STRATEGIES FOR IMPLEMENTING TECHNOLOGY

These classroom-tested strategies enable teachers to make the best use

of available technology to enhance students’ mathematical success.

Karen d. campe

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Copyright © 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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Mathematics teachers can use a broad range of technologies—calculators, computers, display systems, and others—as teaching and learning tools. Although actual access is influenced by

budgets and demand, the important thing is to make the best use of the technology available. Whether you have one computer station for demonstration, a classroom set of graphing calculators, or a fully wired classroom, you can take steps to make your technol-ogy implementation most effective and successful.

Following are examples of real situations that good teachers have experienced:

• Youhavetaughtsystemsofequationsusinggraphing calculators. During the test, a student asks for help: “I know I need to find the point of intersection of the graphs, but I don’t remember what buttons to press.”

• Yourclassusedgeometrycomputersoftwareto complete an exploration on similar triangles. The next week the students do not remember the concepts they investigated in the lab.

• Astudentcomesforextrahelp.Youaskhimtorecall the lesson you taught, and he says, “Well, I remember you used the SMART Board,” but he has forgotten the content of the lesson.

In these situations, simply using technology is not enough; well-intentioned activities can fall flat or pose roadblocks to student understanding, and practical issues can interfere with student success and lasting concept retention. The classroom-tested strategies described here allow technology not just to be used but to be used well, enabling you to make the best use of the technological tools available.

BEFORE THE LESSONLearn the Tools Be sure that you are proficient with the specifics of the technology needed to complete the activity. If you do not know how to use the calculator or computer program, precious instructional time will be wasted. However, this does not mean that you must be an expert and know everything about the technology to use it in your classroom. Be willing to say to students, “I don’t know, but I will find out and get back to you.” In fact, when you acknowledge to students that you learned something new yourself (from a confer-ence or a workshop or on your own), they see you in a new light and identify with you as a learner.

Plan for GroupsDo not leave partnerships to chance; rather, take into account both students’ familiarity with the technol-ogy and their mathematical ability. Decide whether the activity is better suited to groups of students with

similar skill levels or whether you want more bal-anced, heterogeneous groups. Some students work well with good friends, but other friendships distract the students from the task. If you want students to work individually, consider assigning a buddy to compare results and help with trouble shooting.

Use Ready-Made ActivitiesTake advantage of resources from calculator and computer product Web sites, textbook ancillaries, professional publications, conference workshops, and colleagues. However, always work through the material yourself and edit as necessary to make it appropriate for your students.

Provide Directions and Demonstrations Once, when my class began an activity in the com-puter lab, a dozen students immediately raised theirhandstoaskaquestion,eventhoughIhadexplained the steps orally. A clear, specific, written handoutavoidssomeofthosequestions.Further,while I assist one student, I can remind others to read the directions. If possible, include keystrokes and screenshots in the handout. Let students know what keystrokes they must remember and have them put those instructions in class notes. Dem-onstrate the tools to the class before the activity begins and highlight likely trouble spots.

Consider Prerequisite SkillsDo not assume that students recall what has been taughtearlier;usewarm-upquestionstobesurethat they are ready for the mathematics concepts in the technology activity. Make connections to prior knowledge and related mathematical ideas to aid comprehension. Review any critical technology pro-cedures to be certain that students are proficient.

Design to Build UnderstandingI incorporate the following elements into investiga-tion activities: • Warm-ups—Warm-ups review necessary defi-

nitions and properties and remind students of material previously taught.

• Procedure—A clear procedure details the requiredstepsandkeystrokes.

• Datacollection—Have students display numeri-cal data, measurements, diagrams, or graphs.

• Conjectures—Have student groups record observa-tions (correct or not) and discuss their conclusions.

• Applicationofthemathematics—How might the concepts appear in homework, assessments, or real contexts?

• Extension—An extension takes the mathematical principles further. Although optional, I find this essential for fast finishers or high-ability learners.

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Be Willing to ExploreCapitalize on the potential of technology as a dis-covery tool in two ways. One type of technology investigation guides students to focus on a particu-lar concept, and another format allows more open-ended exploration (McGraw and Grant 2005).

As an example of the first type of lesson, stu-dents can relate the shape of graphs of polynomial functionstothedegreeoftheequationsandtheirfactored forms (see fig. 1). An example of the sec-ond type of lesson is analyzing connected midpoints intriangles,quadrilaterals,andpolygons(see fig. 2),whichrequiresstudentstousemathemati-cal reasoning and decision making as key compo-nents of their investigation.

Teachers may be surprised by the direction this classinquirytakes—itmayseemthatthelessonhasgone off track—but the discussion can yield rich dividends in terms of students’ understanding and reasoning. Welcome multiple strategies and solu-tions in the technology activity; even though this type of lesson can take more time, you will cover more ground. On the other hand, it is okay to sim-plify the investigation to lead students to the results you desire, especially if time is short. Both types of lesson structures are effective, so match the level of exploration with the lesson objectives.

Scaffold Students’ Learning Youcanloadapremadefileontocalculatorsorpro-vide it on the school network drive, thus making it available to absent students. If a geometry figure is too complicated for weaker students to construct

themselves, create the figure in advance. This step enables access to significant mathematics for learn-ers who sometimes get short-changed when we assume that higher-level concepts are beyond their reach. Or use an interactive tool such as a slider bar for students to manipulate and observe the conse-quencesonagraphorothermathematicalobject.In figure 3, students see how the action of moving the slider changes the values of the parameter and the graph; they are then prompted to consider the result and reason about the underlying mathemat-ics concepts.

Teachersshouldcarefullyplanquestionsequencesthatencouragestudentstoexplain,findpatterns, and make predictions. This action-conse-quence-reflectioncyclecanamplifyunderstandingand achievement for all learners (Dick, Burrill, and Brady Gill 2007).

Use What You HaveDo not feel held back by limited technology; just leverage your resources for the greatest impact. If you have only one of the particular tool, get (or borrow) a projector for demonstrations. Provide students with a thoughtful handout to accompany the demonstration so that they can collect data and make conjectures as if they were using the technology individually. If you have a class set of calculators, you can arrange labs and other learning experiencesthatdonotrequirethatthetechnologybe taken home. If your students own the technol-

Graph the following polynomial functions one at a time in appropriate windows and make sketches.

Y1(x)=x2 + 2x – 8Y2(x)=x2 + 2xY3(x)=x3+3x2–x–3Y4(x)=x3+3x2 + 2xY5(x)=x3+3x2

Y6(x)=x3

Y7(x)=x4 + 2x3–13x2–14x+24Y8(x)=x4–4x3+4x2

Y9(x)=x4 + 2x3–4x2 – 8xY0(x)=x4

1.Whatdoyounoticeaboutthedegreeofthepolynomial, the number of terms, and the shape of the graph?

2. How is the factored form of the function related to the graph?

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Points M and N are midpoints of sides AB and AC, respectively.

1.ExplorethepropertiesofthemidsegmentMN.2. Create the third midpoint and explore the

midsegment triangle. 3.Extendtheexplorationtomidsegmentsinquadrilateralsandpolygons.

4.Decidewhichquestionsyouwillaskaboutthis situation and what methods you will use to investigate. Can you prove your conjectures?Fig. 1 in this lab, students learn to relate graph shapes to

equations.

Fig. 2 Mathematical reasoning and decision making are

key components of this investigation.

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ogythemselves,besuretorequireitforhomeworkand any out-of-class projects, so that the tool is not a novelty used only rarely.

DURING THE LESSONDo Not Press Buttons It is crucial for teachers to let students move the mouse and press the buttons themselves. I some-times have to clasp my hands behind my back to avoid the temptation to handle the tools, but students learn the technology by doing, not just by watching someone else. Discuss alternative keystrokesequencesandthedistinctionsbetweencalculator syntax and mathematical notation. Oth-erwise, students may overlook details of the sym-bolism, an oversight that can mask misconceptions.

Change RolesAllow students to run the display system and rotate that responsibility. This distribution of responsibili-ties creates a classroom environment that is less teacher directed and makes students more account-able and engaged. The pace will be better suited to students’ working speed, and their opportunity to take on teaching roles will promote understanding.

Clarify ExpectationsHold every member of the group responsible for work done in the activity. Avoid the problems that occur when one strong student is permitted (or

persuaded) to do the majority of the group’s work. Instead, collect individual lab reports or assign roles thatrequireeachpersontocompleteadifferentsec-tion of a group project or discussion. Make clear to the entire class what must be turned in and what will be graded. Students tend to discount tasks that arenotgraded(Wilson1994),so,ataminimum,give a participation grade to make sure that stu-dents value the activity. Outline the technological and mathematical objectives that students will be achieving in the lesson.

Adapt How You Give HelpWhenstudentshavequestions,keepinmindthatprovidingassistanceforinquirylessonsisdiffer-ent from providing assistance in teacher-directed instruction. Try to guide students to discover the concept and make connections on their own or within their group. Their learning will be more of an accomplishment, and they will retain it better than when they just copy down what you tell them. For example, using geometry software, students recorded measures of interior and exterior angles in a triangle but got stuck while searching for addi-tional conjectures (see fig. 4). Rather than telling students which angle pairs were noteworthy, I suggested, “Try adding up pairs of angle measures from your chart.” Pretend that you do not know the answer,givehintssparingly,andaskquestionsthatsupport the students’ reasoning and sense making.

1.Dragtheslidersform and b and observe the results on the graph.

2. Make a prediction about what will happen given the following: •mis4andbis–1 •mis–1andbis4 •m is negative •m is zero •b is negative •bothm and b are zero

3.Explainhowchangingthevaluesofm and b affects the graph of the line y=mx + b.

Fig. 3 students use a slider to learn how the parameters

m and b affect the line.

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1.Constructatriangleandmeasureitsinteriorangles and one exterior angle.

m∠BCD m∠ABC m∠BAC m∠BCA

2. Collect data for the chart by dragging vertex B. 3.Whatdoyouobserveabouttherelationships

of the angle measures in the chart?

Fig. 4 students might need a nudge to determine the

sought-after relationship here.

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Have Informal or Formal CheckpointsManage the activity by walking around so that you can gauge students’ progress and adjust your instruction. Pause the technology activity and discuss an intermediate result with the class to be sure that central mathematical ideas are clear. In the handout, prompt students to “check your work with the teacher before moving on.” If you observe that students are confused about how to use the tool, redirect the class with clear technology instructions. Help students develop self-assessment techniquestomonitortheirwork.

Exploit Technology’s PowerGraphing, geometry, statistics, and display tech-nologies offer speed, accuracy, and robust oppor-tunities for visualization and interactive learning experiences. Tap these capabilities by graphing functions and examining multiple representations, gathering and analyzing large data sets, and using dynamic and animated geometry figures; in each of these instances, go beyond what students can do by hand. Illuminate important concepts and proper-ties that students usually memorize by connecting numeric, algebraic, geometric, and coordinate envi-ronments. For example, figure 5 shows the typical unit circle as an interactive geometric construction linked to numerical trigonometry values, algebraic identities, and graphs of trigonometric functions, thus allowing students to develop a deeper under-standing of the fundamental concepts and rich con-nections involved.

Complement and extend the study of inverse functions and conic sections by using parametric equationsasagraphingtool.Drawonthefullcapa-bilities of an interactive whiteboard in class; then save and distribute notes and electronic lessons for absent students and test review. A further benefit of using technology is that it affords opportunities fortheteachertoaskdifferenttypesofquestionsasthe lesson unfolds, such as these:

• “Whatwillhappenif…?”• “Howis…affectedby…?”• “WhatmustIchangetomake…happen?”

Thistypeofadvancedquestioningstimulatesstudents’ higher-order thinking.

Be Ready for TroubleHave a spare calculator with fresh batteries or an extra computer already booted up and logged in. Know how to save to a flash drive or network drive—and save often. If you are using preloaded files, have students “save as” with a new file name so that the original file is still accessible. Just in case, have a no-technology back-up plan for the les-

P is a point on the unit circle centered at the ori-gin. PX and TU are ⊥ to the x-axis. The rotation angle in degrees of point P is q.1.RotateP around the circle with the

con-

trols and examine the values of the coordi-nates and expressions.

•Whatchangesandwhatstaysthesame? •Whatpatternsdoyouobserve?2. Determine the relationship between POX

and TOU. Prove your conjecture.3.Whatelsecanthisgeometricconstructionshow?

Values from the geometric construction have been captured into the spreadsheet above.

4.Whathappenstopxlistandpylistvaluesasangsize increases?

5.Canyoufindavalueinpxlistthatequalsavalue in pylist? Formulate a general rule for when this occurs and connect it to the geo-metric construction.

The graph above will plot the rotation angle of point P and the y-coordinate of P.

6.AnimatepointP around the unit circle and interpret the resulting graph.

7. What other graphs based on the geometric construction are possible?

Fig. 5 An interactive geometry construction introduces

students to circular functions.

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KAREN D. CAMPE, karen.campe@yale.edu, who previously taught high school mathematics, teaches math-ematics methods at Yale University

in New Haven, Connecticut. A Teachers Teaching with Technology national instructor, she is inter-ested in using technology to enhance student understanding. Wilfred herNANdez

son. Finally, know how and where to get support in your building or from online technology sources.

AFTER THE LESSONSummarize the Learning After the exploration, either the teacher or the students should review the activity so that the sig-nificant ideas are made salient for the class. This is an excellent opportunity for a mathematical con-versation about which observations were fruitful and which conjectures turned out to be erroneous, and discussion can lead to generalization, justi-fication, and proof when appropriate. Insist that students record key concepts in their notes and not just submit the lab report. These teacher actions indicate to students that the technology investiga-tion comprises important knowledge for the class; otherwise, teachers run the risk that students will think that the investigation was just “that fun thing we did on the computers.”

Assess Student UnderstandingCollect lab reports that contain any of the follow-ing: written explanations, sketches, printouts, con-ceptquestions,applicationquestions,open-endedquestions,or“make-your-own”questions.Havestudents present results to classmates or submit projects. Grade students on what they produce. To differentiate for diverse learners, consider grading on their process and progress as well.

Onquizzesandtests,includeitemsrequiringcalculatorsandquestionsthatcheckwhetherstu-dents comprehend and can apply the mathemati-cal concepts taught with the tools. For instance, after examining transformations on functions with graphing technology, students can be asked to pre-dictwithoutthecalculatorhowchangingtheequa-tion will affect the graph.

Finally, use noncalculator assessment items so that students can demonstrate the fundamental skills that you want them to be able to perform without the calculator. For example, they should be able to select an appropriate window for a data set without relying on the automatic ZoomFit feature.

Reflect on Your LessonWhen the technology activity is complete, take notes for the next time about what did and did not work. Make revisions to the task and to the handout to better meet your goals and match your students’ needs. Ask yourself if using the technol-ogy tool actually fostered the mathematical under-standing that you were teaching; if the technology does not promote learning, do not use it for its own sake. Find a colleague with similar technol-ogy goals with whom to share experiences and lesson plans.

CONCLUSIONNothing revitalized my teaching more than my excitement about using graphing calculators, geom-etry software, and interactive display systems in my classroom.Iquicklyrealized,however,thatresolv-ing to use these powerful tools was not enough to ensure that I harnessed their full potential. These visual and dynamic technologies can dramatically affect the mathematics learning environment only if we as teachers make careful decisions about their implementation.

Keep in mind two primary goals: The technol-ogy must support the mathematics concepts, and it must enhance students’ learning of those concepts. I have observed that technology, when used effec-tively, can strengthen conceptual understanding and aid long-term retention. Strive to have your students formulate conjectures, make connections, and substantiate conclusions for themselves, so that they take ownership of the knowledge and spend less time as passive listeners. When you use tech-nology in your classroom—and use it well—you open the door for learning and make mathematics more meaningful and motivating for your students.

REFERENCESDick, Tom, Gail Burrill, and Lisa Brady Gill. “New

TechnologiesOfferNewWaystoEngageStu-dents.” NCSMNewsletter38(Fall2007):19–21.

McGraw, Rebecca, and Maureen Grant. “Investigating Mathematics with Technology: Lesson Structures ThatEncourageaRangeofMethodsandSolu-tions.” In Technology-SupportedMathematicsLearn-ingEnvironments,2005YearbookoftheNationalCouncil of Teachers of Mathematics (NCTM), editedbyWilliamJ.Masalski,pp.303–18.Reston,VA: NCTM, 2005.

Wilson, Linda. “What Gets Graded Is What Gets Val-ued.” MathematicsTeacher87,no.6(September1994):412–14.