This is the write-up of Assignment #7
Brian R. Lawler
EMAT 6680

Tangent Circles

The Problem

This investigation begins with the following problem:

Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.

<<click here to read the full text of this directed investigation>>

Prepare a retrospective summary on your experience with this assignment. The summary might take a mathematical bent, stressing the underlying theorems and relationships. It might take a pedagogical bent, stressing the exploration and discovery. It might take a "here is something interesting I found" bent. Or . . . be creative . . .

Discussion (1)

I am finding myself very interesting in considering the learner when working through these explorations. I think my work on the InterMath project at UGA has me thinking along these lines. The project has many problems posted on a website. In it's most basic form, the intent of the website and associated workshops are to further develop teachers understanding of mathematics _and_ technology _and_ how to design student experiences that will enhance student learning.

While I worked through this set of problems posted on Dr. Wilson's website, I noticed many aspects of what I suspect were his intentional development of demonstrations, discussion, and question-posing. Dr. Wilson wrote in a more discussion type of format early. For me, this developed some confidence that I understood the materials and could solve the early problems. There were both visual examples and well-written statements to aid my understanding of the mathematics and help me perform my own constructions.

Another repeated technique was to encourage exploration, recording of observations, as well as some tips and direct instruction. This encouraged me to get started and try things, but also identified goals to complete. In a sense, it generated an environment that I could feel free to explore, but still made it clear what I needed to accomplish.

I was also impressed with the strong, coherent theme that ran throughout the problem investigation. There was the notion of creating tangent circles. Additionally, midway through the activity it became natural to wonder about the construction of a parabola. A bit later, the investigation hinted at it. And then the concluding problem yielded insight into what was necessary to construct the parabola - or maybe better said to identify the loci of the trace of the centers of the tangent circles. {{{ By the way, the solution to the prompt "12. Is the locus ever a parabola? Is it a parabola in some limiting case?" is YES - it is when one of the two original circles degenerates into a line.}}} Dr. Wilson has repeatedly used this technique of establishing a theme and concluding presentations or investigations with a tremendous aha! or connection among topics that creates an strong amazement and interest in understanding the mathematics further. And these connections also serve to help the learner appreciate the beauty of mathematics as well as develop an understanding that mathematics is not some tremendously disjoint set of problems and numbers and symbols.

However... I have some concerns about applying my learning experiences as they were for other learners, children or adult. First and foremost is a lack of feedback inherent in the structure of the investigations and posting of products. I felt like frequently I did not get the feedback I may need to proceed successfully on rather difficult problems. The investigations are rather lengthy and certainly build upon themselves. I believe in giving creating an open learning environment, allowing the learner the plan their use of time, and other beliefs about learning that Dr. Wilson had. In fact, I tend to believe I put to action many of these same beliefs in my high school classrooms. However, it was a good experience for me to understand the potential negatives that go along with a learning environment such as this.

To bring my discussion of these struggles full circle, I feel that the design - scaffolding, prompts, coherent theme - of the investigations created a wonderful environment for exploration and discovery. However, I greatly desired discussion, presentation, and more frequent feedback on my thinking (not necessarily or only from an instructor). I believe I could progress much further in awareness of some relationships that I missed or towards the development of some proof I was unable to create. It is a bit unsettling to be left with that awareness.

Maybe this isn't bad. Maybe this does lay the groundwork for me to pursue further mathematics in a manner that will make it more personally meaningful. Maybe this gives me a better understanding of what knowing and learning are. hmmm.... I am a bit surprised with myself. I think I am holding onto to a "school" definition of learning more tightly than I thought. Something to meditate about.


Although that was my conclusion to my reflection (for now), I will include the table below in which I brainstormed some of the positives and negatives I feel are inherent in the investigation and nature in which I experienced it.

learning can go many directions, learners will not all end at the same place learning can go many directions, learners will not all end at the same place
relies on self-motivated learner relies on self-motivated learner
develops self-directed learning can be frustrating for learner
created opportunities to feel pride in accomplishments I wonder about social comparison issues... did other learners feel intimidated by others work? I have read that the multidimensionality of the this learning environment reduced social comparison. The assumption was that since people explored work of their choosing at a level they selected, that social comparisons would diminish.
I believe that a tremendous amount of discovery occurred....--> But at what level? It feels less spectacular as I feel like I was led by my nose right to a watering hole the instructor knew of, as opposed to me creating something truly new.
Exploration was certainly greatly promoted Was the discovery "correct"? mathematically correct? correct for the learner? Will their be misconclusions or incorrect understandings propagated. Opportunities for the learner to test the new knowledge via practice or application to new problems is lacking... to a degree
the previous negative reminded me of another good aspect... because the problems followed a mathematical theme, there was often built in feedback or opportunity for disequilibrium if a learner came to an erroneous conclusion. Later down the road, this error was likely to create a pertebution that could not be resolved, motivating the learner to go back through problems and revisit. Yet I can't get away from feeling a need for additional feedback, whether from an instructor or via peers - pairwise or group work, presentations, ... (or many other models)
  Proof and the communication of proof (I think) is inherently a 2-way street. It cannot be a complete proof unless the audience is convinced. (But I think this may not be the standard view of proof - something mathematical that exists outside the learner.)
Promoted many avenues and motivation for further exploration and understanding. Will the learner continue???


Discussion (2)

While working on this series of investigations, I not only was able to solve many of the problems, but also continued learning much about the technological tools that we are using in the course. Below are links to webpages that have Quicktime movies of animations of some of the work I have done in Geometer's Sketchpad. A **warning** before clicking to them - the movies are quite large and I imagine would be very cumbersome for a slower connection or processor.

While working on "4. Discuss the loci of the centers of the tangent circles for all case you construct.", I sought to consider the various relationships that could exist between the original circles in order to develop an idea about what all the possible cases could be. Once I felt I had a fairly good idea what the possible cases could be, I developed a single Sketchpad document in which a reader could explore the various situations in a directed manner. Although I did not put the necessary polish to the document, I created an environment in which the user can click "move" to go to various cases and click animate in order to observe the loci of the various cases. Click here to see this Geometer's Sketchpad document. Also, click here to download a script that will construct the two circles tangent to two given circles.

Also while working on number 4., I came across a command to "Film Button" under the menu: Edit/Action Button/. So, I learned how to film an animation and present it here. You will see a movie of the generation of the loci of the centers of the two circles tangent to two given circles.

Question 4
GSP sketch
GSP script
Quicktime movie


I also will share a movie I created to demonstrate my work on "14. Given a line and a circle with center K. Take an arbitrary point P on the circle. Construct two circles tangent to the given circle at P and tangent to the line." Again, my presentation is by no means polished, but it is a nice demonstration of what the Geometer's Sketchpad tool can do to aid presentation of findings. Click here to see this Geometer's Sketchpad document. Also, click here to download a script that will construct the two circles tangent to a given circle and line. And finally, click here to view the Quicktime movie I created to demonstrate the loci of the centers of two circles tangent to a given circle and line.

Question 14
GSP sketch
GSP script
Quicktime movie

Comments? Questions? e-mail me at

Last revised: December 28, 2000

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