|
Journal Entries January 1999 |
Friday, January 8
In class we explored the three transforms; namely, translation, rotation, and reflections using GSP. By investigating how each GSP transform worked we were able to explore the transforms without formal definitions. As a result of our explorations we were able to define each transform informally. This has implications for the teaching of transforms in the secondary school setting. By having students work with the transform menu, you can introduce concepts of parallel, perpendicular or use them to introduce transforms. Either way you can discuss transforms in an informal manner without using any formal definitions. This helps to promote discovery learning on the students part.
Monday, January 11
We explored further the what is meant by a rotation both informally and formally and discussed how important order is when marking your angle using GSP.
Wednesday, January 13
We discussed in detail what the definition of a dilation both informally and formally. It was decided through class discussion that to determine a dilation, we need a point, called a center, C, and a positive real number, called the ratio, k. Given a point, X, the dilation, X' is determined as follows: X' lies on ray CX and the ratio CX'/CX = k. Then a discussion ensued about how you could define a dilation using Euclidean construction tool. We determined that similar triangles would have to be used. The discussion also led us to talk about Euclidean construction of a rotation. The last few minutes of class we were introduced to the research project dealing with transformations from a function perspective. We learned that a transformation is a function (takes points to pionts). We have X'=F(X) that is F: P arrow P. P is the plane. We have four types of transformations and want to investigate their products or compositions. Then we reviewed the notion of what exactly is a composition. For the project we will restrict our multiplication table to translation, rotation, and reflection. We will construct a 3x3 composition table. We did one example, reflection * reflection. From our exploration on GSP, we decided that you would get a rotation where the center is the point of intersection of the two mirrors of the reflection and the angle is the angle that the two mirrors make when they are intersected.
Friday, January 15
This class we elaborated further on the project involving the compositions of transformations. The project was explained in more detail using the example reflection(reflection). This was similar to our investigation in the previous class. The difference in today's example was centered around what happens if the mirrors are parallel. Thus, we have an additional case of the composition: reflection(reflection). When the mirrors intersect, the composition is a rotation. However, if the mirrors are parallel, the composition is a translation, if the the marked vector is twice the length of the segment perpendicular to the two given parallel mirrors whose endpoints are on the mirrors. Then we discussed what happens if the product of two of the transformations is not one of the three given transformations, ie. reflection, rotation, and translation. Then it must be a glide reflection which is a special product that is an isometry. In a glide reflection, you need a marked vector for the translation and a mirror for the reflection. The order of the transformation does not matter. You can translate then reflect or vice versa. Overall, this class helped lay the foundation for beginning the project and understanding the dynamics involved in definining the new transformation based on the original data given.
Wednesday, January 20
This class allowed us the opportunity to work individually on the compositions. The key question in doing this project was,
How does the given data for the two transformations determine the product?
Once you know the relationship between the given data and how that is used to produce the composition, then the problem is solved. But as we have found out through our initial investigations, this is the most difficult part of the assignment. It was helpful to work on the project in class to receive guided instruction from you. Until I really began working on the project in depth, the amount of work and involvment in the project needed was not apparent.
Friday, January 22
This class time involved investigating the composition: rotation(translation)=rotation; one of the most difficult parts of the project.
Today's class was extremely helpful in understanding the teaching/learning process of how you are given a challenging problem(s), asked to investigate, explore, and discover, engage in a high degree of frustration and futile attempts with a small degree of success, then are guided through a solution of one of the more challenging parts of the project which then allows you, the learner, to finally understand the answer through the teacher's guidance and their "teacher" lenses. This guided in-class discovery proved to be an important part of the learning process. By having invested a great deal of time, effort, and intellectual ability into the assignment, the in-class exploration proved to be more valuable and mean more. It had a greater weight than if we had not struggled and were simply given the answer in class.
This project also demonstrated the fine line between giving students enough information and guidance to investigate a challenging problem and telling them too much so that they can do the problem without struggling. Some students can deal with futile attempts at solving a problem, not find success, and be fine. Other students, however, want the answer without the struggle or frustration. These students often are difficult to teach, since they want to take a more passive role in their learning. However, as educators, it is still vital to provide them with these experiences even if they do find a great deal of hardship.
Today also served to clear up a geometrical misconception I had in regards to finding the center of a circle using the perpendicular bisector of two chords of the circle. I think I have it right now:
Given two pairs corresponding points of a transformation, C and C" and F and F", we can find the center of the circle(s) that passes through these points by constructing the segments CC" and FF", finding their respective midpoints and the perpendicular bisectors through those midpoints. The center of the circle will be the intersection of the two perpendicular bisectors. I had approached this differently in the project and was incorrect in my approach in finding the new center.
To summarize, the intellectual struggles and challenges involved in this project demonstrated how important it is to provide the students we teach ample opportunities to go beyond their perceived threshhold of knowledge and to encourage new thought, reflection, and learning based on those learning experiences.
Monday, January 25
Today's lesson focused on a discussion revolving isometries. It focused more on the theoretical part of transformations including definitions. The notes are found on the web.
I am clear on the definition of an isometry and the types of transformations that are categorized as isometries. I still need alittle more work on the property that "An isometry is a bijection." I suppose we will discuss and prove this later.
The class discussion centered on the associative and commutative laws was extremely helpful. I was also unsure of these concepts as they related to the products of isometries. The class discussion helped clarify the difference between the two. Pedagogically speaking, this reinforced the idea that these two properties can be confusing. Thus, when we teach secondary mathematics students, it is vital that we encourage a discussion around these two laws and stress their importance. Taking time earlier in the mathematics careers of students would help when they enter more formal mathematics courses in college such as the one we are in.
One of the most interesting ideas that we touched on today was that "Isometries preserves circles." I am still thinking about what this means. I suppose it means that no matter what transformation we apply to a circle, as long as it is an isometry, the circle will simply move in location. But the radius will still be the same. Hence, the circle will be the same size as the original circle.
To summarize, we touched on several important theoretical ideas and definitions pertaining to isometries. We also discussed the assignment 3 that is due Friday 1/29.
Wednesday, January 27
As we ventured into proofs of the properites surrounding isometries, I understood the proof we did in class today about the product operation on the set of isometries is not commutative. The extra credit problem you mentioned in class seems rather muddy to me at this moment. I am hoping it will become clearer as we discuss proof as it relates to isometries.
Friday, January 29
We continued with the topic of proofs particularly the proposition that states, "Every reflection is an isometry." I enjoyed hearing alternative ways to approach this proof. It affirms the notion that when students see different approaches, we as mathematics educators must confirm and validate their ideas. By hearing alternative approaches gave me insight into the proof and made it much easier to understand. The muddied waters became clearer. We must always remember to listen to our students ideas, even if they are different, and acknowledge them. This will create a safe learning environment where students will feel free to discuss mathematics and take risks.
Return to Journal webpage to read more journal entries