At this point, I am finished with my project - a translation of Book III of the elements with links to GeoGebra manipulatives (no proofs at this point, however). Please take a look, and feel free to leave feedback.
You can use the link to the right under Pages, or click here.
If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, click here.
As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:
Get students asking questions.
As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas. On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers. I have increased my wait time, too. On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly. I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum. This can be a step toward helping students develop arguments to defend their thinking. Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.
GeoGebra is an excellent tool and resource.
I knew what GeoGebra was before Math 641, but I had not taken the time to try it out. Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes. I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well. It can be used to illustrate and explore such a wide variety concepts. It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.
There are far more ways to present and explore mathematical ideas than I am currently aware of.
That is an obvious statement. What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources. I had never used Twitter before this class, and I had never done any blogging. I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog. But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course. Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs. Rather than sticking only to publications like Mathematics Teacher, resources like the mathtwitterblogosphere are another great place to find new ideas.
Geometry and mathematics in general are far more broad than most students get a chance to see.
Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore. As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass. Modern Geometry was a refreshing contrast. There were many times during the semester when I thought, if I could spend three days doing this with my students, they would be way more engaged, and a few more of them might even decide they like math. I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical. Maybe I'll have to keep up a good bulletin board for once.
Working on my project for Math 641 has forced me to learn a lot more about GeoGebra. Some of the manipulatives for the propositions in Book III were quite simple, but others had a lot going on. Here are some of the new things that I learned:
How to define a figure made up of two or more other objects
I first ran into this issue when I was trying to make a picture for a circular segment. I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color. After a few Google searches, I found instructions for how to create a list. In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example
This would define the figure bounded by line segment AB and the arc on circle c between B and A. It showed up as a list in the algebra pane, which I could select and give a color to for emphasis. Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.
More varied ways of assigning conditions to show objects
In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures. For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same. I learned how to put measures other than a slider value in the conditions to show object line in object properties. I used the distance between points,
frequently. I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles. If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example
As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added. Most of the time, I had constructed completely separate segments with the different colors or marks. It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!). However, it was still nice to be able to change line styles and thicknesses in addition to color.
I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now. They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.
If you know of easier ways to do the things described above, I'd love to know! Please leave a comment.
The basic idea is this: How to motivate students to help one another improve their understanding.
In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes. Here are some basics about the structure of our homework that I need to mention before I describe the system further:
- We have two-day homework. That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.
- Actual homework (what is done out of the book) is not part of students' calculated grade. On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn. They may use their homework as a reference on daily assessments.
- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points. They have to complete some additional practice problems first. We call the retakes "reworks".
At the start of a unit, my colleague creates groups of 3 to 4 students. He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together. On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson. While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups. So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.
The especially interesting part of this group system is the incentive my colleague attaches to it. He calculates each group's overall average grade at the start of the unit, and again at the end of the unit. If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.
The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty. After all, the students with the lowest grades are the ones that have the most room for improvement. Alternatively, it's hard to bring a 98% up a whole lot. Helping a struggling group member is the most promising way of earning those extra credit points. Teaching is the great way to learn, so there's a benefit to the excelling students also. My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.
I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores. In some form or another, though, this sounds like a great idea. Does anyone else out there have a good method for getting students to invest in each others' learning?
For my Math 641 project, I am working on a "translation" of Book III of Euclid's Elements. As my source for the Elements, I am using Joyce's Elements webpage. It is obviously in English already, so by translate I mean that I am trying to use more modern terminology for the things described in Book III. It has been a challenge though, particularly for some of the longer, multi-faceted propositions. It is difficult to know how to state some of those more clearly, without getting much too wordy (as I'm prone to do).
For visuals to accompany the definitions, and for manipulatives to accompany each proposition, I am using GeoGebra. I am still a rookie when it comes to GeoGebra, so it is probably taking me longer to develop these things than most, but I have been surprised at how time consuming making these manipulatives has been. I've been learning a lot about how to set conditions to show objects, and having to think carefully about what conditions I can set in a diagram to emphasize each proposition. I had originally imagined that I would make an illustrated step-by-step GeoGebra file for the proofs of each proposition as well, but that might be an ongoing task for some time in the future.
To organize and display my work, I am adding an additional page to this blog per Dr. Golden's suggestion. I had considered using Google Sites, but after looking that over it seemed just as well to lay out the definitions and propositions in the blog, with links to the GeoGebra pages for each manipulative.