Creating Art Using Geometric Transformations

We expect a lot from students. In math, there is an emphasis on thinking, problem solving, communicating thoughts using mathematical language, trying, making mistakes, evaluating, trying again and re-evaluating. It is all part of the mathematical process, but, I often find that students get frustrated by the long bouts of concentration required of them.






I've begun to put more of an emphasis on art in my math classes. The art, of course, is directly related to the unit of study.





Recently, I had my class investigate Transformations through art. They had two objectives. The first was to use Rotations (Turns), Reflections (Flips) and Translations (Slides) to design a piece of art. The second was to explain their artwork using mathematical language during a student/teacher conference.







Before beginning, students investigated art from different cultures and found many examples of geometric art where the patterns relied on transformations of congruent shapes.

The examples shown are from Grade 6 students.

There is a great deal of math thinking that goes into well composed artwork. Shapes need to be spaced out evenly and centered. That involves measuring, planning, rough work, making
mistakes and trying again.








Students demonstrated a variety of examples of each translation. For example, some rotations involved centers of rotation within the shape, others had centres of rotation on a shape's vertex,  while  centres of rotation outside of the shape were also used.







This project took my class about three weeks to complete. They did not work on it from start to finish. Instead, it was something that they would return to over and over again.







Sometimes they used the project for down time - colouring with a friend - when they needed a break. Other times, an idea might pop into their head and they would put down what they were doing to incorporate their thinking before they forgot about it.




Spending a long amount of time on the project had many benefits. It allowed students to review ideas in their head after the instructional component of the Transformations Unit was over. As mentioned above, it gave students a chance for "productive down time" and an outlet for their creative spark. It also gave students an opportunity to really investigate what each other were doing. Ideas were talked about informally as students shared their artwork and their thinking when they wanted to.

Art and Math share a long history. The combination gives students opportunities for projects that really do connect to "the real world."

Puzzles in Middle School Math and Language Centres

I have begun using puzzles in my Math and Language centres. I have found that they can be used to demonstrate the need to plan and to assist in developing students' understanding of what planning looks like. I have used puzzles tio integrate cross-curricular ideas into math and language. I have also noticed that they offer some down time where students can socialize while coming together for a common goal.

PUZZLES USED TO PROMOTE PLANNING AND ORGANIZING

In my language classes, students learn to use the "Writing Process." Similarly, in my math classes, they use the "Four Step Problem Solving" method. Both require students to plan and organize. Puzzles teach this as well. While it might seem rudimentary, the BIG IDEA is that, to accomplish something complex, it is best to chunk it up into attainable bits. Furthermore, the chunking requires logical rationale. Most learn that the first thing that needs to be done is to separate puzzle pieces into groups; particularly edge pieces and pieces of the same colour or design. 

Organizing puzzles into edge pieces and pieces with similar colours and designs.

After the planning stage comes the application stage. This is where calculations are attempted, a rough draft is worked on or, in the case of a puzzle, the piles of pieces are connected. All three share a common point; that, despite the planning and organizing, putting it all together is not simple. It takes effort. Pieces that a student might think connect don't. Sometimes this is solved simply, by rotating a piece and trying again (do you see how I used that math word - integration). Sometimes, one must accept that the original plan wasn't quite right and that the piece needs to be re-categorized. After awhile, everyone comes to a point where a systematic approach is necessary - just picking up one piece at a time, trying it in every possible rotation and then putting it aside to try another piece, Genius is one percent inspiration and 99 percent perspiration (Thomas Edison).

After students complete a few puzzles, the lessons about planning, organizing and persistence become a symbolic point of reference that they all share.

While this has been as far as I have connected the planning process to puzzles thus far, I have considered manners of extending the learning. For instance, I have contemplated stopping students as they organize the class puzzle to ask them how many pieces they think make up the perimeter (estimation based on the proportional size of individual pieces) or to explain what percentage of edge pieces they think they have to locate first before they begin putting them together (demonstrating an understanding that being exactly correct isn't necessary in the planning stage).

PUZZLES TO INTEGRATE OTHER SUBJECT AREAS

Puzzles symbolically make a connection for students about the need to plan, organize and show resilience. While doing so, they allow students to think about the images in puzzles. Consider these puzzles about space and the ocean.

While studying the Biodiversity of Living Things students might work on a puzzle about living things. As they do so, they are likely to notice interesting features that would enable them to ask inquiry based questions. Why does a dolphin have a long, pointy nose? Is that thing a plant or an animal?

Likewise, while studying Space, students might also work on a puzzle about space. Questions are inevitable. Did the Space Shuttle really get close to the moon? What are those objects that aren't stars and planets? If the sun is so bright, why is space dark?

I particularly like puzzles that are maps. As students organize a map puzzle, they must think about what they know. 

The lakes all go together in Ontario. Oh wait, this long, big lake is actually in Manitoba. The state of Montana has to go beneath Alberta and Saskatchewan.  I wonder if Montana trades a lot with those provinces?

The more they think, the deeper their learning.

At some point during a unit, students can be asked carefully chosen questions about the puzzle meant to lead into thinking and learning activities.

Why were the images in the puzzle chosen? Is the overall  image realistic or unrealistic? Would you take out an image or add an image to make the puzzle match what we are learning in class more closely? Design your own puzzle to illustrate an important idea that we are learning about.

SOCIALIZING AND COMING TOGETHER TO ACHIEVE A COMMON GOAL

Something that I have become aware of as a teacher and a parent is that students cannot be thinking all day long. They need some downtime to recharge and even to socialize in positive ways. Puzzles offer an opportunity for this while also bringing the class together to achieve a common goal. As I have tried to point out, puzzles promote active learning (planning and organizing) and incidental learning (noticing small details). Students sitting down together to socialize sometimes find themselves learning without really knowing it. They find themselves talking about a subject area in unexpected ways; this coral reef is beautiful, remember how we were studying that global warming is destroying them, and, they help each other out; here, I think this piece might fit. 

Puzzles can be difficult endeavors that take a lot of time. Completing them gives a class a goal and a sense of shared success. As students work together, they begin to see hidden talents in each other. Sometimes, the best puzzle-ers are those who don't always get praise.  Kids who might have a difficult time focusing sometimes find themselves focusing on a puzzle for long periods of time. Kids who might not get noticed for being the most industrious might be the best at finding the hard to find pieces.

Of course, there are some downsides to puzzles in the classroom. They can be expensive. Pieces go missing or get bent and dog eared over time. They require space to complete and they take up that space for extended periods of time. However, I have found that the positives outweigh the negatives. I hope that you do to.

On Line Book Talks in Middle School

Using Social Media in the Language Classroom

Reading for Homework

When I taught younger elementary students, creating an "At Home" reading program was simple. A newsletter was sent home explaining to parents and caregivers how important it is to read with their children and how important it is for children to read every day. The newsletter was accompanied by a reading log for students to fill out and for their parents to sign.

In Middle School, this does not work so well. Students read longer books, so reading out loud with a parent or caregiver is no longer a viable option. Reading Logs may be used to monitor student progress, but many students learn that their is no way for adults to truly know whether they actually read what they said they read or not.

Too many times, I noticed that Reading Logs filled out by students followed a formula. A long chapter book was picked and each entry listed a new chapter. The exact same pen was used to fill out each entry. The entries were scrawled messily in barely legible handwriting, while the log itself was crisp and clean, as if it had never seen the light of day - an odd dichotomy to say the least. Before collecting the Reading Logs, I would watch students madly writing on a form that they were vainly trying to keep hidden from my field of view.

What I felt was that the students who were avid readers were following the spirit of the Reading Log while those students who most needed to read every day had learned to "game" the system.

On-line Book Talks

My solution is for students to read a book and then review it on-line as a Book Talk. Other students read the review and then ask questions. The reviewer responds to the questions and then continues the conversation.

Start With a Short Book Review

The format of the book review is simple. Students state basic information about the book (e.g., title, author, genre, number of pages), they summarize the plot, they describe their favorite part and they give a recommendation. I point out to students that one way to show how smart they are is to write concise reviews. I have students read some examples of book reviews from past years. We discuss how long reviews are difficult to read and I provide some mini-lessons on how to write concisely (e.g., by not repeating a thought, by staying on topic etc.).

Set Up the Proper Environment

Setting up the proper environment for good on-line Book Talks takes an equal amount of effort.

First, I set up "Small Groups" and determine which students will be a part of them. For Book Talks, I place students into Homogeneous groups.

Then, I create a framework for how the "Small Groups" will be used. My class and I co-create agreements to promote healthy on-line interactions. Agreements always include ideas like "No Put Downs," and the use of "Appreciations Statements." Students often include other ideas to ensure that their Book Talks run smoothly. For instance, a group might decide that all posts need to have actual content (a question or an answer to a question) to keep the thread uncluttered and easy to follow.

Some groups like to include posts just to offer Appreciation Statements.
Other groups find that this creates too many replies and makes following a conversation more difficult.
They prefer embedded Appreciation Statements.

Introduce Students into How to Ask a Question

When I introduce Book Talks, I provide examples of Book Talk questions for the students to use. I found this on-line source that I developed my questions from:

http://theliterarylink.com/bloom_questions.html

Later, I have my class investigate a Q-Chart so that they can formulate their own thinking questions.

Here is an example of a short conversation between students.

In the beginning, students use questions that I create for them.
Later, they create their own using a Q-Chart.

The real thinking occurs when students answer
the questions and then continue the conversations.

Getting Involved

As a teacher, I take a small role in the Book Talks. Sometimes I schedule the talks; but I have found that students are quite competent at scheduling their groups independently. I monitor posts to ensure that comments are on topic and positive; however, I have also found that students monitor each other very well. I check that everyone completes a Book Talk and offer feedback. I have found that when students know that I am interested in their ideas, they put more thought into their responses.

Offering feedback lets students know that you're interested in their ideas.

Final Thoughts

I have been running on-line Book Talks for three years. I organize them as homework. Instead of just "reading for 30 minutes" each night, students know that they have to read to prepare for their Book Talk.

I have found that students are sometimes able to hold each other accountable more effectively than teachers can. For the Book Talks to work, everyone must participate and the students will call each other out for not doing so. Unlike oral book talks, I can see who is participating and who is not. Students running a Book Talk also know that if their group is not participating, their Book Talk will not look very good. So they motivate each other.

Many students look forward to presenting their ideas on-line and look forward to their Book Talk much more than if they had to write a more traditional book review. They get enjoyment out of the responses that they receive from their fellow students.

Lastly, you might have noted that the students' responses that I have provided do not always demonstrate good grammar. I am not too concerned about that. If a point is not made clearly, students ask each other for clarification or point out that their question was not responded to properly. In reality, it gives the students a real life understanding of the need for good grammar.

Try on-line Book Talks for yourself and let me know how yours worked out. I always check my blog for responses.

For today's blog, I how I'm using "Edmodo" in my Language Class, but everything I point out can also be accomplished on "Google Classroom."