This week, I read the article "To Know and to Teach: Mathematical Pedagogy from a Historical Context" by Frank Swetz. In the article, the author notes that particular techniques for teaching mathematics, such as the use of instructional discourse, a logical sequencing of problems and exercises from simple to complex, and the use of manipulatives is much the same as it was thousands of years ago. He goes on to expand on each of these by pulling examples from different times in history, including some geometry problems from the Old Babylonian period approximately 3000 years ago! I actually worked through some of the problems myself and found them to be quite fun.
The authors' section on mathematical exercises was particularly intriguing to me, as I was constantly thinking of current school mathematics texts and the structure of exercises in those book. What struck me the most was the total amount of exercises. Within these old texts, the number of exercises seems to be capped around 20. In regards to current calculus texts, one section could have at least 80 problems!! Most of the problems are essentially the same, with some slight variations here and there. In regards to some Chinese problems from 100 BC, the author states that in order to solve them "thinking is required." This made me stop to think. In the current school system, how much "thinking" do our students actually do? Even if problems in textbooks go from simple to complex, how much thinking are students doing on these problems?
Moreover, do current mathematical textbooks encourage students to think critically? I am tempted to say no for the most part. Of course, I'm running off of my own personal experience at this point. I remember the majority of my homework in math class being the same problem over and over, tests being variations of those problems, and this evolved to the point where I "didn't have to think about it." If I am remembering correctly, there was a Schoenfeld article I read recently about students' beliefs about mathematics. He was observing a high school geometry class and when the teacher was referring to studying geometric construction problems for the state exam, he told the class "they shouldn't even need to think" (or something along those lines). Unfortunately, I think I'm also guilty of this as a teacher, in telling my students that certain types of problems should be essentially automatic. This is certainly not what I want for my students, but it is what expected in the system.
Does performing 40 variations of the same problem enhance student learning and understanding of particular mathematics concepts? Why did texts evolve in such a way? I am left with many questions as to why textbook exercises are the way they currently are. I look forward to the discussion on Wednesday.
Sunday 29 March 2015
Thursday 19 March 2015
Reflections on "Learning angles through movement"
This week, I read Smith, King, and Hoyte's 2014 article "Learning angles through movement: Critical actions for developing understanding in an embodied activity." In the study, the authors worked with students in grades three and four, while they explored a task developed for a Kinect for Windows program. The program detected angles that students made with their bodies, while projecting an abstract visual of the angle on the screen. The abstraction was done in four phases. The first associated only colour to different classifications of angles. That is, if the student made an acute angle in any orientation, the screen would appear pink. Obtuse, right, and 180 degree angles were associated to their own colours as well. The second phase projected the color as well as two line segments (of equal length) connected at a point that mimicked the angle. Since research has shown that many students seem to make decisions about the size of an angle based on the length of the line segments, the third phase projected two line segments of non-equal length with the corresponding background colour. Finally, the fourth phase took all the aspects from the third, but added a protractor that changed with the orientation of the angle.
The initial sample consisted of 20 students who took a pre-test, participated in the task-based interview, and then were administered a post-test. The authors report that half of the students had increases in test scores from pre- to post-test, two scores decreased, and eight scores stayed the same. The authors also found that increase in test scores was statistically significant. Besides presenting this quantitative data, the authors also presented a case study of two students who had low pre-test scores. Ian, who scored low on the pre-test but scored 100% on the post-test, was very engaged in the task. He had a strong connection between the movement of the arms with the visual images on the screen, as well as an openness to exploration. Kara, on the other hand, seemed quite reluctant with the task. She was not as open to exploration and did not have a strong connection between the movement and the visual images. I do wonder how Kara was feeling coming into the interview. Just from the pictures of her in the article, her gestures seemed quite timid, especially compared to Ian's large, confident gestures. Could this have attributed to her openness to the task?
This paper was my first exposure to any of the embodiment literature, so I came in a bit skeptical. I have to say that I was very excited by the study once I started to read. Movement has been a large part of my life from a very young age. Even in my teaching, some colleagues have described me as "a mover." I love moving in my entire teaching space, using my arms and hands to demonstrate certain concepts, as well as just using my hands for speech in general. The authors concluded that it appears exploration with the body can facilitate exploration of the mind. When I put this into the context of my ballet training, I couldn't agree more. I, like Ian, often experimented with my body, particularly when it came to turning. I realized that shifting my weight slightly in one direction helped my balance immensely, so I stored that into my mind for the next time I needed to try it. Slight adjustments to my body, assisted in my overall understanding of how to execute a movement. As a side note, I was very open to making these adjustments. I really didn't care if I fell, if anything I learned something from it. On the other hand, many of my friends were very timid when it came to pirouettes and turning, since they didn't want to slip and fall. This brings me back to the openness of Ian and Kara and how this might have affected their involvement in the task.
As an extension to this study, I wonder how this could be related to students' understanding of radians. In the calculus curriculum at UBC, degrees are used sparingly. In fact, I avoid them entirely within my lectures and I will go as far as to say that most people in the department do as well. I have found that the majority of my North American students prefer to use degrees and generally avoid radians. Could a similar technology be used to help students understand radians for angle measurement?
The initial sample consisted of 20 students who took a pre-test, participated in the task-based interview, and then were administered a post-test. The authors report that half of the students had increases in test scores from pre- to post-test, two scores decreased, and eight scores stayed the same. The authors also found that increase in test scores was statistically significant. Besides presenting this quantitative data, the authors also presented a case study of two students who had low pre-test scores. Ian, who scored low on the pre-test but scored 100% on the post-test, was very engaged in the task. He had a strong connection between the movement of the arms with the visual images on the screen, as well as an openness to exploration. Kara, on the other hand, seemed quite reluctant with the task. She was not as open to exploration and did not have a strong connection between the movement and the visual images. I do wonder how Kara was feeling coming into the interview. Just from the pictures of her in the article, her gestures seemed quite timid, especially compared to Ian's large, confident gestures. Could this have attributed to her openness to the task?
This paper was my first exposure to any of the embodiment literature, so I came in a bit skeptical. I have to say that I was very excited by the study once I started to read. Movement has been a large part of my life from a very young age. Even in my teaching, some colleagues have described me as "a mover." I love moving in my entire teaching space, using my arms and hands to demonstrate certain concepts, as well as just using my hands for speech in general. The authors concluded that it appears exploration with the body can facilitate exploration of the mind. When I put this into the context of my ballet training, I couldn't agree more. I, like Ian, often experimented with my body, particularly when it came to turning. I realized that shifting my weight slightly in one direction helped my balance immensely, so I stored that into my mind for the next time I needed to try it. Slight adjustments to my body, assisted in my overall understanding of how to execute a movement. As a side note, I was very open to making these adjustments. I really didn't care if I fell, if anything I learned something from it. On the other hand, many of my friends were very timid when it came to pirouettes and turning, since they didn't want to slip and fall. This brings me back to the openness of Ian and Kara and how this might have affected their involvement in the task.
As an extension to this study, I wonder how this could be related to students' understanding of radians. In the calculus curriculum at UBC, degrees are used sparingly. In fact, I avoid them entirely within my lectures and I will go as far as to say that most people in the department do as well. I have found that the majority of my North American students prefer to use degrees and generally avoid radians. Could a similar technology be used to help students understand radians for angle measurement?
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