Our team micro teaching project went well. Mina and Sam were excellent teaching partners and it was a well planned lesson (we are thankful to Sam for all his hard work). We combined three basic portions: Mina gave an introduction, Sam furthered this with a proof and further explanation and I used bouncing tennis balls to show a real world example. I believe most of our 'students' enjoyed the presentation and were able to follow the reasoning (see feedback to follow). Our weakest element was our use of time. We went longer than we had planned to and so felt rushed. It was great experience, though, and has given us good exerience teaching this subject in front of math students.
Averages on Feedback
Clarity
Structure was clear - 4.14
Verbal & Visual were clear - 4.07
Math ideas were clear - 4.14
Active Learning
I was engaged in learning - 3.57
A variety of activities offered - 3.64
Instructors showed engagement was valued - 3.93
Offered activities connected to other math areas - 3.93
Connected to other areas of life and culture - 3.43
Some comments included: good classroom management, very engaging, clear instructions, good humour, clear review, concise, fun activity,good easy examples, it's good to have a real life problem at the end, very well organized.
Also, on the not so positive: bit much time in review, short on time, the board was blocked (couldn't see), wait for students to respond, little long and drawn out, the pace is too fast, needed examples of geometric series at first, define terms clearly, relax when speaking.
Friday, October 16, 2009
Wednesday, October 14, 2009
Geometric Series
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.
thanks to Sam Douglas for this lesson plan
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.
thanks to Sam Douglas for this lesson plan
Citizenship and Democracy Finding a Place in the Mathematics Classroom
Probably one of the last things the average person thinks about when considering what should be taught in the mathematics classroom is social issues. There are so many branches of the tree that make up teachable mathematics, how is it possible to include citizenship education amongst them? Elaine Simmt makes the argument that it is possible and should be done.
Simmt argues that because numbers and statistics are so vital and interlocked into our daily lives, it would be irresponsible of our leaders at all levels not to incorporate the teaching of them to all who enter school. She suggests, “mathematics education and citizenship education need not be distinct tasks of the teacher; rather, appropriate mathematics teaching also prepares the student for citizenship”.
While a noble cause and one that most involved with mathematics would think an excellent new use for the age-old study, this is quite difficult in practise. Simmt does not suggest, as the reader might originally think, that debate and direct tackling of social issues is what should be taught. She argues instead that slight changes should be made in the way that teachers teach the same things they have always taught. This in turn will better prepare the future citizens of the world. It is a great goal that should be welcomed by all math teachers, regardless of how improbable it seems.
Simmt argues that because numbers and statistics are so vital and interlocked into our daily lives, it would be irresponsible of our leaders at all levels not to incorporate the teaching of them to all who enter school. She suggests, “mathematics education and citizenship education need not be distinct tasks of the teacher; rather, appropriate mathematics teaching also prepares the student for citizenship”.
While a noble cause and one that most involved with mathematics would think an excellent new use for the age-old study, this is quite difficult in practise. Simmt does not suggest, as the reader might originally think, that debate and direct tackling of social issues is what should be taught. She argues instead that slight changes should be made in the way that teachers teach the same things they have always taught. This in turn will better prepare the future citizens of the world. It is a great goal that should be welcomed by all math teachers, regardless of how improbable it seems.
Friday, October 9, 2009
What if Not?
The Strengths of this method are seemingly quite obvious, despite being seldom used in ‘real world’ problem solving. This strategy enables the problem poser to consider a great many alternative situations to the current problem, in order to see the current problem in the clearest possible light. As with any tangible object, if one can pick it up and rotate and flip it and hold it in different shades of light, one is going to know much more about what it looks like than if it were simply stared at from across the room.
There are unfortunately limitations to this strategy. Time constraints can become a factor if a multitude of other situations are considered instead of focussing on the job at hand. Confusion and loss of focus can result for many people if they delve too deeply into other explorations. Some ‘what if not’ questions, as can be seen with some of the examples in the book, are simply a waste of time and do not help with the solving of a problem at all. Another limitation to consider is that many problem solvers may be thrown off their ‘rhythm’ by allowing their minds to wander around a problem.
In our microteaching assignment for Wednesday, we have been asked to incorporate the ‘what if not’ strategy. I think this will be a fun a way to ‘teach’ the concept of Arithmetic Series, especially considering the students will already know the concept. Questions come to mind such as “what does it mean for a series to be arithmetic and what would it look like if it were not arithmetic?”, “what if we could not generalize an arithmetic series?”, “what would a series look like that had no sum?” It should be a fun and thought provoking way to introduce a concept.
There are unfortunately limitations to this strategy. Time constraints can become a factor if a multitude of other situations are considered instead of focussing on the job at hand. Confusion and loss of focus can result for many people if they delve too deeply into other explorations. Some ‘what if not’ questions, as can be seen with some of the examples in the book, are simply a waste of time and do not help with the solving of a problem at all. Another limitation to consider is that many problem solvers may be thrown off their ‘rhythm’ by allowing their minds to wander around a problem.
In our microteaching assignment for Wednesday, we have been asked to incorporate the ‘what if not’ strategy. I think this will be a fun a way to ‘teach’ the concept of Arithmetic Series, especially considering the students will already know the concept. Questions come to mind such as “what does it mean for a series to be arithmetic and what would it look like if it were not arithmetic?”, “what if we could not generalize an arithmetic series?”, “what would a series look like that had no sum?” It should be a fun and thought provoking way to introduce a concept.
Monday, October 5, 2009
Take 10 on "The Art of Problem Posing"
This book begins with a witty and occasionally funny style of writing. I enjoyed the repeated fifth reason for why a repetition of key ideas is used throughout the book.
I question why the introduction needed to be so long and incorporate revisals from previous editions. Perhaps this could have been done as a seperate part of the book.
I like how the book can be enjoyed by all levels of mathematical ability.
I enjoyed the vast array of different questions that were given as possible topics of thought. Some I just may use in future problem solving.
I thought it was interesting to involve history as a possible exploration when thinking about problem posing.
I enjoyed the First Phase of Problem Posing being Accepting. I think this is the same first phase in addictions counceling.
I enjoyed each of the five examples for Sticking to the Given.
I found it interesting that two offbeat examples of questions were given at the end of the Supreme Court Judge handshake example.
I am looking forward to the second phase of problem posing.
I am still left wondering if being better at problem posing will help my abilities (and those of my students) in problem solving.
I question why the introduction needed to be so long and incorporate revisals from previous editions. Perhaps this could have been done as a seperate part of the book.
I like how the book can be enjoyed by all levels of mathematical ability.
I enjoyed the vast array of different questions that were given as possible topics of thought. Some I just may use in future problem solving.
I thought it was interesting to involve history as a possible exploration when thinking about problem posing.
I enjoyed the First Phase of Problem Posing being Accepting. I think this is the same first phase in addictions counceling.
I enjoyed each of the five examples for Sticking to the Given.
I found it interesting that two offbeat examples of questions were given at the end of the Supreme Court Judge handshake example.
I am looking forward to the second phase of problem posing.
I am still left wondering if being better at problem posing will help my abilities (and those of my students) in problem solving.
Friday, October 2, 2009
Two Future Letters
June 30, 2019
Dear Mr. Whitbread,
Thank you for the time you took to teach my Math class over the last school year. I began taking your class with a dislike for Math - I had had previous bad experiences. I'll never forget your first class where you encouraged us all to drop our fears of math and learn again to appreciate the beauty of the subject. You told us that most of us would never love everything about math, but all of us could learn to love certain aspects - much the same as all our other subjects. I liked that and it has stayed with me. I will always remember how I looked forward to your classes. Sure there were boring parts of classes, but you always gave me something that would make me think.
Dear Mr. Whitbread,
After spending a large number of hours in your classroom, I can't help but look back and think it was a large waste of time. It wasn't all your fault - math is just boring and I could never understand why we needed to learn it. I did everything I could to not fall asleep in class while you were working on examples. I really wish you hadn't called on us for examples and made us do group work. It made me feel stupider than I already had felt I was when I didn't know the answers to your questions. Why couldn't you just teach us like my previous teachers and make us learn things at home? It would have made me feel way more comfortable.
Maybe you can be a better teacher in the next ten years.
While I hope more letters conform to the first, I do expect some negative feedback as well. I know I am going to be teaching a subject that carries a lot of dislike by a great many students (and people in general), but I hope that my positive, good natured demeanor is able to win most over. I hope I can help those that already love mathematics to continue to learn its graces.
Dear Mr. Whitbread,
Thank you for the time you took to teach my Math class over the last school year. I began taking your class with a dislike for Math - I had had previous bad experiences. I'll never forget your first class where you encouraged us all to drop our fears of math and learn again to appreciate the beauty of the subject. You told us that most of us would never love everything about math, but all of us could learn to love certain aspects - much the same as all our other subjects. I liked that and it has stayed with me. I will always remember how I looked forward to your classes. Sure there were boring parts of classes, but you always gave me something that would make me think.
Dear Mr. Whitbread,
After spending a large number of hours in your classroom, I can't help but look back and think it was a large waste of time. It wasn't all your fault - math is just boring and I could never understand why we needed to learn it. I did everything I could to not fall asleep in class while you were working on examples. I really wish you hadn't called on us for examples and made us do group work. It made me feel stupider than I already had felt I was when I didn't know the answers to your questions. Why couldn't you just teach us like my previous teachers and make us learn things at home? It would have made me feel way more comfortable.
Maybe you can be a better teacher in the next ten years.
While I hope more letters conform to the first, I do expect some negative feedback as well. I know I am going to be teaching a subject that carries a lot of dislike by a great many students (and people in general), but I hope that my positive, good natured demeanor is able to win most over. I hope I can help those that already love mathematics to continue to learn its graces.
Dave Hewitt Video
Our MAED 314A class got to watch a short video featuring two lessons being taught by Dave Hewitt, a grade 8 and 9 Math teacher in the UK. The enjoyable part of watching Mr. Hewitt teach is that he seems to make the learning very enjoyable for his students. There is a definite theatrical style incorporated into his teaching method that seems to hold the interest of everyone in the classroom, and those watching the video. The idea of using a meter stick to make a tapping sound and to use space on the walls beyond the chalkboard are two tools that he uses to keep the students engaged and actually learning in a non-boring, ‘outside the box’ way. Once he has established the students basic understanding of how numbers fit on the number line, he begins to introduce algebra without anyone realizing variables have been introduced. When many math students see their first x or y as part of a mathematics problem a wave a fear goes over. The students in Mr. Hewitt’s class had nothing to fear when he was simply “thinking of a number that…” When an x was then introduced, it was a seamless progression. It is entertaining and engaging methods such as those of Mr. Hewitt that I think would make the Secondary Mathematics classroom a much more enjoyable place for so many young minds and I hope to teach in that classroom.
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