The days leading up to this week's class, I found myself going ahead in my posts to get a better understanding about what we would be discussing. This week the math mindset module had me exploring the connection between answering math problems quickly and student success in the classroom. When I viewed this video prior to class it made sense to me and I could understand the reasoning behind giving students a chance to ponder over math problems instead of requiring them to give speedy answers to display their ability. I had this in mind when I came to class this past Thursday. The first activity we performed involved mental math operations and required one person to figure out their answer after having a word problem read to the class. They would then reply that they have that number and would read their word problem so that the next individual who had that answer could continue the cycle. Interestingly, there was nothing mentioned about giving quick answers to these problems but you could tell from the very beginning that some of the future teachers in the room wanted to show everyone that they knew the answer first and couldn't stop themselves from blurting out the answer. This continued to happen even after being told not to yell out the answer. I had a little ah-ha moment as the individuals in the room who needed a little extra time to figure it out felt belittled by their peers yelling out their answer before they could figure out that they were holding the next piece to the cycle. I was reminded of the mindset module video and how it is better to give our students a chance to figure out the problem so that they can make deeper connections with the mathematical concepts.
As a future educator, one pedagogical concept found in the growing success resource really struck home for me. In class we had a chance to discuss differentiated content. I kept getting confused that differentiated content was modified curriculum, where teachers give students who are at a lower learning level different work with a different set of big ideas and expectations. Dr. Khan made sure to go over this thoroughly as he could see that there was some interest about this in the class. One point he made was that differentiated instruction involves educators giving their students parallel tasks that "meet the needs of students from different developmental levels but that get at the same big ideas and are close enough in context that they can be discussed simultaneously" (Khan, 2016). At first I even debated with another class mate about this because I felt that this still involved modification but he helped me to see how the overall idea and expectations are still the same with differentiated content. He said that the teacher anticipates the problems that the different learners in the room will encounter and creates other problems that will help those students to see the concept from a different perspective. They are still learning the same information but from a different view. This conversation with the instructor and that student has helped me to put a finger on the kinds of things that I can do to differentiate my content. This can be done by using graphic organizers, varied supplementary materials, compacting the material or flexible groupings for those students that struggle at math. All-in-all I had a lot of great ah-ha moments in this class and I have made the time to write these down in a journal where i can go back and view this information during my teaching block and future career.
Resources Viewed
Khan, S. (2016). EDUC 8P54: Parallel Tasks [PowerPoint Slides].
Retrieved from
https://lms.brocku.ca/portal/site/EDBE8P54D12FW2016LEC003/page/39971b0a-3b0e-4242-98b5-ad5cd21dbec1
Monday, September 26, 2016
Friday, September 16, 2016
My weekly report and reflection 2
In this week's math class I found myself doing three important things that really struck home with me. I revisited an old concept that frustrates me and I was performing some creative in class activities. To begin this post I will address the concept that frustrates me the most in mathematics. That concept is fractions. In our last meeting we were discussing the ways of approaching this topic with students and how to help them discern the difference between 5/8 and 6/10. The question was asking students to decide which fraction is bigger. I know it's 5/8, I can completely understand why it's 5/8 but trying to explain this concept in an intelligent and concise way to students is a struggle for me. I even did my best to stand in front of the class and explain how in the fraction 5/8 you have 8 whole parts (I used a pie) with 5 of them coloured in, leaving you with only 3 pieces lefts over. On the other hand, with the fraction 6/10 you have 10 whole pie pieces with 6 of them coloured in, leaving you with 4 pieces left over. Thus, the first fraction (5/8) is bigger because you had less parts left over. Well, by the time I was finished speaking I probably used about 50-75 words too many and had the ugliest looking set of scribbled pies on the board. I learned from that experience that less is more and that I needed to redefine the way that I think about fractions and refine my explanation. Our instructor then showed us the same two fractions represented by squares. For the fraction of 5/8 there were 8 squares with 5 coloured in and for 6/10 there were 10 squares with 6 coloured in. The difference between his diagrams and mine was that both diagrams took up the same amount of overall space, however, the pieces drawn in the fraction 6/10 were smaller. This was because there were more parts that needed to be placed in the same amount of space as 5/8. Visually, it was easy to tell from the very beginning that 6/10 was smaller because the pieces were smaller. He then reminded me of another way to demonstrate the difference between two fractions. Translate them into decimals and use a number line. Personally, I like this way better because it makes more sense to me. Just divide 5 by 8 and 6 by 10 and you get 0.625 and 0.6 respectively. Place those two numbers on a number line and show how 0.625 is closer to 1 than 0.6. In the future I will probably do it this way because it related better to the way that I process information but I won't dismiss the idea of showing the square diagrams because every student thinks differently and what works for me might not work for every student.
During our in class activities we were asked to look at three separate rotational views of the same picture containing owls on a 4x4 square platform. The question asked us to figure out how many owls were standing on the platform. We were given connecting cubes as manipulatives to help visual/kinetic learners to solve the problem more easily. I chose to not use the connecting cubes but to watch what others were doing and to discuss the problem with one of my classmates. Through community-based learning I was able to better understand the problem and got the correct answer. Just having that one student mention to me which way the one picture was rotating helped me to understand the question much quicker and easier than having to build the entire 4x4 platform. By reading the curriculum document I discovered that this problem relates to the grade 6 specific expectation of "build[ing] three-dimensional models using connecting cubes, given isometric sketches or different views of the structure" within the geometry and spatial sense strand (Ontario Ministry of Education, pg. 92).
The second in class activity that I really liked and that I will definitely incorporate into one of my classes in the future was a finger counting activity. The task was to count on our fingers from 1 to 1000, starting at the thumb with 1. The catch (and there's always a catch) was that once we reached the pinky finger at 5 we needed to then count in the opposite direction on our hand with the ring finger being 6 and the thumb consequently becoming 9. We then continued counting on our fingers in the same direction that we first started making the pinky 13 once we had returned again. The question asked us what finger we would land on once we reached 1000. What I really liked about this task was seeing the different strategies that people were using to get their answer.
As you can see here in this picture, the diagram on the right shows the group trying to use division to reach a solution and the group on the left was trying to decide which numbers would fall on each finger. Our group started to count each finger to establish which fingers were showing a common pattern.
We discovered that the index finger and the ring finger had a pattern involving groups of ten. At first we thought that those groups of ten were going back and forth from the index finger to the ring finger and back again (i.e., 10 on the index, 20 on the ring, 30 on the index, 40 on the ring). Later we found out that the pattern was a little different were 10 was on the index finger, 20 was on the ring finger and then 30 would also be on the ring finger before placing 40 on the index finger. We finally came to the conclusion that 1000 landed on the index finger and other groups came to the same conclusion after some time and by using various strategies. It would be interesting to see how younger people would try to figure out this problem and would provide some really rich conversations about mathematical strategies and the way that each individual is different and will approach problems in different ways. All in all, this last math class was a really good learning experience and I felt as though I walked away with some good teaching strategies and felt a little more confident in my ability to teach fractions in the future.
Resource Viewed
Ontario Ministry of Education. (2005). The Ontario curriculum grades 1‐8: Mathematics [Program of Studies]. Retrieved September 16, 2016, from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf.
During our in class activities we were asked to look at three separate rotational views of the same picture containing owls on a 4x4 square platform. The question asked us to figure out how many owls were standing on the platform. We were given connecting cubes as manipulatives to help visual/kinetic learners to solve the problem more easily. I chose to not use the connecting cubes but to watch what others were doing and to discuss the problem with one of my classmates. Through community-based learning I was able to better understand the problem and got the correct answer. Just having that one student mention to me which way the one picture was rotating helped me to understand the question much quicker and easier than having to build the entire 4x4 platform. By reading the curriculum document I discovered that this problem relates to the grade 6 specific expectation of "build[ing] three-dimensional models using connecting cubes, given isometric sketches or different views of the structure" within the geometry and spatial sense strand (Ontario Ministry of Education, pg. 92).
The second in class activity that I really liked and that I will definitely incorporate into one of my classes in the future was a finger counting activity. The task was to count on our fingers from 1 to 1000, starting at the thumb with 1. The catch (and there's always a catch) was that once we reached the pinky finger at 5 we needed to then count in the opposite direction on our hand with the ring finger being 6 and the thumb consequently becoming 9. We then continued counting on our fingers in the same direction that we first started making the pinky 13 once we had returned again. The question asked us what finger we would land on once we reached 1000. What I really liked about this task was seeing the different strategies that people were using to get their answer.
 | |||
| (Dekker, 2016) |
As you can see here in this picture, the diagram on the right shows the group trying to use division to reach a solution and the group on the left was trying to decide which numbers would fall on each finger. Our group started to count each finger to establish which fingers were showing a common pattern.
 |
| (Dekker, 2016) |
We discovered that the index finger and the ring finger had a pattern involving groups of ten. At first we thought that those groups of ten were going back and forth from the index finger to the ring finger and back again (i.e., 10 on the index, 20 on the ring, 30 on the index, 40 on the ring). Later we found out that the pattern was a little different were 10 was on the index finger, 20 was on the ring finger and then 30 would also be on the ring finger before placing 40 on the index finger. We finally came to the conclusion that 1000 landed on the index finger and other groups came to the same conclusion after some time and by using various strategies. It would be interesting to see how younger people would try to figure out this problem and would provide some really rich conversations about mathematical strategies and the way that each individual is different and will approach problems in different ways. All in all, this last math class was a really good learning experience and I felt as though I walked away with some good teaching strategies and felt a little more confident in my ability to teach fractions in the future.
Resource Viewed
Ontario Ministry of Education. (2005). The Ontario curriculum grades 1‐8: Mathematics [Program of Studies]. Retrieved September 16, 2016, from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf.
Tuesday, September 13, 2016
My weekly report & reflection 1
Hello everyone, I'd like to welcome and thank you all for taking the time to read this first blog post about the pre-service mathematics course in year 2 of the consecutive education program at Brock University. Sadly, the purpose of this post and those to follow will not be to grind and inundate you with a plethora of new mathematical concepts or to even help you finally solve those pesky Diophantine equations. I know, now that you've heard the bad news please take a moment to compose yourself before you continue reading. In all seriousness, the purpose of these posts will be about helping myself and hopefully the reader to successfully find the power in the art of self-reflection. The process of self-reflection is an important one. It allows not only pre-service educators but educators in all stages of their career to improve their teaching practice and to reflect on the strategies they use to foster wonderment and discovery in their students. In our first class, 'wonder' and 'discovery' were two buzz words that came up repeatedly.
The first in class activity that really had me exploring these two ideas (more so wondering than discovering) was a sorting activity that had our table groups looking through a series of printed squares containing the math curriculum expectations of grades one to eight in a single strand. Our strand was geometry and spatial sense. From the very beginning I was honestly overwhelmed by the amount of information that we had to discern. Doctor Khan made sure to give us just enough information in that particular moment to leave us wondering about what we were doing with our lives. He told us to sort these squares in which ever way we saw fit. Well this just went against everything that I had ever been taught in math. There is only one way to solve any math problem and I did not have the key to solving this equation. I was left wondering whether or not the information needed to be sorted by the similar words in bold or by the content that each presented. I expressed my displeasure to Doctor Khan and he simply said "well good, write down what you are wondering". At that moment I kind of went "oh, that's the whole purpose of this, I'm suppose to wonder and struggle with the problem". Our group's final product looked very ordered and chronological.
After sorting our information we had a chance to do a gallery walk of the other group's work and I still had difficulty seeing the solution to my problem. We discussed what each group did and by reflecting I noticed that some of the content we placed at the end in grade eight could have gone before grade two. Such information like demonstrating the applications of geometric properties in the real world as described in GSS 022. In grade two students need to sort and classify quadrilaterals based on the geometric properties. At the very least an educator could have their students sort and classify real world geometric shapes. Throughout my investigation of our group's final product I found many other instances where the orders could have been changed based on the curriculum's suggested tools of engagement. I then realized that educational practitioners need to know the space that they are working in and to look across the curriculum to use sensibilities found in later grades to help encourage students in earlier grades. The guidelines given are not as cut and dry as I once thought.
During the week that followed we needed to watch a series of videos discussing the commonly propagated attitudes towards math that unfortunately create stereotypes and form math myths. You might be 'wondering' what this is all about so let me shed some light on the situation. In today's classrooms some of the common attitudes towards math are that it's only for those who are math people, that it's a difficult subject because a question can only be answered one way, and that because of these reasons it has no useful application in the majority of people's lives. To make things worse the media commonly propagates these attitudes throughout all forms of entertainment. It has become such a problem that one of the common stereotypes to arise from all of this is that boys are good at math and girls are not. This stereotype is one that I heard while growing up but was difficult for me to believe in (despite being helped by my Grandfather in the subject while my Grandmother helped with English). Many of the female students in my elementary and secondary classes have actually been quite good at math and on quite a few occasions have surpassed my ability completely.
One math myth that I did believe in while growing up was that people are either left brained or right brained and that people who fell into one category or the other had difficulty performing tasks associated with the opposite side. I was actually reminded about this myth by a fellow classmate of mine while responding to one of their forum posts. As a musician I believed that I fell into the right sided category that relates to creativity and artistic sense. I believed that being in this category meant that I could never be a mathematician or form a career involving mathematics because that belonged to the left brained people. By watching the required videos this week I learned that every individual can have a math brain.
Much like any muscle in the body, the brain can grow and be strengthened by repetitive problem solving. The key is that educators create lessons that fuel student interest and inquiry. Students need to have the opportunity to 'wonder' about an equation and then take the time to struggle with that problem so that they can 'discover' the solution on their own. Just like the sorting activity that I first struggled with and later understood its purpose. I learned that it's good for students to sometimes feel uncomfortable while inquiring about a problem and then experience the sense of relief and accomplishment that follows when discovering the solution. In doing so it helps strengthen the brain by forming new synapses to aid in the transfer of information through an electrical impulse between neurons. Hopefully during my teaching block I can encourage my students to have all of their synapses firing through lesson that promote 'wonder' and cultivate 'discovery'. At the end of the day I just hope that I'm able to see through all of the smoke that those synapses create.
The first in class activity that really had me exploring these two ideas (more so wondering than discovering) was a sorting activity that had our table groups looking through a series of printed squares containing the math curriculum expectations of grades one to eight in a single strand. Our strand was geometry and spatial sense. From the very beginning I was honestly overwhelmed by the amount of information that we had to discern. Doctor Khan made sure to give us just enough information in that particular moment to leave us wondering about what we were doing with our lives. He told us to sort these squares in which ever way we saw fit. Well this just went against everything that I had ever been taught in math. There is only one way to solve any math problem and I did not have the key to solving this equation. I was left wondering whether or not the information needed to be sorted by the similar words in bold or by the content that each presented. I expressed my displeasure to Doctor Khan and he simply said "well good, write down what you are wondering". At that moment I kind of went "oh, that's the whole purpose of this, I'm suppose to wonder and struggle with the problem". Our group's final product looked very ordered and chronological.
 |
| (Dekker, 2016) |
During the week that followed we needed to watch a series of videos discussing the commonly propagated attitudes towards math that unfortunately create stereotypes and form math myths. You might be 'wondering' what this is all about so let me shed some light on the situation. In today's classrooms some of the common attitudes towards math are that it's only for those who are math people, that it's a difficult subject because a question can only be answered one way, and that because of these reasons it has no useful application in the majority of people's lives. To make things worse the media commonly propagates these attitudes throughout all forms of entertainment. It has become such a problem that one of the common stereotypes to arise from all of this is that boys are good at math and girls are not. This stereotype is one that I heard while growing up but was difficult for me to believe in (despite being helped by my Grandfather in the subject while my Grandmother helped with English). Many of the female students in my elementary and secondary classes have actually been quite good at math and on quite a few occasions have surpassed my ability completely.
One math myth that I did believe in while growing up was that people are either left brained or right brained and that people who fell into one category or the other had difficulty performing tasks associated with the opposite side. I was actually reminded about this myth by a fellow classmate of mine while responding to one of their forum posts. As a musician I believed that I fell into the right sided category that relates to creativity and artistic sense. I believed that being in this category meant that I could never be a mathematician or form a career involving mathematics because that belonged to the left brained people. By watching the required videos this week I learned that every individual can have a math brain.
Much like any muscle in the body, the brain can grow and be strengthened by repetitive problem solving. The key is that educators create lessons that fuel student interest and inquiry. Students need to have the opportunity to 'wonder' about an equation and then take the time to struggle with that problem so that they can 'discover' the solution on their own. Just like the sorting activity that I first struggled with and later understood its purpose. I learned that it's good for students to sometimes feel uncomfortable while inquiring about a problem and then experience the sense of relief and accomplishment that follows when discovering the solution. In doing so it helps strengthen the brain by forming new synapses to aid in the transfer of information through an electrical impulse between neurons. Hopefully during my teaching block I can encourage my students to have all of their synapses firing through lesson that promote 'wonder' and cultivate 'discovery'. At the end of the day I just hope that I'm able to see through all of the smoke that those synapses create.
Subscribe to:
Comments (Atom)