Battleground Schools: Summary and Reflection
The chapter of Battleground Schools which focuses on mathematics education begins by highlighting the ideological schism which has been the source of many of the changes to math curricula in North America. The two main camps vying for influence in this conflict are the traditionalists (lecture, homework and assessment based) and the progressives (focused on understanding, inquiry and learning through experimentation). The article then gives an overview of the history of math education from the beginning of the 20th century to our present day. The Progressivist Reform (circa 1910 – 1940) sought to reinvigorate school math teaching by giving preference to the ‘why’ aspects of problem solving. Through this effort, they aimed to shift emphasis away from inexplicable procedural computation, and focus on improving students’ understanding. Following the launch of Sputnik in 1957, North Americans became intensely afraid of falling behind the Soviet Union technologically. The ‘New Math’ movement of the 1960s was a result of this insecurity, and its aim was to produce a generation of rocket scientists. What this led to was a massive overhaul of North American math curricula through the addition of many abstract topics, such as set theory and linear algebra. Ultimately, this approach was scrapped, as it neglected the importance of basic skills and the material was beyond the understanding level of many teachers. The article concludes with a section outlining the current conflicts which are raging due to the implementation of nationally standardized curricula, and it emphasizes the importance of working toward a consensus between the two ideological positions.
I really enjoyed this article. I found it to be a very good overview of the different directions math education has been taken in over the past century or so. Particularly, upon reading about the ‘New Math’ movement, I was reminded of an idea I had several years ago about potentially including an abstract algebra class in high schools. I think that exposing more gifted students to some of the elements of pure mathematics is an important idea, and worthy of consideration; however, I do not believe that these elements could ever serve as the bedrock for an entire set of K – 12 curricula. Concern for foundation skills would need to be addressed. Additionally, I was very interested to see the correlation between traditionalist views about math teaching, and evangelical Christian lobby groups. It would seem that even in math, there are rigid, dogmatic thinkers who attempt to push their views upon others. Overall, I feel like this article is an excellent source of background information for any prospective math instructor.
Wednesday, September 30, 2009
Monday, September 28, 2009
Assignment 1 Reflection
Assignment 1: Individual Reflection
I found this assignment to be quite interesting and useful to me. In particular, John Yamamoto’s input went a long way toward alleviating some of my concerns about specific aspects of teaching. Listening to the presentations of other groups was equally illuminating. Many groups asked their teachers intelligent and specific questions regarding how much homework to assign, or how to encourage discovery and discussion. These responses furnished me with some rough ideas which could help me during my first experiences in the classroom. Additionally, I noticed that there was a wide variety of different opinions represented among the surveyed teachers. Some seemed quite authoritarian, and they approached teaching in a very instrumental way; however, I got the impression that most teachers utilized a mixture of approaches, and they aimed to build rapport and appeal to different learning styles. On the other hand, I was slightly dismayed by the responses from the surveyed students. It seemed as though very few of them had any appreciation for mathematics beyond its use as a calculation tool. In general, I left with the feeling that most students would prefer a step by step method, or an algorithm that they could apply, so that they would be able to get an answer and move on. I hope that throughout my career as a teacher I will have the opportunity to force feed an appreciation for proper mathematics into at least a few of my students.
I found this assignment to be quite interesting and useful to me. In particular, John Yamamoto’s input went a long way toward alleviating some of my concerns about specific aspects of teaching. Listening to the presentations of other groups was equally illuminating. Many groups asked their teachers intelligent and specific questions regarding how much homework to assign, or how to encourage discovery and discussion. These responses furnished me with some rough ideas which could help me during my first experiences in the classroom. Additionally, I noticed that there was a wide variety of different opinions represented among the surveyed teachers. Some seemed quite authoritarian, and they approached teaching in a very instrumental way; however, I got the impression that most teachers utilized a mixture of approaches, and they aimed to build rapport and appeal to different learning styles. On the other hand, I was slightly dismayed by the responses from the surveyed students. It seemed as though very few of them had any appreciation for mathematics beyond its use as a calculation tool. In general, I left with the feeling that most students would prefer a step by step method, or an algorithm that they could apply, so that they would be able to get an answer and move on. I hope that throughout my career as a teacher I will have the opportunity to force feed an appreciation for proper mathematics into at least a few of my students.
Conversation Summaries
Conversation Summaries
My group consisted of Rory, Amelia and myself. For our conversation, we collaborated to come up with a list of ten questions; five for a student, and five for a teacher. The student we interviewed was my sister, a 10th grade math student, and John Yamamoto was kind enough to participate during the teacher portion. For the teacher portion, our questions were:
1) What did you find to be your biggest challenges with your early teaching experiences?
John indicated that, initially, he had the greatest trouble managing his classroom effectively. He indicated that he would “work around the chatter,” rather than clamping down on it. His recommendation for dealing with this problem was to set very clear expectations, and be consistent about enforcing them.
2) What accommodations have you made to help students with learning difficulties?
With regard to students with learning difficulties, John stated that his work with deaf students at Burnaby South Secondary made him very aware of the pacing of his lessons. Most challenging for John, were the students who spoke little to no English, and he said that, if possible, he would liked to have had more experience working with ESL kids. Most importantly, John stated that it was necessary to be patient and mindful of kids’ histories, as “their problems [in the classroom] often have nothing to do with you.
3) How do you engage students with low motivation?
Here, John emphasized the importance of using humour to break down barriers. Also, he said that it was necessary to acknowledge that math was not their favourite subject, but hopefully they would learn to enjoy it a little bit. Additionally, John made a habit of taking an interest in his students’ interests, in order to relate to them more easily and build rapport. His main goal with these kids was to get them to come to class.
4) How do you vary your explanations when students have difficulty understanding?
Here, John’s main message was simply: “don’t worry about it too much.” Occasionally he would make use of colleagues when stuck with a particular lesson plan, or utilize the more gifted students to help collaborate on a different explanation. Mainly, he indicated that getting comfortable with different approaches to lesson planning was something that came with experience.
5) What do you enjoy most and least about teaching?
John said that it was amazing to form relationships with the students and to watch them grow, but it was often very difficult to see them leave and move on to new phases in life. Also, while he had some complaints regarding how certain topics were presented, and unbalanced curricula, he said that the good vastly outweighed the bad.
I asked my sister the next set of questions in a conversation over the phone:
1) Why do you think it is important to teach math in school?
After some initial uncertainty, she said that she felt math was important for many future careers, and that it helps to improve your reasoning power.
2) How do you develop your first impressions of a teacher?
She said that she judges mainly based on the teacher’s sense of humour and the level of organization in the classroom. She feels that a highly organized classroom is indicative of a strict teacher.
3) Think of a math lesson that you found particularly memorable. What made it unique?
In 7th grade, she had a student teacher give a lesson about Pi by dressing as a chef and feeding the class apple pie.
4) How would you feel about incorporating more group activities in your math class?
Initially, her reaction was quite negative. She indicated that explanations from other students often only contributed to her confusion. She did, however, seem open to the idea of doing group projects.
5) What is an effective way that a teacher has helped you to understand a tricky concept?
In 8th grade, her math instructor explained equation solving using a unique analogy involving negatives being bad and getting sent to the basement until they come out positive on the other side.
Overall, we found the experience to be quite positive and illuminating.
My group consisted of Rory, Amelia and myself. For our conversation, we collaborated to come up with a list of ten questions; five for a student, and five for a teacher. The student we interviewed was my sister, a 10th grade math student, and John Yamamoto was kind enough to participate during the teacher portion. For the teacher portion, our questions were:
1) What did you find to be your biggest challenges with your early teaching experiences?
John indicated that, initially, he had the greatest trouble managing his classroom effectively. He indicated that he would “work around the chatter,” rather than clamping down on it. His recommendation for dealing with this problem was to set very clear expectations, and be consistent about enforcing them.
2) What accommodations have you made to help students with learning difficulties?
With regard to students with learning difficulties, John stated that his work with deaf students at Burnaby South Secondary made him very aware of the pacing of his lessons. Most challenging for John, were the students who spoke little to no English, and he said that, if possible, he would liked to have had more experience working with ESL kids. Most importantly, John stated that it was necessary to be patient and mindful of kids’ histories, as “their problems [in the classroom] often have nothing to do with you.
3) How do you engage students with low motivation?
Here, John emphasized the importance of using humour to break down barriers. Also, he said that it was necessary to acknowledge that math was not their favourite subject, but hopefully they would learn to enjoy it a little bit. Additionally, John made a habit of taking an interest in his students’ interests, in order to relate to them more easily and build rapport. His main goal with these kids was to get them to come to class.
4) How do you vary your explanations when students have difficulty understanding?
Here, John’s main message was simply: “don’t worry about it too much.” Occasionally he would make use of colleagues when stuck with a particular lesson plan, or utilize the more gifted students to help collaborate on a different explanation. Mainly, he indicated that getting comfortable with different approaches to lesson planning was something that came with experience.
5) What do you enjoy most and least about teaching?
John said that it was amazing to form relationships with the students and to watch them grow, but it was often very difficult to see them leave and move on to new phases in life. Also, while he had some complaints regarding how certain topics were presented, and unbalanced curricula, he said that the good vastly outweighed the bad.
I asked my sister the next set of questions in a conversation over the phone:
1) Why do you think it is important to teach math in school?
After some initial uncertainty, she said that she felt math was important for many future careers, and that it helps to improve your reasoning power.
2) How do you develop your first impressions of a teacher?
She said that she judges mainly based on the teacher’s sense of humour and the level of organization in the classroom. She feels that a highly organized classroom is indicative of a strict teacher.
3) Think of a math lesson that you found particularly memorable. What made it unique?
In 7th grade, she had a student teacher give a lesson about Pi by dressing as a chef and feeding the class apple pie.
4) How would you feel about incorporating more group activities in your math class?
Initially, her reaction was quite negative. She indicated that explanations from other students often only contributed to her confusion. She did, however, seem open to the idea of doing group projects.
5) What is an effective way that a teacher has helped you to understand a tricky concept?
In 8th grade, her math instructor explained equation solving using a unique analogy involving negatives being bad and getting sent to the basement until they come out positive on the other side.
Overall, we found the experience to be quite positive and illuminating.
Wednesday, September 23, 2009
Robinson Article Response
Robinson Article: Reflection/Response
I quite enjoyed this article because it gave some rough guidelines describing ways to go from an adequate math instructor, to an instructor who inspires. Robinson’s initial approach to teaching is reminiscent of just about every high school math teacher I’ve known. They are burdened by content – laden curricula, and they feel forced to teach in a very instrumental, skill based manner. Unfortunately, this leads to a situation where students are virtually unable to think critically, and they have great difficulty adapting their learning to unusual problems. This is precisely what Robinson experienced in her class. Initially, I was skeptical about the effectiveness of incorporating the types of activities suggested by Robinson. The only experience I have with group work in a math class is using integer tiles to solve worksheets in Math 8. I found this to be a colossal bore, as I was fluent in integer operations and I saw no need to ‘waste my precious time’ with these ridiculous blocks. However, after reading Robinson’s description of how she set up her function jigsaw groups, I found my attitude toward the idea of collaborative activities to be somewhat improved. I think I would be somewhat nervous about implementing such a strategy myself, until I had a firmer grasp of how to achieve all the PLOs for a given course; however, I would be interested to try it after becoming more acclimatized to my role as a teacher.
I quite enjoyed this article because it gave some rough guidelines describing ways to go from an adequate math instructor, to an instructor who inspires. Robinson’s initial approach to teaching is reminiscent of just about every high school math teacher I’ve known. They are burdened by content – laden curricula, and they feel forced to teach in a very instrumental, skill based manner. Unfortunately, this leads to a situation where students are virtually unable to think critically, and they have great difficulty adapting their learning to unusual problems. This is precisely what Robinson experienced in her class. Initially, I was skeptical about the effectiveness of incorporating the types of activities suggested by Robinson. The only experience I have with group work in a math class is using integer tiles to solve worksheets in Math 8. I found this to be a colossal bore, as I was fluent in integer operations and I saw no need to ‘waste my precious time’ with these ridiculous blocks. However, after reading Robinson’s description of how she set up her function jigsaw groups, I found my attitude toward the idea of collaborative activities to be somewhat improved. I think I would be somewhat nervous about implementing such a strategy myself, until I had a firmer grasp of how to achieve all the PLOs for a given course; however, I would be interested to try it after becoming more acclimatized to my role as a teacher.
2 Memorable Teachers
Notes About Two Memorable Teachers
Mr. Norm Cheng (gr. 12 math teacher):
· Not the greatest teacher ever.
· Tended to be somewhat cranky, particularly in the mornings.
· Often made fun of his students, and was generally quite hilarious.
· Did a pretty good job of pointing out links between different concepts.
· Quite devoted to helping students succeed.
· Helped me teach myself calculus.
Dr. Joel Feldman (3rd year Real Analysis instructor):
· Possibly the most intelligent, well rounded math instructor in the history of math instruction.
· Genius at finding ways to explain very abstract concepts.
· Extremely challenging assignments that forced you to think creatively.
· Very helpful, and cared about student success.
· Conveyed an aura of total competence and mastery of his subject.
Reflection:
I think I admire both of these men. Mr. Cheng was the first instructor to teach me the awesome power of mathematics, and he gave me my first glimpse at what the subject was really about. Dr. Feldman was not typical of an instructor in an honours course, because he treated all of his students equally whether they were geniuses or not. Also, he would make time for you beyond the scope of his regular office hours. In addition to all this, he is a brilliant mathematician. The combination of genius and competent instructor is exceedingly rare, but that is exactly what Joel Feldman is. In my teaching, I would strive to be similar to both of these men by being considerate of my students’ needs, and encouraging them to think in creative ways.
Mr. Norm Cheng (gr. 12 math teacher):
· Not the greatest teacher ever.
· Tended to be somewhat cranky, particularly in the mornings.
· Often made fun of his students, and was generally quite hilarious.
· Did a pretty good job of pointing out links between different concepts.
· Quite devoted to helping students succeed.
· Helped me teach myself calculus.
Dr. Joel Feldman (3rd year Real Analysis instructor):
· Possibly the most intelligent, well rounded math instructor in the history of math instruction.
· Genius at finding ways to explain very abstract concepts.
· Extremely challenging assignments that forced you to think creatively.
· Very helpful, and cared about student success.
· Conveyed an aura of total competence and mastery of his subject.
Reflection:
I think I admire both of these men. Mr. Cheng was the first instructor to teach me the awesome power of mathematics, and he gave me my first glimpse at what the subject was really about. Dr. Feldman was not typical of an instructor in an honours course, because he treated all of his students equally whether they were geniuses or not. Also, he would make time for you beyond the scope of his regular office hours. In addition to all this, he is a brilliant mathematician. The combination of genius and competent instructor is exceedingly rare, but that is exactly what Joel Feldman is. In my teaching, I would strive to be similar to both of these men by being considerate of my students’ needs, and encouraging them to think in creative ways.
Monday, September 21, 2009
Micro Teaching: Summary & Reflection
Peer Evaluation Summary:
I found my classmates to be very receptive participants in my activity. They enjoyed my introductory anecdotes, saying that they were useful and provided a clear motivation for the specific technique I demonstrated. Additionally, they wrote that the practice with partners of disparate sizes was a good way to display the effectiveness of leverage and joint manipulation when neutralizing an opponent. They also indicated that I did a good job of monitoring their progress throughout the activity, and correcting their form as necessary. Although they found the initial technique to be too complicated, they were pleased at my ability to adapt the lesson to their skill levels. Some wrote that they would like to have seen some different ways to apply the technique, although they understood that the time limit was quite constraining. Finally, they stated that I needed to focus on improving the organization of my summary.
Self Evaluation and Reflection:
Overall, I thought the activity was carried out successfully. I felt like my colleagues understood the main points I was trying to get across; namely, some principles of self defense, and the idea of using joint locks to overcome a physically superior attacker. I was quite concerned by the difficulty my classmates were having with the technique I initially taught, but I felt like I did an effective job of modifying the lesson based on their lack of martial arts experience. Effectively summarizing a lesson is an area of my teaching that I have been working to improve, and I was not surprised that my classmates pointed it out. Also, I had wanted to demonstrate one or two other ways to apply the technique I showed, and with better time management I feel like this could have been accomplished. All in all, I’d give myself a B+.
I found my classmates to be very receptive participants in my activity. They enjoyed my introductory anecdotes, saying that they were useful and provided a clear motivation for the specific technique I demonstrated. Additionally, they wrote that the practice with partners of disparate sizes was a good way to display the effectiveness of leverage and joint manipulation when neutralizing an opponent. They also indicated that I did a good job of monitoring their progress throughout the activity, and correcting their form as necessary. Although they found the initial technique to be too complicated, they were pleased at my ability to adapt the lesson to their skill levels. Some wrote that they would like to have seen some different ways to apply the technique, although they understood that the time limit was quite constraining. Finally, they stated that I needed to focus on improving the organization of my summary.
Self Evaluation and Reflection:
Overall, I thought the activity was carried out successfully. I felt like my colleagues understood the main points I was trying to get across; namely, some principles of self defense, and the idea of using joint locks to overcome a physically superior attacker. I was quite concerned by the difficulty my classmates were having with the technique I initially taught, but I felt like I did an effective job of modifying the lesson based on their lack of martial arts experience. Effectively summarizing a lesson is an area of my teaching that I have been working to improve, and I was not surprised that my classmates pointed it out. Also, I had wanted to demonstrate one or two other ways to apply the technique I showed, and with better time management I feel like this could have been accomplished. All in all, I’d give myself a B+.
Thursday, September 17, 2009
Microteaching #1 - Wrist Grab Defense
Basic Single Wrist Grab Defense:
Symmetric Stance
Bridge:
First, I will survey my students to find out what prior martial arts experience they have. Also, I will explain that in feudal Japan it was common to attempt to prevent an opponent from drawing his weapon by seizing his wrist. As a result, many basic techniques in traditional styles of Japanese Jiu-Jitsu are initiated by wrist grabs.
Teaching Objectives:
· To improve my comfort level when teaching kinesthetically, and when supervising groups.
· To demonstrate that effective application of a technique can help compensate for physical limitations based on size, strength, etc.
· To introduce my students to some basic principles of self defense.
Learning Objectives:
Students will demonstrate an understanding of the historical motivation behind studying a wrist grab defense.
Students will display a basic understanding of how to utilize leverage and joint manipulation to overcome a physically superior adversary.
Pretest: To be done during bridge phase.
Participatory:
I will demonstrate the technique, and students will partner up into groups of 2-3 to practice with one another. I will encourage students to practice with someone similar to themselves in size and strength (group composition permitting).
Post test:
I will switch up the practice groups so that students can practice with opponents who have a significant size or strength difference, in order to help demonstrate the effectiveness of a properly applied technique.
Summary:
Give a quick rerun of the technique, and a brief glimpse at some similar defenses utilizing the same technique.
Symmetric Stance
Bridge:
First, I will survey my students to find out what prior martial arts experience they have. Also, I will explain that in feudal Japan it was common to attempt to prevent an opponent from drawing his weapon by seizing his wrist. As a result, many basic techniques in traditional styles of Japanese Jiu-Jitsu are initiated by wrist grabs.
Teaching Objectives:
· To improve my comfort level when teaching kinesthetically, and when supervising groups.
· To demonstrate that effective application of a technique can help compensate for physical limitations based on size, strength, etc.
· To introduce my students to some basic principles of self defense.
Learning Objectives:
Students will demonstrate an understanding of the historical motivation behind studying a wrist grab defense.
Students will display a basic understanding of how to utilize leverage and joint manipulation to overcome a physically superior adversary.
Pretest: To be done during bridge phase.
Participatory:
I will demonstrate the technique, and students will partner up into groups of 2-3 to practice with one another. I will encourage students to practice with someone similar to themselves in size and strength (group composition permitting).
Post test:
I will switch up the practice groups so that students can practice with opponents who have a significant size or strength difference, in order to help demonstrate the effectiveness of a properly applied technique.
Summary:
Give a quick rerun of the technique, and a brief glimpse at some similar defenses utilizing the same technique.
Wednesday, September 16, 2009
Skemp Reflection
Relational vs. Instrumental Understanding
After reading Skemp’s article, I am convinced of the general superiority of relational teaching in mathematics instruction. That being said, I believe there are some areas where an instrumental approach can have certain benefits.
Skemp initially describes instrumental knowledge as “rules without reasons.”
The first effect this statement had on me was to remind me of the vast majority of my experience as a math student. Virtually all of my teachers throughout high school taught mathematics in a very rigid, mechanical way, with little thought to deeper analysis or explication.
The second quotation which stood out to me was Skemp’s statement that some students just want “some kind of rule for getting the answer.”
What this reminded me of was some of the experiences I have had as an instructor at Sylvan. Many of my students have had learning difficulties in mathematics. The simple ‘rule to get the answer’ method is helpful for them, as it helps improve their confidence by making them feel successful. Unfortunately, I believe this approach also promotes intellectual laziness and the tendency to look for an ‘easy button’ whenever possible.
“… instrumental understanding […] usually involves a multiplicity of rules rather than fewer principles of more general application.”
Upon reading this, I thought of the analogy of a tool box. Instrumental teaching requires you to carry around a giant box, with many specialized tools. Conversely, an instructional method which promotes relational understanding lets you carry around a lighter box with fewer, more efficient tools.
Writing as a Devil’s Advocate in favour of instrumental instruction, Skemp writes: “They will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects …”
Upon initial inspection, I thought this point had some validity. However, after further reflection, I decided that it would be insulting to mathematicians everywhere. This would have us consider math as a mere tool for accomplishing tasks, rather than an independent discipline worthy of study in its own right.
The final quotation which struck me most was that relational instruction is “easier to remember.”
I agree wholeheartedly with this. Throughout my entire undergraduate career, I would always loathe to memorize formulae. Instead, I would learn to derive them, and pore over proofs in my notes until I fully understood how and why they worked. The memorization would then occur almost by accident.
After reading Skemp’s article, I am convinced of the general superiority of relational teaching in mathematics instruction. That being said, I believe there are some areas where an instrumental approach can have certain benefits.
Skemp initially describes instrumental knowledge as “rules without reasons.”
The first effect this statement had on me was to remind me of the vast majority of my experience as a math student. Virtually all of my teachers throughout high school taught mathematics in a very rigid, mechanical way, with little thought to deeper analysis or explication.
The second quotation which stood out to me was Skemp’s statement that some students just want “some kind of rule for getting the answer.”
What this reminded me of was some of the experiences I have had as an instructor at Sylvan. Many of my students have had learning difficulties in mathematics. The simple ‘rule to get the answer’ method is helpful for them, as it helps improve their confidence by making them feel successful. Unfortunately, I believe this approach also promotes intellectual laziness and the tendency to look for an ‘easy button’ whenever possible.
“… instrumental understanding […] usually involves a multiplicity of rules rather than fewer principles of more general application.”
Upon reading this, I thought of the analogy of a tool box. Instrumental teaching requires you to carry around a giant box, with many specialized tools. Conversely, an instructional method which promotes relational understanding lets you carry around a lighter box with fewer, more efficient tools.
Writing as a Devil’s Advocate in favour of instrumental instruction, Skemp writes: “They will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects …”
Upon initial inspection, I thought this point had some validity. However, after further reflection, I decided that it would be insulting to mathematicians everywhere. This would have us consider math as a mere tool for accomplishing tasks, rather than an independent discipline worthy of study in its own right.
The final quotation which struck me most was that relational instruction is “easier to remember.”
I agree wholeheartedly with this. Throughout my entire undergraduate career, I would always loathe to memorize formulae. Instead, I would learn to derive them, and pore over proofs in my notes until I fully understood how and why they worked. The memorization would then occur almost by accident.
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