Team Teaching Reflection - "What's My Shape?"

Sara, Courtney, and I worked together for our team teaching assignment, and I think we'd all agree that the most difficult part of the process was actually finding a great problem to introduce to the class. We spent days looking through children's text books, teacher's guides and online websites to no avail. We we becoming quite frustrated until I remembered some of the great activities we came across in our text book throughout the semester. I don't know why we didn't think of it before, but there was no need to search random websites trying to figure out if they had any credibility when we had an excellent resource in our hands the whole time. As I flipped through the pages of our text book, I remembered that Mary had said in class that she wanted us to present a wide range of mathematical problems to the class so that we would gain a knowledge of different mathematical concepts. I decided to look through chapters we didn't study during the semester because I wanted our classmates to become familiar with topics we haven't extensively covered in class. As I flipped through the geometry chapter, I noticed that nobody had done a problem with shapes yet, so I found one that looked both fun and interesting.

After discussing the problem that I found with my group members, our initial worry was if it was "open ended enough". We worried it wasn't a suitable question because it didn't follow the same sort of pattern many of the other problems presented in class followed. Everyone thus far had been presenting problems with no straight forward or "right" answer so we struggled with the concept that ours would have one correct answer. In the end, after much discussion, we came to an agreement that mathematics will have correct answers sometimes. The main point of this course wasn't to teach us that math shouldn't have one correct answer, it was to show the importance of understanding how we get to that one answer, understanding the different strategies we can use to get there, and about being able to explain and justify the methods we use.

We knew that students would have fun playing "What's My Shape?" because we did too! While preparing our presentation, we played a few trials of the game to get an idea of how long it would last, and we loved it. Too often math becomes boring - it doesn't need to be! We learned that if we find an activity fun/boring, chances are students will too. Why supply activities that we ourselves would hate to complete? It just doesn't make sense. We also liked the activity because it can be modified for any age/grade level or ability level. Younger/less able students could play the game using fewer shapes with more distinguished characteristics, while more advanced children could use a larger number of more similar shapes for a challenge. The game could also be adapted so as to include 3D shapes instead of 2D. As children advance through the geometry unit and gain more skills with regards to the language they develop and the ideas they learn, rules of the game could easily be modified to enhance the challenge.

For some reason, I wasn't as nervous for this experience as I was for previous team teaching episodes in other courses. I think that my love for mathematics and my comfort level with our topic/activity is what made me feel confident. I know that my classmates enjoyed our activity as we received many positive comments afterwards. Overall, it went well, and I know that it is probably because we provided a student centered, cooperative learning experience with fun and interesting mathematics intertwined.

That said, I know there will always be room for improvement and a huge part of being an effective teacher is having the ability to look back at our performance and reflect on ways that we can better our methods and approaches. If I were to do this over again, I would allow "students" to have more time to complete the activity - OR - go deeper into the class discussion at the end. Our lesson only lasted somewhere around 13 minutes, and I know that was due to nervousness. I know with practice we will eventually get better at managing time and conducting class discussions, but I also understand that time is valuable and not a minute should be wasted! As a second change, I would include in our class discussion a question about the language students used when distinguishing shapes. We didn't quite know at what grade level or area in the unit that specific language should be brought up. However, after speaking with Mary, we realize it is important to speak about the language children use in mathematics - especially in a geometry unit. (ex: sides, angles, round, (a)symmetrical, etc.) Again, with time and experience, I know we'll improve. Lastly, looking back, I realize that we should have written down the methods that students used to get their answers so that they could be discussed further, the methods could be understood by everyone, and so that students could be reminded in the future of the questioning strategies that proved effective with this activity.

Overall, it was a great experience. Team teaching allows us to experience what it feels like to be both a teacher and student and really allows us to reflect on these roles. It is one of the most valuable ways to learn while sitting in a university classroom and I have throughly enjoyed taking on both roles. Way to go class - yet another accomplishment!

Math Fair Reflection

A brief synopsis:

The problem Sara M & I presented was called "The Sword of Knowledge" whereby a dragon with 3 heads & 3 tails had to be slayed by cutting off all its heads & tails. Using the sword of knowledge, with 1 swipe, you could either cut off 1 head, 2 heads, 1 tail, or 2 tails. Sounds simple enough. However, there was a catch. When cutting off certain heads & tails, as a consequence, some usually grew back, depending on what chops you took (they were listed in a table). The goal was to find a way to slay the dragon by following the given rules.

The mathematics:

Hmm, I actually had to think a little bit about this one. I never knew that finding the mathematics in a math problem could be tricky! Here's what I have come up with: the mathematics in the problem had to do with number concepts and operations. Solvers had to choose whether they wanted to chop either 1 or 2 heads or 1 or 2 tails at a time. Then they had to add/subtract the number of heads/tails that were removed/grew back in its place (if any). The mathematics also required students to use logic to come up with a strategy/the knowledge to beat the dragon. For example, the only time nothing grew back after a swipe of the sword was when 2 heads were cut off. Therefore, using logic, problem solvers would realize that you need to find a way to cut off all tails and be left with an even number of heads at the end so that 2 at a time can be chopped off without anything growing back! The mathematics also involved showcasing the necessary skills to draw mathematical information from the supplied charts/tables and often required an ability to use trial and error to spot patterns and find ways to overcome problems as they faced them.

How was it received?

When it was my turn to stay by our board and present the problem, I found that classmates were very determined to find the solution to our problem. Everyone seemed to love it, and nobody walked away undefeated. Although some people found the solution right away, there were many that were finding it tricky. However, nobody gave up. They all had the urge to keep trying and it felt great to know that it was interesting enough to keep people's attention and determination even when it got tough. I have heard a couple classmates say that it was one of their favorite problems to solve, so that makes me happy! :)

If given the chance, would I do anything differently?

Realistically, probably not! I loved the problem, the way we presented it, I loved the manipulatives we made, and how everyone really seemed to genuinely enjoy it. I am happy we made the manipulatives - the heads, tails, and sword - because classmates really liked doing the chopping motions and actually seeing what heads and tails were left after each chop. I think the visual element kept interest and made the task more enjoyable and solvable for everyone - versus just writing in charts on lined paper.  

As facilitator of the problem, I:

found out just how difficult it was to NOT give hints & suggestions to struggling problem solvers. Knowing the solution made it difficult to just stand there and let classmates figure out their mistakes themselves. It took some conscience effort to stand by and not say something when the problem was so close to being solved and watching them make just one last mistake. It was especially difficult to not give in when classmates straight up asked for help/solutions. It made me aware that it will be difficult as an educator to not give into pleading eyes when future students are struggling and asking for help - but I also understand the importance that I allow them to grow and become independent, active problem solvers on their own.

When solving problems, I learned:

that as a mathematical thinker, I am very persistent. I never want to give up when a problem interests me. I never like to feel "defeated" and will try and try again until I get it right. It brought back memories of completing mathematics homework while in grade school. When I would find something difficult I would not, no matter if it took me hours of just staring at the paper, give up. I would get so frustrated and I can still hear my mom saying "that's enough Amy, just take a break and start over fresh. Sometimes you just need a change of scenery and it'll come to you". I would never listen though, and somehow, in the end, I always figured it out. I never felt that determination in a long time, and all of a sudden, when completing the problems at the math fair, there it was again.. that deep desire to solve each problem that I attempted.

What I enjoyed the most:

I enjoyed the atmosphere of the math fair the most. I loved how interactive it was. The problems were presented in such appealing ways that they invited you in. The drive to solve the presented problems was much higher than if the problems were just written on paper. The interactive environment (working with others, working with manipulatives) made for such a peaceful and fun mathematics experience - without all the unnecessary pressure that normally accompanies math classrooms.

My challenge:

as mentioned earlier, was definitely keeping quiet as facilitator of the problem. I wanted so badly to chime in and offer hints as soon as classmates came to a block - however, I think I did a pretty good job at resisting that urge. I tried my best to let them work it out, and it was fun to see the different ways that people's minds work. Everyone had a different strategy and approached the problem differently, it was neat to see.

Would I consider doing a math fair with future students?

Of course! As I already mentioned, the interactive atmosphere of the math fair is so inviting. I loved it! I think children would find a math fair much more exciting than writing out and solving problems on worksheets. I think with such a fun environment, children would learn to let go of their mathematical insecurities and just have fun with it. I would definitely recommend having one to any teacher of mathematics!

My Shifting View of Mathematics

As our course progresses and I continue learning more about mathematics and how it should look in the classroom, I find myself contemplating concepts/ideas that I had never really given much thought to before. Before reading chapter two in the text and attending class on Thursday, I would have been one of the first people to say math is a "black and white/right or wrong" kind of subject. I guess that opinion stemmed from my experiences with mathematics growing up. There was only one right answer and one way to get to that answer. Any "workings" that didn't support the teacher's method and any answer that didn't match the one in the back of the book was automatically wrong. Honestly it never really bothered me though, because I was one of the lucky ones who easily got the right answer, showed the "right" workings and never really asked questions. However, after reading the text, math is not about regurgitating facts without even thinking about the process or why we use it. As our text states:

"Doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions..." (pg. 10, 3rd ed)

Because of my past experiences, I am still programmed to worry about the right answers of problems instead of on the strategies and processes created to solve them. When completing the problems in chapter two, it was actually really bothering me that the answers were not in the book. I especially wanted to know the answer to the "Two Machines, One Job" problem on page 13. It really bothered me that the two different answers given could be backed up so well... how could they both be right? Or could they? I guess this is where it finally clicked for me that the creating of strategies and being able to "back up" the process is equally (or more) important than the answer itself. We all have to use mathematics everyday in our lives to solve problems. When we try to actively solve our day to day problems there are no easy methods/answers - there are only strategies and approaches that we have to come up with and test on our own. Students should be taught to solve problems in the classroom like they will have to do it in the real world. Therefore, "right" answers shouldn't be everything, nor should one method be considered the ONLY method to get there.

As a final thought, I believe that mathematics can indeed be considered a humanity. Math is always changing. New methods and theories are constantly replacing old ideas and it isn't as black and white as some might believe. With new teaching methods requiring students to think about, analyze and test possible solutions to problems, it suddenly becomes less of a repititive 2+2=4 "just because" but rather a way to solve everyday problems in more than one "right" way.

Video Response

Hello everyone!

This entry is in response to Sir Ken Robinson's video about the role of schools in nurturing creativity within students (watched in class). I'll start out by stating that I agree with Robinson's view that the education system today extinguishes the creativity that children naturally possess. From my own experiences growing up and from what I've seen/learned so far in the education faculty, I feel as if young students start out thinking creatively and channeling it through everything that they do. But as they age and progress through elementary and jr/sr high one can very quickly see most of that creativity vanish. I think that imigination can be used powerfully but for the most part schools don't value imigination, they value facts and correct answers. 

When I was young, I didn't show much creativity in my work for fear of being wrong. I felt as if being 'wrong' was the worst thing that could happen when completing schoolwork. As a result, I never took chances, I always completed my work in the most straightforward way. I would assume that many students feel this way. One thing I feel that we should do as future educators to promote creativity is to encourage risk-taking in our classrooms. Students shouldn't have fear of making mistakes. Mistakes are proof that an effort was made. I also believe that a lot of creativity can show through the arts, but many times the importance of arts is neglected in school. I believe that creativity would have a larger chance of persisting with children if more emphasis was given to music, drama, dance, etc. 

Although I have little experience in the subject of mathematics, to promote creativity  in my math classroom, I would try to:

-Allow students to use different materials to solve problems/demonstrate their understanding of concepts.

-Allow input. Students should make some of their own choices in the classroom. Don't constantly give a lot of restrictions with assigned work.

-Be open-minded. (Don't assume there is only one way to do something correctly.)

-Encourage risk-taking - mistakes are ok to make!




"Every child is an artist. The problem is staying an artist when you grow up."
                                                                                 -Pablo Picasso
                      

Mathematics Autobiography

When I reminisce on my mathematics experiences from primary/elementary, I realize I actually don't remember all that much. What little I do remember from primary is sitting at tables with a group of classmates filling in brightly colored workbooks that used a lot of visuals/pictures to help us out. I can remember using manipulative materials such as blocks, counters, and card games to help us solve problems and learn new concepts. (I would get so excited to see my teacher pull materials from the shelves to hand around during a lesson!) I can also remember being asked by my teacher to help peers who needed a little extra time with assigned work when I finished mine early.  I liked being able to help my peers but, being shy, I didn't enjoy being put on the spot. Math was always interesting and enjoyable to me, fun even, in the primary grades. However, as we aged, and progressed through the elementary years, math became more boring. It became less interactive, as most work was done individually, "in our heads", on "boring" lined exercise books, and manipulative materials became a rare treat. I found that attitudes in the classroom about math quickly went downhill by late elementary/junior high.

It's hard to remember much about my primary/elementary years, but it seems as if my mathematics teachers enjoyed teaching it. It is hard to tell how they really felt about it because they were responsible for teaching everything - but if they didn't enjoy it, they hid it well. High school was different though, teachers were specialized in the area and it was very obvious that they loved it! Some of the math teachers I had in senior high would get so happy when teaching their favorite units/concepts. In high school, I found a new appreciation for math in my last couple years. I transferred from NL to AB in grade 11 and they put me in pure mathematics. It was so different from the math I had taken in NL and it was difficult for me to adjust to a completely different curriculum in the middle of high school but I loved the challenge.  

Math was always something that came easy for me. I knew I was "good" at it because I always excelled without having to put much effort into it. I know I was lucky in this respect, many of my friends couldn't say the same. That doesn't necessarily mean I loved it though. Math was always that subject that I neither loved nor hated, I never really thought much about it - it was just something we had to do. I enjoyed the feeling of success I would get from making my teachers and parents proud when I'd bring home my 90's & 100's, but it wasn't the subject itself that brought me joy, it was knowing I was good at something that not everyone could be good at.

As for assessment, it was always the same. Worksheets, open-book assignments (which were basically the same as worksheets), unit tests, and as we got older: finals. Tests and exams usually had some multiple choice and some short answer questions. It was always the same and generally straightforward. I'm sure there was probably on-going informal assessments in the classroom such as observation and checking of homework etc. but as a child I can only remember worrying about things I knew were being marked: the assignments and tests. 

Since university, I've only taken two mathematics courses: the recommended 1050 & 1051. For my focus area a few years ago I was debating between math and english. In the end I chose english, but I've always kind of regretted it.

Overall, I've had good experiences with mathematics. However, no particular memories stand out in my mind from primary/elementary. I do think that mathematics is an extremely important subject to teach children because it's all around us: we use it to count money, to buy things, to tell time & dates, for baking, for clothing sizes, for weather, and so on. Numbers are everywhere, I consider it a literacy all on its own.We use it everyday without even thinking about it - so as an adult, I can see why it is emphasized so much in grade school.

Welcome to my Blog!

Hi everyone!

My name is Amy and I'm a forth year student at Memorial University of Newfoundland, currently completing my B.Ed (primary/elementary). This blog is required for my Ed 3940 course - Mathematics in the Primary and Elementray grades. I've always enjoyed math and have been looking forward to this course since the beginning of my program! I'm quite interested to learn about approaches to take when I begin teaching mathematics to my future students. I've never written a blog before so please bear with me as I try to figure it all out!

:)