Creative Constructions

Recently my Geometry class were studying parallel and perpendicular lines. We discussed how to construct parallel lines (using a transversal and constructing congruent corresponding angles). Then on a quiz I showed my students this picture and asked them to construct a line parallel to line AB through point C.

I had one student who turned in this as an answer. 

  

Now I am the kind of teacher who is fine with my students doing math in multiple ways. I will not require my students to do it my way, but it is important to me that the methods that are used will always work. I asked this student to explain what he did and he couldn't. All he would say is "But they're parallel" and I asked "Why?" Finally I said "Unless you can explain to me why it will always work, I won't give you full credit.

But I was really intregued by this problem. So I sat down and sketched it using Geometer Sketchpad and I found that, in fact, it did always work. It wasn't until I saw the shape moving that I realized what he had constructed and why it always works. Here is a picture.

This student constructed a rhombus, which is a parallelogram. So the line he constructed actually was parallel. So we discussed why it worked and I gave him full credit.

Then later, on a test. I gave my students this picture and asked them to construct a line perpendicular to l through point A.

The method we covered is by drawing a large arch through point l and then bisecting the segment. But this is what that same student constructed.

This was much easier for me to see why it works and I gave him full credit. I am still not exactly sure how to describe why this works. I think the best explanation is by using an isosceles trapezoid.

These creative constructions have made me consider coming back to constructions when we cover congruent triangles and quadrilaterals. There are some great connections to be made here.

Test Them Before They Forget

I am still reflecting on my visit to Kanagawa Sohgoh High School last week. As I have reflected I am impressed at how the Japanese education system encourages students to retain and remember what they learn. I feel that in many ways the American education system indirectly encourages students to forget. We have a “Test them before they forget” mentality. First, we have frequent quizzes because we are afraid that they will forget. Second, we review the content before tests because we have assumed that they have already forgotten the material and we need to remind them before they take their test. Third, the curriculum that students are being exposed to is disjointed enough that they really can forget what they learned after the test. Finally, our summer is two and a half months long and students will forget much of what they learn if they don't use it for this long of a time period.

How the Japanese Education System Encourages Retention 

1. Integrated Curriculum: I mentioned in my last post about how integrated the Japanese curriculum is. If you are constantly pulling in previous ideas, you will not only remind students of these ideas, but through more connections the ideas will be deeper and be seen as more important to remember.

2. The teacher that I talked to only gives one test per semester (they have three semesters per year). Therefore the students are expected to remember and retain what they learn for many months instead of a few days.

3. The summer break is a few weeks long, not a few months. I don't know what the exact length is. But when I mentioned to a fellow Japanese Science teacher that we have two and a half months off in the summer his jaw dropped.

My Reflection

From the first point, I am going to find ways to integrate my curriculum. Not only within units, but also within the curriculum and past year's curriculum. This probably means that I will need to pull away from my textbook because it doesn't provide enough of this kind of integration.

This second point is really interesting. I don’t know if I am ready to adopt this assessment practice. I am embarrassed to say that I spend almost as much time reviewing and assessing as I do teaching. If I tested less often then I would be requiring students to retain and remember the material longer and I would also have more time in class to not feel rushed through the curriculum and time to make connections.