Geometric Series: Finite and Infinite cases
Bridge/Pretest: Recall the geometric sequence (consecutive terms have a constant ratio). Recall that a series is the sum of a geometric sequence. Give students the bouncy balls for initial investigation.
Learning Outcomes: Understand connection between sequences and series. Be able to compute finite and infinite sums. Begin to formulate an understanding of series convergence.
Teaching Outcomes: Incorporate multiple teaching modes. Prove the formula for calculating a geometric series.
Participatory: Have students bounce different balls. Investigate how successive bounces decrease in size. Lead into the idea that the heights of consecutive bounces form a constant ration = geometric progression. Discuss how to add the total length of 20 (or so) bounces. Prove formula for sum of geometric series. Raise the question: ‘what if the ball never stops bouncing, but continues forever with each subsequent bounce scaled by the same ratio?’ Discuss how powers of r, for r < 1, will approach zero if the exponent is sufficiently large. Segue into infinite series, and display how the formula for the sum will be modified.
Post-Assessment: Have students break into small groups (3-4) to collaborate to complete a problem set. Make some problems tricky by manipulating indices.
Summary: Recap relationship between sequences and series. Discuss infinite series, and the issue of convergence.
Wednesday, October 14, 2009
Citizenship Reflection
Citizenship Education in Mathematics: Reflection
I found Simmt’s article quite interesting, and I felt that it correlated quite well with the previous readings on problem posing. I also appreciated the list of goals for math education, as it highlights the importance of mathematics as a part of cultural heritage and a way to self-empower. I also found the description of mathematical modeling to be quite interesting. The way Simmt describes the ‘formatting power’ of mathematics, made it seem as though the relationship between math and societal phenomena is more intertwined than one might think. The image which came to my mind was of a feedback loop driven by information, where output of a model is utilized and reconfigured into new input. Although teaching to promote responsible citizenship has not been central to my thinking as a mathematics instructor, I do feel that attempting to mold young people into positive members of society is a sacred and important duty. Additionally, I like the idea of creating a classroom environment where students have the opportunity to collaborate and pose their own problems. In this way, students can feel like they are more involved in their personal learning processes. Moreover, by emphasizing the importance of multiple solution methods, students will be forced to reexamine their hypotheses and look at problems from different points of view. In general, I feel like education for citizenship in a math class would not be accomplished in an obvious way. Rather, it would be accomplished by stealth, and by properly manipulating the learning environment.
I found Simmt’s article quite interesting, and I felt that it correlated quite well with the previous readings on problem posing. I also appreciated the list of goals for math education, as it highlights the importance of mathematics as a part of cultural heritage and a way to self-empower. I also found the description of mathematical modeling to be quite interesting. The way Simmt describes the ‘formatting power’ of mathematics, made it seem as though the relationship between math and societal phenomena is more intertwined than one might think. The image which came to my mind was of a feedback loop driven by information, where output of a model is utilized and reconfigured into new input. Although teaching to promote responsible citizenship has not been central to my thinking as a mathematics instructor, I do feel that attempting to mold young people into positive members of society is a sacred and important duty. Additionally, I like the idea of creating a classroom environment where students have the opportunity to collaborate and pose their own problems. In this way, students can feel like they are more involved in their personal learning processes. Moreover, by emphasizing the importance of multiple solution methods, students will be forced to reexamine their hypotheses and look at problems from different points of view. In general, I feel like education for citizenship in a math class would not be accomplished in an obvious way. Rather, it would be accomplished by stealth, and by properly manipulating the learning environment.
Friday, October 9, 2009
"What-if-Not" reflection
At the risk of sounding somewhat informal, I think that the ‘what-if-not’ strategy outlined in The Art of Problem Posing is really cool. From my initial reading of it, the strategy basically involves analyzing an existing problem, definition or object, making a list of its characteristics, and then omitting certain of them to see what might happen. I think this is a fabulous way to stimulate creative thinking among math students, and I already have some ideas about how I can apply it to the micro-teaching exercise next week. The idea I had for the presentation was to demonstrate a transition from arithmetic sequences to arithmetic series. I plan to do this by posing the problem of how to calculate the sum of the first 100 natural numbers, extending that to the sum of the first n natural numbers, and then applying the results to find the sum of a sequence of the form t(n) = a + (n-1)d. Basically, my plan is to ‘what-if-not’ two characteristics of the sum 1+2+3+…+ 100. Namely, ‘what if the sum does not end at 100, but continues to an arbitrary number,’ and ‘what if the initial term and difference are not 1.’ I think that ‘what-if-not’-ing is an excellent way to promote original and creative thinking. Particularly, I think it provides a great opportunity for teachers and students to collaborate in ways that could foster an atmosphere of discovery in a math class. From a teacher’s point of view, I think that the ‘what-if-not’ strategy is an effective method for generating multiple problems from a single mathematical object. Most importantly, the strategy allows students to experience discovery, which might lead them to be more curious about mathematics. One potential pitfall of using this strategy is that it seems like it might require quite a bit of effort on the part of the student. This is fine for more capable students, but if a student has a poor relationship with mathematics, implementation of the ‘what-if-not’ strategy may require a great deal of teacher intervention. Aside from this concern, I think ‘what-if-not’-ing is a strategy which I could use to great effect.
Monday, October 5, 2009
Art of Problem Posing: Entry #1
There were several aspects of this first reading that I found interesting:
1) I was pleased to see the description of problem posing as a method of inquiry, as it relates to some of the work being done in my other mathematics education course.
2) I have some reservations about how to directly apply this philosophy to math teaching in general.
3) I think that inquiry via problem solving is a useful tool for teaching, but I believe it would need to be supplemented with some conventional (ie: lecture based) methods.
4) I enjoyed how the author describes the significance of a problem as being proportional to its ingenuity and playfulness.
5) By promoting this idea of playfulness, I think it would be much easier to promote a sense of curiosity about mathematics.
6) Also, encouraging students to play around with problems and modify them could go a long way toward combating math phobia.
7) I was also very pleased to see the emphasis placed on shifting contexts and perspectives.
8) This idea of shifting perspectives was also brought up in our other math ed. class as a way of fostering a deeper understanding of a concept.
9) I am curious about the idea of internal vs. external thinking with regard to problem posing, and I hope to see this idea expanded upon in the later sections of the book.
10) I was also interested to read about how lay persons often have an easier time of creating interesting problems than those with more formal math training. It would seem to coincide with Ken Robinson's idea about formal education having a destructive influence on creativity.
1) I was pleased to see the description of problem posing as a method of inquiry, as it relates to some of the work being done in my other mathematics education course.
2) I have some reservations about how to directly apply this philosophy to math teaching in general.
3) I think that inquiry via problem solving is a useful tool for teaching, but I believe it would need to be supplemented with some conventional (ie: lecture based) methods.
4) I enjoyed how the author describes the significance of a problem as being proportional to its ingenuity and playfulness.
5) By promoting this idea of playfulness, I think it would be much easier to promote a sense of curiosity about mathematics.
6) Also, encouraging students to play around with problems and modify them could go a long way toward combating math phobia.
7) I was also very pleased to see the emphasis placed on shifting contexts and perspectives.
8) This idea of shifting perspectives was also brought up in our other math ed. class as a way of fostering a deeper understanding of a concept.
9) I am curious about the idea of internal vs. external thinking with regard to problem posing, and I hope to see this idea expanded upon in the later sections of the book.
10) I was also interested to read about how lay persons often have an easier time of creating interesting problems than those with more formal math training. It would seem to coincide with Ken Robinson's idea about formal education having a destructive influence on creativity.
Friday, October 2, 2009
2019: 10 years gone
Happy student:
Mr. Douglas always encouraged creativity and critical thinking. I felt that this was the aspect of his teaching that impacted me the most. He wasn't always the nicest guy, but you could tell that he cared deeply about the success of his students. Moreover, he tried to foster a sense of curiosity in his students, and this helped a great deal in maintaining my motivation.
Unhappy student:
Mr. Douglas was the worst teacher I ever had. I always dreaded going to class, because I knew it would just end up being 70 minutes of either boredom or anxiety. I felt like there was not enough variety in the structure of the lessons, and he seemed to teach more to the level of the best students without any concern for those of us who struggled.
Reflection: I think this is a pretty good indicator of some of the things I'm concerned with regarding my life as a teacher. The success of my students is very important to me, but I know it will be impossible to reach every single one. I suppose it will become an exercise in optimization.
Mr. Douglas always encouraged creativity and critical thinking. I felt that this was the aspect of his teaching that impacted me the most. He wasn't always the nicest guy, but you could tell that he cared deeply about the success of his students. Moreover, he tried to foster a sense of curiosity in his students, and this helped a great deal in maintaining my motivation.
Unhappy student:
Mr. Douglas was the worst teacher I ever had. I always dreaded going to class, because I knew it would just end up being 70 minutes of either boredom or anxiety. I felt like there was not enough variety in the structure of the lessons, and he seemed to teach more to the level of the best students without any concern for those of us who struggled.
Reflection: I think this is a pretty good indicator of some of the things I'm concerned with regarding my life as a teacher. The success of my students is very important to me, but I know it will be impossible to reach every single one. I suppose it will become an exercise in optimization.
Unorthodox Teaching Video
Dave Hewitt Video – Reflection
I found this film to be an interesting example of unorthodox pedagogical techniques in action, and I must readily admit that I would never have thought to engage an entire class in the way that Mr. Hewitt does. By focusing the entire class on the task of experimenting with a single skill, Mr. Hewitt accomplishes several things at once. In particular, he assesses the initial knowledge level of his students, fosters a participatory environment, and ensures that the students’ knowledge is expanded in a logical and organic manner. I was quite impressed with how Mr. Hewitt accomplished this last task. In one specific example, he segued from a discussion involving integers to a discussion of algebraic expressions, and he did it in a very sneaky way by simply stating that “I’ll just pretend this ‘x’ is like 15. I’ll put a number there later.” In all of my teaching experiences, the introduction of variables is almost always a dreadful shock to students. It was very interesting to see this task accomplished in such an efficient and effective way, and all without spooking the kids. I did have two minor reservations to Mr. Hewitt’s teaching style, and they stem from my own preferences as a student. Generally, I prefer lectures with a quicker rate of information delivery, so the pacing of Mr. Hewitt’s lessons might seem glacial and agonizing to a more capable student. Additionally, while I am impressed by his use of tactile/auditory teaching methods, it would be detrimental to cater exclusively to these styles of learners. Aside from these concerns, I found the video to be a good demonstration of an intriguing and innovative way to teach mathematics.
I found this film to be an interesting example of unorthodox pedagogical techniques in action, and I must readily admit that I would never have thought to engage an entire class in the way that Mr. Hewitt does. By focusing the entire class on the task of experimenting with a single skill, Mr. Hewitt accomplishes several things at once. In particular, he assesses the initial knowledge level of his students, fosters a participatory environment, and ensures that the students’ knowledge is expanded in a logical and organic manner. I was quite impressed with how Mr. Hewitt accomplished this last task. In one specific example, he segued from a discussion involving integers to a discussion of algebraic expressions, and he did it in a very sneaky way by simply stating that “I’ll just pretend this ‘x’ is like 15. I’ll put a number there later.” In all of my teaching experiences, the introduction of variables is almost always a dreadful shock to students. It was very interesting to see this task accomplished in such an efficient and effective way, and all without spooking the kids. I did have two minor reservations to Mr. Hewitt’s teaching style, and they stem from my own preferences as a student. Generally, I prefer lectures with a quicker rate of information delivery, so the pacing of Mr. Hewitt’s lessons might seem glacial and agonizing to a more capable student. Additionally, while I am impressed by his use of tactile/auditory teaching methods, it would be detrimental to cater exclusively to these styles of learners. Aside from these concerns, I found the video to be a good demonstration of an intriguing and innovative way to teach mathematics.
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