Week 12 Reflection

       12 weeks later and we have completed our first semester of teachers college. I cannot believe that we're already a quarter of the way finished our time here at Brock U. Math class with Patricia has been quite the journey, and going back through the course material while curating for my Digital Portfolio I realized just how much we have learned this semester. My whole thought process about math has changed since the beginning of the course. In particular, my views about the ability of students to grasp math has been changed forever. As a student I always thought that there were mathematical minds and minds that were better at language and arts. After this course with Patricia I now know that every student is capable of being successful in Math class and as teachers we have to create a learning environment that supports a growth mindset, and is encouraging, open and supportive to student learning. Pat has also gone through a number of different mathematical concepts with us that have changed the way that I perceive mathematics. For example, she has shown us new ways to divide fractions and to perform subtraction, just to name a few. These new techniques that she has shown us are indicative of the concept that there are no right and wrong ways to understand how to do math. Instead, we should support students and anyway that they understand to come to an answer. Working in groups and using collaborative learning in class was a great demonstration of these different ways of seeing math. In groups, although half of the group may solve the problem one way, the other half may choose another solution. Our EQAO problem assignment was also a good example of this. This little classroom techniques, such as group work and working with physical manipulatives have changed the way I think about teaching math. Instead of just going up in the front of and teaching students how to plug numbers into an equation, its all about having students think deeply about mathematical concepts in order to understand them. It is more important for a student to understand WHY we are doing something, rather than HOW. A great technique to have the students think deeply and make connections with a mathematical concept and real life situations is through guided inquiry. Pat has also taught us the importance of having a deep understanding of what we are teaching students. Like I said earlier, it is more important to be able to teach students why we are doing something rather than how. For example, if you don't know the reason behind why we flip a fraction and multiply when dividing, then maybe you should learn a new technique, or learn the reason behind the technique to explain to students the reasoning. Patricia has also introduced us to a number of online resources that we can use to further and deepen our professional knowledge such as Edugains. These resources will not only help in classroom techniques, but also lesson planning etc. Moving forward in my mathematic and teaching career I hope to exemplify the standards that Patricia has taught us to uphold and if I am able to be half as effective as Patricia in my own class then I know that I am on my way to becoming a great teacher.

Digital Math Portfolio

Week 11 Reflection

In this weeks class we were responsible for reading chapter 3 in the Making Math Meaningful textbook. This chapter is focused on Assessment and Evaluation. As we come to the end of the semester and our time in Math class, it is important that we take a class to look at assessment and evaluation, specifically in a math class setting. As Pat pointed out to us, and as we have been learning in all of our classes, and in particular our Assessment class, the point of assessment is not to rank the children, or to have marks so that we can fill out the report cards. Rather, the point of assessment is to improve and ensure student success, assessment for improvement.

After already completing some of the first semester courses, and reaching the near end in others, we have learned the importance of implementing a formative style of assessment. This style of assessment is ongoing as uses assessment as a tool to encourage and supplement learning, rather than just being the end result of learning. This shift in styles to a formative style of assessment reflects the change in education; there is nothing more important that the student and getting them to learn and retain information. I also really appreciated the way that Pat explained the different type of assessments, OF, AS, and FOR learning. The math textbook also did a great job of explaining the differences between the types of assessment. I actually found the discussion on the types of assessment in math class more useful than any of the other courses I’ve had to take. Pats explanations were very effective to me in terms of understanding what each assessment style meant. For example, Assessment for learning was something that confused me a little, but Pat explained it very simply by saying that it is simply to figure out where your students are, and what they know, so you can plan their learning; this is essentially the minds on section/prior knowledge of the lesson plan.  In addition, Pat also touched on the edugains resource again and how important it may prove to be to us as teachers in terms of lesson planning as well as assessment and guidance. Pats explanation of the achievement chart in the slideshow was also helpful to me as it broke down the mathematical process into the different categories in the chart. I think that this is something that will prove to be useful to me in my lesson planning.

Dawn. 2009. Math [Online Image] http://bit.ly/1InswiW

            We also did some fun activities in class this week that challenged our mathematical knowledge.  Pat grouped the class using Popsicle sticks with different numbers on them. I liked the idea of having the different colored sticks with different numbers handy for when you need to make groups; you can make the groups by color or by the numbers, a nice little technique to keep in mind. After we were grouped off we had to rotate to different tables and take part in different math games. I enjoyed all of the games, but I think that the game with the toothpicks is one that is especially useful for a math class. In the activity you are given a number of clues and a bunch of toothpicks and you have to create the geometrical shape described in the clues. I found these to be quite challenging and they really tested your knowledge of shapes and definitions. It is also a fun way to collaborate with your classmates. In addition, the games with the number charts were also very fun to me. Similarly to the toothpick games, you are given a bunch of clues and must figure out what the secret number is. Again, this game tests your math knowledge, specifically your knowledge of multiples etc.

Week 10 Reflection

      This week we were responsible for a nice little chunk out of the Making Math Meaningful textbook. We were required to read three chapters, 19, Data Display and Analysis, 20, Collecting and Describing Data, and 21, which was focused on Probability. As indicated in the title, chapter 19 is all about collecting, displaying and analyzing data. The textbook goes into detail explaining the many different types of graphs that can be used to collect data, and also the pros and cons of using each different type of graph. We also touched on this idea in class; there is not one type of graph that you should always use. This is something that I think I may have forgotten from my days of creating graphs in school. To be honest, I forgot about just how many different types of graphs we were taught about when we were younger. However, Patricia refreshed our memories and explained to us that different graphs should be used in different situations. For example, a line graph is a good way to show an increase in something over time, such as temperature throughout the day.

bootmi. 2012. graph. [online image] http://bit.ly/1MAPrLC

 To help demonstrate this we did a fun little activity where Pat brought in a tub of cookies and asked us all to guess how many cookies were in the tub. We then plotted our answers on a stem and leaf graph to create a visual. The use of a graph to display the information was useful as it provided a visual display that made the information about the guesses easily understandable. Patricia then introduced us to a great resource called www.tinkerplots.com. This is a great website in which you can enter data and the program displays for you all of the different patterns that are within the data, as well as automatically calculates things such as mean, median, and mode. I think that this resource would be great to use in a class after you have your students collect some data. After inputting their data they would be able to play around with the different filters and discover different patters in their data. In terms of having students collect data, Patricia gave us some good advice for the older grades in elementary school. The advice was that in the older grades, it may be tempting to have your students collect data on a very elementary topic, such as ice cream flavor, but she advised us that if we are going to do a data collection unit it needs to be something that will suit the proper grade level. Patricia did a great job of combing all three chapters into the activity we did in class regarding the tub of Oreos. It is encouraging to me to see how she is essentially able to hit many different curriculum expectations with one activity. For example, with the one Oreo activity she covered all three chapters we were supposed to read for the day, Data Display, Collecting Data, and Probability. We demonstrated probability when we tried to estimate how many cookies there were in the jar. We all used different techniques to reach our estimate and then by calculating the average we agreed on the most probable answer.

 The learning activities for the day were a good display of how to use some of these online resources that I have touched on in earlier Blogs. Padi and Mileena both used gizmos for the presentation and I thought that it was a good use of the technology in an actual lesson plan and can see myself using some of the sets on that website to teach mathematical concepts. In addition, Erlisa used Geometers Sketchpad which I also thought was a fun use of technology, however, I feel as though the sketchpad would not work in a younger classroom because it is too much like Microsoft Paint and allows the kids to just fool around on the computer rather than a website like gizmos which only allows you to use the program for math work.

Week 9 Reflection

This week we were responsible for reading chapter 17, Length and area, and chapter 18, Capacity, volume, Math, Time and Angles, in the Making Math Make Sense Textbook. Patricia started the lesson for the day with an exercise that I found to be very useful. The exercise consisted of Patricia handing out a bunch of little cards that contained the phrases “I have… Who Has”. For example, my hand out had something along the lines of ‘I have Right, Who has an angle smaller than 90 degrees?’ The point of the exercise was to demonstrate knowledge of definitions and shapes etc. So the first person would begin the exercise by reading what they have and then they would pose the Who has question. The next person whose clue was described would then read theirs and the process continues until everyone in the class has read their clue. I thought that this activity could prove to be very useful as it is a good way to test your student’s knowledge of definitions and mathematical concepts. As I described in last week’s blog, many of us do not really know exactly what defines a certain shape, or polygon, and this exercise can be used to identify these gaps in knowledge and give us, as teachers, an idea of what exactly needs to be worked on in class.

            Another useful exercise that we participated in in class was the exercise with the toilet paper roll. The premise of the exercise was that a new school is being built and we needed to find out the amount of steel sheets we would need to construct pillars in the front foyer of the school. We were told that the scale of the pillars would be 1:10 in relation to the toilet paper rolls. We were then responsible for figuring out the circumference and diameter of the toilet paper roll. After this we had to find out the area for the cylinder. We then had to cut the roll in half and again, measure to find out the area. She then asked us to compare the two areas and point out any similarities or differences. Obviously the two areas were the same, but I can see this activity being effective in a classroom lesson where you are trying to demonstrate to the students that the area in these two shapes are the same, even though the shape itself is much different. We then had to do a little bit of inquiry as Patricia posed an OH NO scenario to us where the steel rolls were delivered to the school in a different size. She also explained to us that students usually really enjoy the OH NO scenarios and that they are a good way of consolidating the lesson and letting the students show what they have learned from the first part of the problem.

Jef P. (2007) Pillars [Online Image] http://bit.ly/1PuP5tG

            The lesson plans for the day were also centred on teaching the concepts of Area/Length and Perimeter and Volume/Capacity. Similar to the activity regarding the cylinders and the sheet metal pillars, Sabrina had a great exercise which demonstrated the volume of cylinders in which we were given two pieces of paper, which were the same size, and asked to make two different sized cylinders. She then asked us to decide whether or not each of the cylinders had the same volume even though they were different dimensions. We then filled the cylinders with marshmallows and then compared the amounts to see the volume of the two cylinders. Of course, both cylinders had the same area, but I think that this activity could prove useful in a class lesson in which you were trying to demonstrate volume of cylinders and how even though shapes may have different dimensions, they may still have the same area/volume etc.

Week 8 reflection

This week in math class we were responsible for reading chapter 15 and 16 in the Making Math Meaningful textbook. The two chapters were about 3-d and 2-d shapes and Location and movement. One of the things that I found interesting in the textbook is the different levels of classification in chapter 15. Firstly, the ability to conceptualize shapes classified by Pierre van Hiele and Dina van Hiele-Geldof was interesting to me because it reminded me of theories that we have learned in Cognition and the Exceptional Learner. I had never really thought of math, and more specifically, the ability to recognize and think of shapes, in terms of stages. First the student is able to recognize shapes just by sight, then the student is able to recognize properties of shapes, then the student is able to apply ‘if-then’ reasoning. This is similar to the conversation that we had in class regarding what makes up certain shapes, ill touch on this more later. Lastly, the student is able to identify and classify, as well as deduce properties of shapes. Similar to this theory is Clements stages that young children go through as they combine shapes. In his theory he talks about how children are learning shapes as they play and combine shapes. By playing, students are learning about essential properties of shapes.

Chapter 15 also goes into detail about misconceptions regarding shapes. This is something that we talked about in class that I found pretty interesting as well. Patricia handed us out a sheet with some polygons on it and asked us to classify each shape and give our reasoning. When I looked at the sheet and began working on it, I realized that I don’t actually know all of the rules and properties that every shape must have, other than the obvious ones like triangles, squares, and rectangles. Instead, I think I have just been conditioned to recognize them by sight, much like stage 1 in van Hiele’s stages. Patricia explained to us that the classification of polygons was probably not taught to us very well because our teachers probably did not have a great grasp on the concepts either. This is much like our unit on fractions; we were not taught everything about the concepts because our teachers did not fully understand the intricacies within the concepts themselves. Patricia explained that there is a ton of overlap in the classification of shapes such, as squares are always parallelograms. This made me think that when I have to teach shapes and polygons I will make sure to inform the students of this overlap so they can begin to understand what makes up different polygons, and so they can advance in Hiele’s stages.

            The learning activity presentations this week were also very good. I really liked Shannon’s activity on patterning. I thought that the actual math part of the activity was effective at showing patterns and was not so difficult that students would not be able to figure it out. In addition, her use of the cardboard bridge and the cutout climbers was a great idea. I really think that something like that would engage the kids and really get them into learning the concept. The cutout bridge and hikers could also be used in later lessons to demonstrate other ideas and play math or learning games with students. Eva’s lesson was also really fun. I loved when she played the Mario music as we were using the Miras, as well as when she asked us to come up to the board to draw out the Mario level.

Sean and Steph Mayo. (2015) Mario Mini [Online Image] http://bit.ly/1WD0EDt

Week 7 Reflection

For Math class this week we were responsible for reading Chapter 22 in the Making Math Meaningful textbook. This chapter is based on patterning and algebraic thinking. After reading the textbook and hearing Patricia’s lesson on Patterning and Algebraic thinking I was surprised at how much algebraic thinking goes into patterning. As humans we are great at recognizing patterns and I never really thought about the mental process that happens when we are looking for patterns. However, at the beginning of class Patricia asked us to demonstrate a pattern with some block manipulatives. Recreating the patterns with physical manipulatives was fairly easy, but then Patricia asked us to create a mathematical expression to represent the pattern. Creating the expression itself was not difficult, but I found it interesting how our minds come up with these expressions without us even noticing it. For example, one of the expressions was 2s+1 (where s equals the step number). I personally do not think of the pattern in this expression, but our minds are registering this. I just thought that it was interesting and demonstrated the fact that we all look at math differently and there is no right or wrong way to reach the answer; where one person sees the pattern physically, the other sees the expression.

this is miki. (2009). Algebra. [Online Image] http://bit.ly/1Wpy5cu

            The Learning Activity Presentations for the day also dealt with the creation of expression and patterns and algebraic thinking. In my opinion I thought that Adriana’s activity was the most successful at encouraging algebraic thinking and equation creation. Being able to use the q-tips to physically see the pattern made the creation of the expression very easy. I also enjoyed the Number Tricks presentation by Peter. The only thing I think Peter should have added to his presentation is an explanation of how his expressions worked to end up with the right result every time. Although, a magician never gives away his trick, and I could see an activity like Number Tricks really being a hit in the classroom and something that would excite the kids and get them to wonder about how the trick works.

            This week we also talked again about how to create a good Math question, and the requirements that it must hit. Patricia gave us some foolproof strategies for the creations of open math questions. The tips were simple things such as starting with the answer and removing some numbers from the process to see if students can figure out what the missing numbers in the equation are, or asking for similarities or differences in equations etc., simple strategies to encourage students to think deeper about math. She also provided us with a very useful link that goes into detail about effective question creation that I will most definitely be using when creating math questions. In addition, Pat made some interesting points about posing questions to students. She told us to never frame a question by stating that it is easy or that it is hard because that can cause a reaction and response in the kids before they even get to see the question. For some, hearing that the question is hard will make them shut down and not even want to complete it, while for others, they might try and find some intricate answer instead of the easier answer because they feel as though it was supposed to be a super hard question. She also told us that when a child asks us a question we should never say anything that you can stimulate the student to say; you don’t want to give anything away, make the students think for themselves. These little tricks about how to run a classroom and manage students are a great resource and are great tidbits of information from an experienced teacher.