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.

How Math is Taught in Japan

Today I did something that I have wanted to do for over six years; I visited a Japanese high school. When I was doing my undergraduate studies at Brigham Young University I helped a Mathematics Education professor with his research. The data set was collected on the island of Shikoku in Japan. From this data set and my research I learned a lot about how mathematics is taught in Japan, but I wanted to be there. About four years ago I was offered a job to teach at Nile C. Kinnick High School, located on the Yokosuka Naval Base in Japan. As soon as I knew that I was returning to Japan, I made it a goal to go to a Japanese high school to see how mathematics is taught. Today that came true. I went with a fellow teacher and 16 of our students to Kanagawa Sohgoh High School in Yokohama.

Mathematics Curriculum
There are many similarities to how math is taught in America and Japan. There were still teachers who were giving the students problems to do and the teacher worked out almost all of the problems on the board while they are explaining how to do it. But the kind of problems presented are very different.

1. Their math classes are Math 1 (10th Grade Math), Math 2, (11th Grade Math), and Math 3 (12th Grade Math). They don't have Geometry, Algebra 2, Pre-Calculus, or Calculus topics. Their curriculum is very integrated. For example in the Math 1 textbook there are four units. The first on adding, subtracting, and multiplying polynomials. The next unit is one equations of lines, translations and reflections of functions. The third unit is on trigonometry, like the law of lines and cosines. And the last unit is on histograms, box plots, and scatter plots. One unit doesn't build on the previous one like American textbook do. This requires students to remember what they learn, because they come back to the ideas repeatedly.

2. Each until is broken down into a number of sections. Each section starts off with a very basic concept, and simple problems. The problems then build on each other and connect to other ideas. For example I attended a Math 1 class on using the Law of Sines and Cosines. The first problem covered in the lesson was using the Law of Cosines to find the cosine of an angle, then converted to the sine of that angle, then plugged that into an area formula. The next problem was finding the area of a quadrilateral inscribed in a circle. The quadrilateral was cut into two triangles and the same procedure was done again, but was also connected to inscribed angles and solving quadratic equations. The third problem that was presented was finding the radius of a circle inscribed in circle. The students were introduced to an area formula that used this radius, and they connected this idea with the idea of using the law of cosines to find the area of the triangle. It was clear to me that the problems built on each other by connecting to a variety of new and review concepts.

In my undergraduate research I heard that American curriculum was wide and shallow, where the Japanese curriculum was narrow and deep. I don't know about narrow, but it definitely was deeper than American curriculum.

3. As you probably noticed the level of complexity in the problems stated previously are much more intense than the problems that American students are exposed to. From the three math classes that I attended today (taught by three different teachers) there were only between 5 and 7 problems that were discussed during the 90 minute class period. The Japanese philosophy is to do fewer problems, but do harder problems that connect to more ideas.

4. The last thing I noticed about the mathematics curriculum is it is very textbook driven. Even more than American schools. The problems that the teachers present come directly from the student's textbooks. The teacher uses the exact same problem as the example problem in the textbook. He then asks the students do a "challenge problem" that is printed in the textbook. This "challenge problem" is a very similar problem (changed numbers), but there are only about two of these problems. The teacher teaches through the textbook. Where the teacher ends one class is where they will start next class. There is no homework for students, but they are expected to read their textbook and to study for tests.


Japanese Teachers
Japanese teachers don't have their own classroom. There is are classroom sized rooms that has desks for 5 or 6 teachers. I think all the teachers in one room are in the same department. The teacher then travels to a room to teach, similar to university professors who travel and teach in different rooms at different periods. Japanese teachers also teach about two of the four 90 minute periods a day.

I asked one of the teachers about "Lesson Study." It is an idea that I learned about in my research. What this teacher knew about lesson study was very different from what I learned about in my research. Once a year (I think, I didn't get a clear answer on how frequently this happens), all of the math teachers attend a lesson taught by one of the math teachers. Then after that lesson they discuss the lesson and the things the teacher did well and the things that this teacher can improve on. I loved this idea! I personally think I would get better feedback on my teaching from other math teachers than from the principal. The other math teachers have studied how mathematics is taught, the principal usually has teacher experience, but probably in a different discipline.

Very connected to the idea of "Lesson Study" were "Open Lessons." During the month of November there was a schedule and every period there was one class that any other teacher or student was welcome to go and observe if they want. I also thought this was an interesting idea.

I also asked one of the teachers about grades. He said that he is busy because he is teaching 5 classes, so students grades are determined by the midterm exam, their summer homework, and their attitude in class; that was it. I got the sense that the grade was largely determined by their grade on their midterm exam. He said that there were teachers who assigned and graded homework and gave tests and quizzes, but he didn't have time for all of this.

The last thing that I will point out about the teachers, is every math class has a chalkboard. The teacher brings with them different colored chalk and they color code their teaching. Variables have different colors and those colors are consistent through the lessons. Their writing on the chalkboard is very neat. My students were amazed at how perfect their chalk circles were and how straight their lines were. In my research I learned that part of their lesson plans were a plan for the chalk board. It was fun to see a lessons where I could tell that this was planned out.


Japanese Students
There are a few things about Japanese students that I noticed. First of all every class was set up the exact same. There were about 5 lines of chairs with 5 or 6 chairs in each row. The students were silent through almost the entire class. They never raised their hand, but were called on by the teacher. This was either done in order from the attendance roster, or at random by the teacher. About once a lesson a student was invited to write their answer up on the board. They showed all of their work and then sat down. The teacher then used their work to explain how to do the problem.

One of the teachers also explained that some students only take 2 or 3 classes. Especially seniors may only have classes a few days a week. This freedom would not happen in America.

I also found that there wasn't much that was done by way of discipline. I noticed on student who had her head down on her desk for almost the entire period. Nothing was said to this student.

“Why do students learn math?” and Why this Question is Important

I am at the point in my teaching career that I am asking bigger questions. A very important one is “Why do students learn math?” The answer to this question will drive how math is taught.

If math classes are to prepare students to get into college and then the major of their choice, then the focus of my teaching should be on helping students to understand and do mathematics correctly. Sample questions from the ACT and SAT should be integrated with the curriculum so students get used to answering these kinds of questions. Encouraging students to take as much math as possible in high school will also prepare them to be successful in college.

If students learn math to practice problem solving and to use math in their lives, then I will use a constructivist hands on approach. As much math as possible would be in a real context and students would frequently grapple with real word problems that involve the mathematics in the standards.

My fear is that the answer is both of these. In which case, sadly, my teaching will continue as a focus-less mastery of topics. I hope that there is a more specific answer. And something that I can adjust my teaching to and help my students achieve something. This will be a question that I will be addressing over several blog posts.

Triangle Congruence Card Game

I had an idea today about a card came to practice Triangle Congruence. There would be a game board would look something like this:

Students would have three cards in their hand. Then they would pick up a fourth card, and then discard one card. The first student to have three cards that make a pair of congruent triangles must show their cards to the group and tell them which congruence postulate or theorem they used.

This idea can probably be improved upon, for example adjustments to the number of cards to make the game a little more difficult. For example in a few trial runs I found that it was not uncommon to finish your hand in 2 or 3 turns.

Download the pdf

Solution to the Pizza Problem

Jared Bukarau, a good friend from college, created this video of this pizza problem. I solved it very quickly, but it got me thinking. So I decided to expand and generalize the problem. Here are some possible solutions:

Discoveries:

·         The number of pieces you create will always be 6 more than the number of intersections.

·         If you are allowed to make 5 straight cuts, you can cut any number of pieces between 6 and 16.

o   6 is the minimum, because there are no intersections.

o   The highest number of intersections will be (5 x 4)/2 = 10 because each of the 5 lines intersect with the 4 other lines, but this would double count the number of intersections, so you would need to divide by 2. So the maximum number of pieces is 10 + 6 = 16 pieces. (Although, some pieces are pretty small).

Generalization:

·         If we generalize this problem to n cuts. You will always create n + 1 pieces more than the number of intersections.

·         You can cut any number of pieces between n + 1 and  pieces.

o   Note: is the maximum number of intersections with n lines.