VA: Author.

Reeves, D. B. (2002). f\/laking standardswork: How to implement standards-based assessments in the classroom,school, and district. Denver, CO:Advanced Learning Press.

Wiggins, G., & McTighe, J. (1998).Understanding by design. Alexandria,VA: Association for Supervision andCurriculum Development.

Chris Merrill, Ph.D. isan assistant professorin the TechnologyEducation Program atIllinois StateUniversity, Normal, IL.He can he reachedvia e-mail at [email protected].

Mark Comerford,M.S. is an assistantprofessor inConstructionManagement atIllinois StateUniversity, Normal,

IL. He can be reached via e-amil atcomerford@indtech. it. ilstu. edu.

] 0 October 2004 • THE TECHNOLOGY TEACHER

Lesson Plan Part 1.Integrated lesson plan

Title:Integrated Learning through StairConstruction

Subtitle:Stair Design and Construction:Mathematics and Technology inAction

Standards:Technological Literacy -• Students will develop an under-

standing of the relationships amongtechnologies and the connectionsbetween technology and otherfields of study.

Mathematics -• Students will recognize and apply

mathematics in contexts outside ofmathematics.

" Students will understandmeasurable attributes of objectsand the units, systems, andprocesses of measurement.

• Students will apply appropriatetechniques, tools, and formulas todetermine measurements.

Objectives:At the conclusion of this lesson,students should be able to:• Design two different stairways and

draw them to prescribed scales.• Define individual tread depth and

riser height, stair slope, and stairstringer.

• Use tools to construct two differentstair designs (with treads andrisers) out of cardboard at anappropriate scale, based on a giventotal rise and total run.

• Search various architecturemagazines and the World WideWeb to identify various stairdesigns to assess their purposesfor a given space.

• Describe the historical influence ofstair designs by writing a one-pagepaper on the history and applicationof a stair design.

Equipment/Materials List:Pencil, paper, eraser, architect's scale,calculator, poster board, cardboard, x-acto knife, word-processing program,access to the World Wide Web,

T magazines depicting home designand construction technologies, localbuilding code book (not required, buthelpful).

Introduction:Stairs are a common structure athome, school, and in our society.Stairs are of different shapes andstyles, but the end result is always toprovide a means for movement fromone level to another in residential,commercial, or industrial structures.Stair designs are generally drawn byarchitects who follow building codesand requirements that dictate the stairgeometry, including slope of stairway,minimum tread depth, maximumriser height, and required overheadclearances. Stairs are built on site byconstruction workers, prefabricated ina manufacturing facility and deliveredon site, or are hand crafted bycabinetmakers. The design of stairsdoes not happen by chance, but bytechnological and mathematicalproblem solving, formulas, andtheorems.

Activity:In this activity, students will be usingmathematical formulas, theorems, andtechnological tools to construct twodifferent stair designs, using twodifferent rise and run dimensions. Inaddition, students will have theopportunity to study the history ofstairs and the various styles of stairsby creating a display of their work.

Assessment:There are several assessments thatare both formative and summativethat deal with this lesson and can befound in the activity.

Enrichment Activity:After completing the stair design andconstruction activities, turn yourattention to roof design andconstruction. Roofs are mathe-matically figured the same way asstairs, but the construction processis different. Students should beinstructed on roof designs and, in turn,can use previously learned knowledgeand skills.

T Bibliography:Kicklighter, C. E. {]^55). Architecture:

Residential drawmg and design. SouthHolland, IL: Goodheart-Willcox.

International Technology EducationAssociation. (2000, 2002). Star}dardsfor technological literacy: Content forthe study of technology. Reston, VA;Author,

National Council of Teachers ofMathematics. (2000). Principles andstandards for school mathematics.Reston, VA: Author.

Lesson Plan Part 2.Integrated activity

Stair Construction: Mathematicsand Technology in Action

Overview:Stairs are a common structure athome, school, and in our society.Stairs are of different shapes andstyles, but the end result is always toprovide a means for movement fromone level to another in residential,commercial, or industrial structures.Stair designs are generally drawn byarchitects who follow building codesand requirements that dictate tbe stairgeometry, including slope of stairway,minimum tread depth, and maximumriser height, and required overheadclearances. Stairs are built on site byconstruction workers, prefabricated ina manufacturing facility and deliveredon site, or are hand crafted bycabinetmakers. The design of stairsdoes not happen by chance, but bytechnological and mathematicalproblem solving, formulas, andtheorems.

Introduction:In this activity, you will be usingmathematical formulas, theorems, andtechnological tools to construct twodifferent stair designs, using twodifferent rise and run dimensions. Inaddition, you will have the opportunityto study the history of stairs and thevarious styles of stairs by creating adisplay of your work.

Directions Part 1:# / - Plain Stair DesignAfter receiving the handout from yourteacher, you should:1, Calculate the slope and overall

length of the stringer.

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THE TECHNOLOGY TEACHER • October 2004 11

2. Identify how many risers andtreads will be needed. Note thatin a straight-run stair there isalways one more riser thantreads.

3. Identify the size of the risers andtreads.

4. Use a piece of 8.5x11" paper tolay out and sketch the plan andelevation views of a plain stringer,which has a total rise of 3'91^"and a total run of 4' 4^" using ascale of 1"=^!' 0".

5. Use tools to construct your designfrom pieces of cardboard at halfscale.

6. Turn in your sketches,calculations, and completed stairdesign.

# 2 - L-Shaped Stair DesignAfter receiving the handout from yourteacher, you should:1. Calculate the slope and overall

length of each stringer.2. Identify how many risers and

treads will be needed.3. Identify the size of the risers,

treads, and landing.4. Use a piece of 11 x 17" paper to

lay out and sketch the plan andelevation views of an L-sbapedstair design, which has a total riseof 9' 1Y". using a scale of 1" = 1' 0".

5. Use tools to construct your designfrom pieces of cardboard at thesame scale.

T 6. Turn in your sketches,calculations, stair design, and stairmodel.

Materials:• Architect's scale, paper, pencil,

calculator, eraser, cardboard,x-acto knife, glue gun and glue,and safety glasses

Directions Part 2:Using the World Wide Web, yourschool library, or home/buildingmagazines, search for informationrelated to the history of stairs,different stair designs, constructiontechniques, architects and theirinfluences, and technologicaladvances of materials. Using a pieceof poster board, create a pictorialdisplay of your work. Your displayshould also contain the sketches andpictures of the stair designs youcreated during the first part of thisactivity. Be creative!

Directions Part 3:Compose and format a one-page paperon the history of stairs and thedifferent designs tbat are used inresidential, commercial, or industrialstructures.

T Evaluation:Stair Designs #1 and #2 (each)• Mathematical calculations• Sketches• Stair model

Poster Board Display• Number and quality of pictures• Design and layout• Creativeness

Written Paper• Overall

o Readabilityo Compositiono Spellingo Grammar

Follow-up Questions:1. What is the relationship between

slope and the overall run and riseof a stair design?

2. What materials are mostcommonly used to constructresidential stairs? Why?

3. What tools would be needed tobuild a complete set of stairs?

4. Where else in construction-relatedprocesses would slope andPythagorean Theorem be used?Why?

5. List and name three different stairdesigns and describe theirpurposes.

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