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Change in Thinking?: Reflective Practice in Trigonometry

Contents

1. Where is the practice used?

2. Ref lection as Integrative

3. Ref lection as systematic & disciplined

4. Ref lection as Social Pedagogy

5. Ref lection as a process of guiding personal change

6. Prof essional Development

7. Are Peer Mentors involved with this practice?

8. Prof essional Guidance

9. Supporting Materials

10. Evidence

11. Student work/ePortf olio examples

12. Next steps

Trigonometry at SLCC is currently going through our internal process of becoming a General Education coursef or our Quantitative Literacy designation. In order to do this, the course must have a signature assignment thatincludes ref lective writ ing. Kathy Eppler, who teaches and is the co-course coordinator f or Trigonometry,revised and extended the ref lection component that is typical f or Trigonometry students. The signatureassignment in this courses f irst year of being general education was choses to be the mass-spring modelingproject. This project models the movement of a mass attatched to a spring. An example of a mass-springmodel is a car ’s suspension system. As the car travels over an uneven surf ace, the suspension system movesin order to keep the car stable and the ride smooth. When the suspension system is not working properly, themotion f elt by drivers and passengers is an oscillation. This oscillation motion can be modeled by trigonemetricf unctions. Students are asked to conceptually model this phenomenon and then, in order to prove that theyunderstand the concept, they are asked to f ind real world examples of the motion. The complete assignmentdescription is in the “Supporting Materials” section.

Af ter students in Trigonometry complete their mass-spring project, they are encouraged to conduct someresearch on mass-spring systems in everyday lif e, and ref lect on the connections between the project and itsreal-world applications. The purpose of this ref lection is to of f er students a space to “speak” to what theproject and research suggest about math applications inside and outside of the classroom. A valuablecomponent of the prompt is that it does not assume students’ minds will be changed about math uponcompletion of the project. The ref lection allows Trigonometry students to discuss how the project changedtheir minds OR how it the project reinf orced their ideas about math in the real world. Students are asked tof urther their thinking about math by explaining their ideas and why they changed or didn’t. This is a self -investigative and integrative learning prompt.

Upon completion of the project and ref lection, student work is assessed according to the f ollowing rubric.

Grading Rubric f or Assignment and Ref lection

3 points-Project is posted in the eportf olio and the eportf olio is linked to MyPage by the deadline. Theref lective writ ing addresses at least one example of a mass spring system and the other questionsposed above. The ref lection is at least two paragraphs in length.

7 points-Project is posted in the eportf olio and the eportf olio is linked to MyPage by the deadline. Theref lective writ ing addresses 2 or more examples of mass spring systems and the other questions posedabove. The ref lection is well written. The work should be clear, well organized and at least one page inlength.

10 points (maximum- only awarded for truly outstanding work)-Project is posted in the eportf olioand the eportf olio is linked to MyPage by the deadline. The ref lective writ ing addresses 2 or moreexamples of mass spring systems and the other questions posed above. The ref lective writ ing isexemplary. The work of f ers unique observations and draws insightf ul, caref ully qualif ied conclusionsand is at least one page in length.

Where is the practice used?Course (all sections)

Program

General Education departments/programs

All Trigonometry sections which is approximately 700 students per semester wtill be required to complete thissignature assignment to receive their Quantitative Literacy General Education credit. The math departmentrequires all f ull and part t ime f aculty to f ollow a standardized syllabus and f or Trigonometry, this now includesa signature assignment which must be assigned and checked by the student. The singature assignment mustbe worth between three and ten percent of the students f inal grade.

Reflection as IntegrativeStudents’ ePortf olio ref lections are designed to help them…

Make connections within a course

Make connections among academic experiences, co-curricular & lived experiences

Trigonometry’s new status at a General Education course giving as student credit f or Quantitative Literacyrequires that the student relate the material they learn to a real world problem or situtation. In Fall 2011, KathyEppler ’s Trigonometry course was evaluated to detemine the depth of learning and ref lection in a this course. The evaluation demonstrated that good signature assignments with ef f ective prompts can lead to amazingref lective writ ing f rom the students.

The ref lection prompt that is used in Eppler ’s course was modif ied and expanded f rom a more general promptabout how math assignments/projects change student views or reinf orces their views on math and why. Whatwe are asking our students to do, f ind one or more instances of the motion in real-world application requiresthe student to t ie together the math they learned in trigonometry to the real world. The student mustunderstand the math they are learning enough to be able to identif y where it occurs in either a natural realm orin a man-made technology. Students can then take the connections one step f urther to anotherclass/discipline.

The ref lection prompt specif ically asks students to make connections to real world applications f or the projectand other disciplines.

Trigonometry Ref lection PromptThe ref lective writ ing should include a response to the f ollowing: Give some examples of mass-spring systemsimportant in everyday lif e. Describe why it would be important to understand the amplitude, period, andf requency of these systems. Did this project change the way you think about how trigonometry can be appliedto the real world? State what ideas changed and why. If this project did not change the way you think, write howthis project gave f urther evidence to support your existing opinion about applying trigonometry. Be specif ic.

Integrating the ref lection component into the course was a 4-step process.

1. Ref lective prompts were introduced in a department meeting. Kati Lewis spoke with math f aculty to help usidentif y best practices in ref lective prompts.2. Ref lective prompts were integrated into an already existing project required in the course.

3. Good examples of ref lective writ ing were shown to students.4. The rubric f or the assignment highlighting what students needed to include in a ref lective writ ing piece. Thestudents were shown this rubric prior to the completion of the project.

The transf ormation of the pedagodgy happened within the process of making the course a general educationcourse. The coordinators of the course discussed how this ref lection could best be implemented. Theinstructors spent t ime f ocusing in class on what good ref lections, and bad ref lections, looked like.

The current ref lection prompt came by ref lecting on which prompts led students to good ref lections and whichled to less than good ref lections. The prompts continued to change until students tarted writ ing consistent,in-depth writ ing/ref lection. The current prompt gives a level of depth required to acertain understanding of theconcept. The students are showing examples of mass-spring systems.

Reflection as systematic & disciplinedStudents’ ePortf olio ref lection processes embody…

The ref lective cycle

The ref lective cycle happens once per year within courses at Salt Lake Community College. The process startswith the concept introduction, in this case oscillating motion, during class. The students are then tested onthis concept in a mid-term exam. The students are then assigned the signature assignment as highlightedabove, and are required to work more indepth with the concept of oscillating motion within a specif ic case,mass-spring systems. The students are asked to ref lect on this concept by relating it to a real world motion. The students are then tested on osscillating motion on the f inal exam. Instructors then ref lect on how well thestudent understood this concept through a series of data analysis of each of the assignments. The conceptintroduction in class or the project will be modif ied, if necessary to achieve greater student understanding. This process typically takes place during spring semester. We currently have no data at this t ime asTrigonometry is in its f irst year of this process.

Reflection as Social PedagogyStudents use ePortf olio to share/peer review/ discuss/collaborate, connecting around course work,ref lections, plans, goals, stories, etc.

Reflection as a process of guiding personal changeStudents use ePortf olio f or educational and career development, identity f ormation, by …

Considering their evolving personal relationship to learning and education

The ref lection allows Trigonometry students to discuss how the project changed their minds OR how it theproject reinf orced their ideas about math in the real world. Students are asked to f urther their thinking aboutmath by explaining their ideas and why they changed or didn’t. This is a self - investigative and integrativelearning prompt.

Professional DevelopmentFaculty and staf f using this practice engage in the f ollowing ePortf olio-related prof essional development:

P-D f or single department/discipline

The math department requires that all General Education math instructors attend ePortf olio and ref lectiontraining. Faculty are also invited to attend ePortf olio and ref lection workshops. In the Fall 2011, Kati Lewistrained math f aculty on the importance of ref lction in math where she discussed how ref lection always hasbeen used in math and why it ’s important to include.

Kath Eppler also shared her student work with the department as well as the ref lective prompt given and theprocess used to obtain this prompt during a department meeting in February. The presentation and discussionwas well received by f aculty members and many other course coordinators f or other courses are now lookingat the ref lective prompts in thier signature assignments.

Are Peer Mentors involved with this practice?Yes

There was peer mentoring in as much as there is a co-coordinator of the class helped develop the project andthe build the ref lection. Also, peer mentoring took place on a larger scale at the department meeting when therelective prompts were share with the entire math department.

Professional Guidance

The Center f or Teaching and Learning develops and guides prof essional development related to this practice:

No

Supporting Materials“Math1060Mass-SpringProject“ description

SLCC Math department website

SyllabusMath1060Spring2012 syllabus

EvidenceThe f ollowing evidence associated with this and other similar practices has been collected:

Other

Student ref lections provide promising evidence to support both the purpose of the mass-spring project andthe value of ref lection in a math course.

Student work/ePortfolio examplesBelow are 2 student ref lections on the mass-spring project.

Example 1Mass spring project- Math 1060- Fall 2011- Reflective writing For my Trigonometry class that I took in the fall of 2011 we were assigned a project called a Modeling theMotion of a Spring. The intent of this assignment was to help us to understand some of the ways trigonometrycan be used in everyday life. In this particular problem we had to figure out both a sine and cosine function for thegiven factors. As we worked through the various steps of the process we eventually were able to prove the logicbehind how the two functions are related to one another and work together. At first I was really confused by thisprocess, but in the end it all came together and made sense. To be honest I had no idea how this would apply to the real world, so some extra research was necessary tofind some examples. Once I had the general idea it was easy to apply the concept to other applications. Some ofthe ways this concept is used in real world applications are basically anything that uses a spring as part of itsworking mechanism. The largest example I can think of would be the shocks on a vehicle. Shocks are basicallygiant springs that help to absorb some of the impact caused by driving on a surface that is not flat and smooth.How they work exactly was a little more difficult for me to understand, but as I thought about it more it made moresense. At first I assumed the larger a vehicle is then the more stiff the spring needs to be and the smaller thevehicle the looser the spring. The tutor helped to explain the concept and why it works. If you picture a vehicletraveling down a road you can see that every time it hits any kind of a bump or other uneven area it causes thecar to bounce; this bounce can be computed as a sine wave! The purpose of the spring is to help to minimize theamplitude of the sine wave. Now, if it is a large vehicle like a moving truck, then it is going to bounce less becauseof the large mass. Therefore the spring does not need to be as “stiff”, meaning since there is less of an amplitudeto minimize, then the spring can have more movement. Inversely, if you’re driving a small compact car, it is goingto bounce all over the place when hitting a bump because of the smaller mass size. In this case the spring needsto be stiffer so as to control the bounce the car will encounter. Another example of a way this could be used is called a mass-spring-damper system. This is a sidewaysspring system used to slow something down or to “dampen’ the impact of something to reduce damage. In mymind I picture a large semi truck that needs to back in to a loading dock in order to deliver its goods. These trucksare huge and it would be nearly impossible for them to back up as close as they need to be without a hard impactto both the truck and the dock. The spring acts as a cushion to absorb some of the impact and making it safer forthe drive, truck, and building.

As far as taking the period and frequency into account, this is how you could calculate the “stiffness” of thespring. The period would be the length of the spring and frequency would be how many coils there are in thespring; the more coils in relation to the length of the spring, the stiffer the spring will be and the less “bounce” it willhave. This was a cool project in that it helped me to see how some of the concepts I’ve been struggling with allsemester apply to the world around me.This eportf olio can be f ound at: mandjhaley.weebly.com

Example 2Trigonometry: Reflective Essay

Springs are used in today’s society almost always. Although you may not see it, they are placed in some of themost complex products out there. An example would be the springs found in military issued weapons. Springs areimportant because they allow for our military to defend the nation against certain adversaries in the world. Knowing the correct amplitude, frequency, and period is crucial in guns because if not properly set, the weaponcould jam up or possibly misfire. If the tension is too low or two high, bullets would not be fed into the chambereffectively, and that is the last thing you want occurring the midst of a firefight.

Another example of how springs are prevalent in today’s society would be the use of springs on a train. There arecountless trains in the world and they all depend heavily on springs. There are springs located above the wheelsof the train. Because the cargo and the train itself put pressure on these springs and not directly on the wheels,the spring allows for a smoother and faster ride. In freight trains, springs are also located in the cargo connectors.These allow for the momentum of the lead locomotive to pick up and allow for the train to move forward. Withoutthese springs it would be almost impossible for the lead car to pull the long strain of cargo by itself. Having thecorrect amplitude, frequency, and period in train springs are very important. If not set properly the train wouldmove very slowly or not at all.

By taking a closer look at springs, I have come to realize just how much we take them for granted. I’ll bet amajority of people don’t realize how many are located in their household or how often they use them. People usesprings every time they fall asleep on their spring mattress or use a pen to write. It’s crazy to think how a simpledesign could be so handy in today’s society.

This eportf olio can be f ound at: keelejake.weebly.com/quantitative-literacy-ql.html

The ref lections above demonstrate what we wanted students to learn f rom the project–math is a crit icalcomponent of our everyday lives.

Next stepsThis semester the math department will be collecting the data to determine how well the mass-spring projecthelped students understand oscillating motion and a conntection of Trigonometry to the real world. During thesummer 2012, f aculty will be compiling and analyzing data. This data will be submitted to our collegeassessment department f or accreditation purposes. This data will also be used to modif y how this concept istaught, if necessary, and how the project is presented, if necessary.

Additional Information

Yes, this practice will help our institution assess our quantitative literacy/quantitative reasoning studentlearning outcome.

As stated bef ore, this practice has become a piece of a larger init iative in the Math Department,specif ically Trigonometry. The work done with the f ine tuning of this ref lective prompt will be used in theref lective prompts of all signature assignments f or general education courses within the math department. Although they cannot be identical, the spirit of the prompt and real world connection will be carried through.