Assignment 3

This is the write up:
https://docs.google.com/document/d/1ks9tgVbTqa8PHAGXDSpHz7uZRsaOmD49zjXKOzYVZP8/edit#

Unit plan:
https://docs.google.com/leaf?id=0B9ro15ubkRLQZTc3ODIzNWMtMDMzOC00ODE4LTgxNzUtMGQxMDM3NTU4NmRl&hl=en

lesson plan 1
https://docs.google.com/leaf?id=0B9ro15ubkRLQODM3Y2QwYzUtOTQ4Zi00MTU0LWIzMzYtYTFjODliNDkyMTdm&hl=en

lesson plan 2
https://docs.google.com/leaf?id=0B9ro15ubkRLQOGIyMTQyYzEtNzIzYS00NjAzLTliNmItYjk1NDU1ODk0OTdh&hl=en

lesson plan 3
https://docs.google.com/leaf?id=0B9ro15ubkRLQMzgyYWU0ZDYtZjkyYS00Y2Q2LWIxMmItY2U2NWZiYjI5YjEz&hl=en

Math Projects Assignment

 “Math Projects” Assignment
Shannon Kennedy, Carly Orr, Marija O’Neill

EDCP 342 November 15, 2010

Part 1 & Part 2

Evaluation of Islamic Tiling Project: Tessellations

            We took a look at Susan’s Islamic Tiling project, and for the most part really enjoyed doing it! It is a beautiful mixture of history and culture with art and mathematics. When discussing this project we talked about many things that it does well, and did not have very many things that we would do to change it!

            One of the greatest strengths of this project is the fact that it forces students to discover symmetries on their own. While doing the project the need to identify reflections, rotations and translations comes about very naturally because it will help the student draw the pattern. An understanding of these concepts is also required in order to find the smallest repeating shape. This discovery based on need makes the concepts discovered far more meaningful for the students.

            After having discovered the symmetries within the pattern, students are then asked to describe the pattern in words. This is an interesting step because students are essentially being asked to describe a mathematical concept in English. Not only that, but in an English that they are comfortable with and makes sense to them. They aren’t being asked to talk about axes of symmetry or angles of rotation. They are simply being asked to describe a pattern, and they can do so however they choose. This is very powerful because it encourages the students to make sense of the mathematical concept of symmetries in theirway, and they can express their understanding however they choose.

            That being said, we feel that there is one extra step here that is missing from this project, and that is that the students are never asked to translate this written expression of the pattern into a mathematical expression. We fear that without this extra step, students may not make a concrete connection between what they are doing and what they have learned in math class. For us as mathematicians it is very natural to think about a pattern in terms of reflections and rotations; however this may not be the case for all of our students. They might describe their pattern as “that same piece again only upside down” or “The head of the one lizard fits in between the head and left arm of another lizard”. Some students may need an extra push in order to convert their thoughts into mathematical ideas. It is also very important for students to be able to express themselves in the mathematical language, since that is how most information is conveyed in mathematics as well as many sciences.

            Other strengths of this project include the use of straight edge and compass. These skills are not often taught in the math classroom anymore, but can be very useful both for practical reasons (in design and construction) and as a mental exercise. Using these simple tools forces students to think about how shapes are formed and the relationships between lines and angles. It challenges them in their visual and tactile thinking abilities. As was mentioned in the assignment, this is the method that was used in ancient times, and so makes the historical aspect of this project very real for the students. They are doing something just as it was done hundreds, if not thousands of years ago. A slight warning should be mentioned here, in that some time will need to be taken in class to teach the students how to create equilateral triangles, draw perpendicular lines and bisect an angle with a straight edge and compass.

            The final part of this project brings in the creative and artistic aspects of this project, in that students are required to create their own tiling. This is fairly straightforward since it only requires a slight change to the previous tiling, however there is a wide variety in the possible patterns students could create. Adding creativity to the math class is always a bonus for a few reasons. It encourages the idea that there is room within this subject to create, discover and explore rather than simply following the rules. This is something that all mathematicians know, but which is rarely understood by high school students. Asking the students to create something also gives them ownership of their work and encourages them to be proud of what they have accomplished.

            This project relates almost directly to the section on symmetry in the grade 9 curriculum, so it could be quite easily implemented at that grade level. That being said, it could still be a very interesting project for older students as well! After all, we enjoyed doing it!

Part 3:  Devise our own Project

The following is the description of our new project.    Instead of asking students to try reproducing and figuring out symmetries on their own, we have asked the students to identify the symmetries and rotations from a given pattern.     We assume that the students have already completed a unit on symmetries (Math 9 IRP) and that this is a review, as well as an extension.  

In addition we have also added a step where the students write a reflection after doing the project.   In Part 1&2, we found that it was sometimes non-trivial to draw a tiling pattern with only straightedge and compass!   Perhaps the students might be surprised, start to wonder how amazing it must have been for artists long time ago (without the aid of computers or other tools) could have constructed such intricate and accurately repeating patterns.     It would be interesting to see their reflections.     We hope that students will gain a deeper appreciation for the rich historical and artistic and mathematical value behind these tessellations, as well as be able to notice interesting patterns that might appear in our modern life.   

MATH 9 PROJECT

MODERN WAY TO CREATE MEDIEVAL ISLAMIC TILING PATTERNS

Purpose

In these assignments, students will explore mathematical concepts behind some ethnic tiling patterns.   Students will examine patterns in terms of rotation and reflection.    Students will identify smallest repeating tile in a pattern, make variations, and reconstruct tiles with a compass and straightedge. 

Description of Activities

Step 1.  Choose one of the given Islamic tiling patterns (see attached handout).

Step 2.    Describe the pattern mathematically by answering the following questions.

  a).   Rotational Symmetry

i)              Find all the points of rotation in the pattern.  

Label these on the pattern.

ii)               For each point, what is the order of rotation, and what is the angle of rotational symmetry?   Label these on the pattern.

  b).   Lines of Reflection (Mirror Lines)

Do you see any lines of reflection?    

Label these on your pattern. 

          

You may wish to sketch or trace a larger simplified version of the pattern on a piece of paper, so you can clearly label on the page. 

Step 3.   Using the discoveries from Step 2, find the smallest (most minimum) repeating pattern, or the most basic tile shape which, if replicated will give the complete wallpaper pattern.

Step 4.  Using only straightedge and compass replicate this minimal repeating shape (by drawing) on a piece of paper.   Document your steps. 

Use the straightedge and compass constructions learned in class:

a)    the equilateral triangle,

b)    the perpendicular line,

c)     bisection of an angle and

d)    circumference of a circle.

Step  5.   Make a small change to your basic minimal tile shape.

Find a way to create your new minimal tile shape using compass and straightedge only.     Document your steps.   Make a pattern for your new tile shape from cardboard or heavy paper and cut it out.

Step  6.   Trace your new tile pattern repeatedly onto a large sheet of paper to find out what new tiling pattern you have now created.

Step 7.   Label the points of rotation, lines of rotational symmetry, and any lines of reflection clearly on the pattern.  

Step 8.   Write 100-150 word reflection on what you have learnt from this project (your learning process – for example, what surprised you? What did you find new or interesting? What part was hard, or easy, or fun for you?).    Give some examples of when you might come across tiling patterns in your daily life. 

Sources

You can see other more ethnic patterns from this website.

http://www.sfu.ca/~mdevos/303/wallpaper/

Some are quite intricate and complicated.   For this assignment, we will only be looking at patterns with rotations and reflections.

What Students Are Required to Produce and Marking Rubrics

Deliverable	Marking	Total
1.    One page display of original pattern showing rotations and reflection information.	Correctly identify all the points of rotation.  1 point Correctly identify all the angles of rotation.  1 point Correctly identify all the orders of rotation   1 point Correctly identify all the lines of symmetry 1 point Clear labeling 1 point	5 pts
2.    Step-by-Step account of how you made the minimum tile shape with compass and straightedge.	Clear, logical thought process.  2 points Correct minimal tile.   2 points Clean sketch of minimum tile.   1 point.	5 pts
3.    Pattern of your new tile,  cut-out on paper or cardboard.	Having a cut-out piece.   2 points.   New tile is slightly modified from old tile.  2 points. Cleanly sketched lines.   1 point.	5 pts
4.   Step-by-Step account of how you made your new file shape with compass and straightedge.	Clear, logical thought process.  2 points Correct minimal tile.   2 points Clean sketch of minimum tile.   1 point.	5 pts
5.   A picture of your new overall tiling pattern, made by repeatedly tracing your new tile pattern.	Correctly identify all the points of rotation.  1 point Correctly identify all the angles of rotation.  1 point Correctly identify all the orders of rotation.   1 point Correctly identify all the lines of symmetry.  1 point Overall pattern appeal.  1 point	5 pts
6.  Reflection write-up	Right number of words (100-150 words)  1 point Thoughtfulness of reflection.  1 point Learnt something new.    2 points Examples of tiling in daily life.   1 point	5 pts
TOTAL		30 points

Handouts

See attached handout with patterns to choose from.

Reflection on Selter article 'Creativity, flexibility, adaptivity, and strategy use in mathematics'

The timing for this article could not be better for me because it is related to my practicum experience. While I was on my short practicum, I was working with a boy in grade 8 who insisted on figuring out a new way to multiply or divide two 2-digit numbers, which was very different from the traditional approach. The fact that he was coming up with a new way to do multiplication/division was great except that some of his supporting methods were not leading him to the right answers and were extremely time consuming. In addition, this activity or this part of math was not the focus of the lesson rather that was the supporting part of the lesson. I was teaching the Pythagorean Theorem and discussion around multiplication was diverting attention from the main focus.
My SA was very frustrated with him as well. In fact, one day after school, he talked to him about perhaps taking Math essentials next. I had a completely different opinion. I thought he should just learn the algorithm and speed up a bit and, if he has time to invent new methods, he can as long as he is up to date with the course material.

So, it is fascinating for me to read this article now as I am also puzzled about this issue of how much freedom to promote and at what point. From the article, we can see that prior knowledge plays a role in the development of flexibility. In the Case of Computational Estimation, the students with greater knowledge demonstrated greater flexibility. In the example of my student above, I should show him a couple of different approaches and provide more information on background knowledge so that he can have more freedom in his strategy selection. Then he will be able to choose a strategy from a selection given or, based on provided supporting methods, he can either invent a new strategy or modify the given ones. The resolution of the matter regarding this student is similar to the article's discussion in the sense of strategy selection.

From the article, strategy uses can be about creativity, flexibility or adaptivity. In addition, there are three different teaching approaches mentioned. One of them is traditional in nature, the routine approach, and the other two are investigative and problem-solving approaches. The last two are more conducive to adaptive strategy use. Their difference lies in the accuracy of the final answers. The problem-solving approach is about inventing new while rejecting the given strategies. As a result, there are more errors in the process which lead to wrong answers 31% of the time.

Problem Solving Question:

A census taker came to a house where a man lived with
three daughters.  "What are your daughters' ages?" he asked.
The man replied, "The product of their ages is 72, and the
sum of their ages is my house number."
"But that's not enough information," the census taker insisted.
"All right," answered the farmer, "the oldest loves chocolate.

What are the daughters' ages?

I like the seemingly humorous element of this problem.  How does the oldest daughter's taste in chocolate help the census taker figure out the age of the farmer's daughters?

The other problem with this question is that it has two possibilities:

8,3,3

6,6,2

They both add to 14. Now, we see the relevance of the 'oldest daughter loving chocolate' - one of the three is the oldest.

First part of the question would include adding up ages to match to the farmer's house number - but the students are not given this number where as the census taker would have that information.

Desert Crossing - Problem Solving

Practicum Stories

The best things about the practicum:

At the beginning of my first lesson, I gave the students apple shape sticky notes to write their names and something about themselves that they would like me to know. Then, I put the notes up on the board to demonstrate the pythagorean theorem of a 3, 4, 5 triangle. So, for every apple note of their personal info, which formed the two squares of the legs of the triangle, I gave them a heart that formed the big square on the hypotenuse. This activity demonstrated the math we were learning and provided me with valuable information about the students. Reading their feedback later at home, was touching and reminded me of the reasons that I wanted to teach; it is the kids and their wonderful take on life. One particular note said: 'I am a good kid and I am looking forward to learning from you. Thank you for your future help.'

The other best thing about the practicum was to see so many different teaching styles. For me, it was a long time ago since I was in high school. Also,  as corny as this may sound 'you get better with practice' - it is true. Actually teaching a few lessons and seeing how much you learn each time is fascinating. It also reduces pressure on us thinking that we have to be awesome right away, this was a good opportunity to see that is not true. So, the microteaching experience showed that our talent can come through once we have had some practice and not expect to have flawless lessons right from the start.

I realized how important this job is and how many lives you touch as a teacher. The work with kids is so delicate and constant reflection is necessary. The connection with kids is hard to explain, amazing.

Reflection on Microteaching

I think I learned more from this lesson than any other so far because it was by far the worst lesson I have ever done. It is amazing how much more learning one can have from mistakes than from doing things right. I thought the feedback from other teacher candidates was very useful and somewhat uplifting as I thought our group lesson was much worse than they did. Here are the things that went wrong:
1. technology - too much of it for a short time like this, did not work (used 10 minutes of class time to get going), lost slides, students did not know how to use graphing calculators (which we did not budget time for) and so on.
2. never once rehearsed, real problem with group flow
3. Esther did the slides for my part so I had no idea what I was doing till I saw the problem just briefly before the lesson and then while presenting. I assumed because the problem is so simple I could just do it. But because I was thinking about how to teach where the information is coming from, whether it was going to appear or not (as we did loose some material), stressing about the timing, and all these things that come with being a new teacher, I did not focus on the problem itself and, therefore, made math related errors, which are the core of the lesson.
I lost focus and was just thinking this is the same as riding a bike I can just do it, but if I tried to eat the ice cream and talk on the phone while riding a bike then it's not the same bike ride... tip for next time: drop the ice cream and the cell phone and ride the bike.
I lesson example, if we can stop worrying about other lesson related things we should have notes (no matter how simple the material is) to refer to in case we loose focus like I did. Also, a good rehearsal would reduce a lot of the other anxieties that I had about timing, technology and general flow.
4. I anticipated a question that Matthew asked but knew would not have the time to explain thoroughly because I was dealing with other group related things before class and did not have the time to think what the best answer would be. Should have said: more specifically how we are going to show the shift of the fn even if we did not have the time to demonstrate rather than just say that we'll talk about it. next class. That kind of answer would give students partial answer and give them something to look forward to in the next class.
5. I knew the intro was the only sure thing before the lesson but the rest I knew would have to be tight for time. Obviously this is a huge lesson: rehearse till we're ready and know will work.
6. Lots of group issues - but mainly timeline of when things were done - it was all last minute. So, when I think of my good lessons I've had I had them "mocked" (in front of friends, family) well before the day of the lesson. That was not the case here.
For next time, if we work with anyone to produce a lesson whether that be an SA or teacher candidate we will have to make sure to work on a well established time line, and allow members to jump in whenever they are ready, but keep going with the program so that we are ready on time.