This is an interesting tool I plan on using when introducing trigonometry next semester in Principles of Math 10. Make sure to turn up your speakers so that you can hear the resultant frequency, which keeps the "pure tone" sound no matter what changes are made.
http://www.mindspring.com/~j.blackstone/dist101.htm
I figure that I'll let the kids play around with this program for 10min, then when I have their curiosity peaked, I'll discuss what we are actually seeing, and how sine waves appear in nature, such as earthquakes: http://www.classzone.com/books/earth_science/terc/content/visualizations/es1002/es1002page01.cfm?chapter_no=visualization .
Then I'll discuss how sine waves appear in mathematics: http://www.youtube.com/watch?v=HFOPb7Ub1e8
I figure this'll be better than when I learned it as a kid, where I never saw an application until Physics 12. ^_^
Wednesday, November 18, 2009
Monday, November 9, 2009
Sunday, November 8, 2009
Short Practicum Stories
STORY #1 - "Quiz Day"
One of my sponsor teachers, who teaches Physics 12 and Mathematics 9-12, told me a story of a time when his whole class was acting up. After multiple attempts to settle the children down, he informed them that he was officially declaring a "Quiz Day".
He handed out the first quiz, and instructed the children to stay seated and raise their hands when they had completed the quiz, and he would come to their desk to mark and record the quiz. When the first student finished his quiz, my sponsor teacher marked the quiz and recorded the mark, then handed the student his second quiz. When the first student finished the second quiz raised their hand, my sponsor teacher marked the quiz and recorded the mark again, then handed that student their third quiz.
By this time, students were asking my sponsor teacher what would happen if they didn't finish all three quizzes, to which he responded that he would give that student a "zero" mark for any quizzes that they didn't completely finish.
Word of my sponsor teacher's "Quiz Day" have become an urban legend in his school, and he has never had a class act up in the last few years.
STORY #2 - Volunteering in the Resource Room
I was volunteering in the Resource Room during my short practicum, and there was a girl that was struggling with solving single-variable algebra problems. I helped her understand the concept by teaching her using similar examples to the ones that she was assigned, and she seemed to clearly grasp the subject material. When I returned to help her a few minutes later, she was struggling with the homework assignment and getting very frustrated. When I checked her work, she had been incorrectly copying the problems from the textbook to her paper, inadvertently mixing up two questions and making her assignment questions multi-variable.
I worked with her to double-checking her work and correcting the copying errors, then reported to the full-time teacher in the Resource Room that the student's main struggle was incorrectly copying the homework questions into her book, and suggested that if the homework were assigned as a handout with room to solve the questions below each problem, she would not be having this problem. The Resource Room teacher said that he would mention this possibility to the mathematics teacher, and they would discuss it next week (after I finished my short practicum).
One of my sponsor teachers, who teaches Physics 12 and Mathematics 9-12, told me a story of a time when his whole class was acting up. After multiple attempts to settle the children down, he informed them that he was officially declaring a "Quiz Day".
He handed out the first quiz, and instructed the children to stay seated and raise their hands when they had completed the quiz, and he would come to their desk to mark and record the quiz. When the first student finished his quiz, my sponsor teacher marked the quiz and recorded the mark, then handed the student his second quiz. When the first student finished the second quiz raised their hand, my sponsor teacher marked the quiz and recorded the mark again, then handed that student their third quiz.
By this time, students were asking my sponsor teacher what would happen if they didn't finish all three quizzes, to which he responded that he would give that student a "zero" mark for any quizzes that they didn't completely finish.
Word of my sponsor teacher's "Quiz Day" have become an urban legend in his school, and he has never had a class act up in the last few years.
STORY #2 - Volunteering in the Resource Room
I was volunteering in the Resource Room during my short practicum, and there was a girl that was struggling with solving single-variable algebra problems. I helped her understand the concept by teaching her using similar examples to the ones that she was assigned, and she seemed to clearly grasp the subject material. When I returned to help her a few minutes later, she was struggling with the homework assignment and getting very frustrated. When I checked her work, she had been incorrectly copying the problems from the textbook to her paper, inadvertently mixing up two questions and making her assignment questions multi-variable.
I worked with her to double-checking her work and correcting the copying errors, then reported to the full-time teacher in the Resource Room that the student's main struggle was incorrectly copying the homework questions into her book, and suggested that if the homework were assigned as a handout with room to solve the questions below each problem, she would not be having this problem. The Resource Room teacher said that he would mention this possibility to the mathematics teacher, and they would discuss it next week (after I finished my short practicum).
Thursday, October 22, 2009
Division by Zero: A Poem
Disclaimer: I haven't written poetry in over 13 years, and I wasn't any good at it back then either.
Division by Zero
I have a proof that two equals one.
It's mathematical.
No, it's mathtastical.
I use it to challenge the students.
They'll check the algebra,
and question the logic.
But the secret should be obvious,
Division by zero.
Reflections
Timed writing and poetry as a pedagogical exercise helped me to reflect on how difficult mathematics may be for some children, since poetry and "free writing" are some of my weakest skills. Aside from putting me outside of my comfort zone, I do not see this as an effective tool for teaching mathematics or sciences, but I am open to further interpretation of this exercise by others.
Division by Zero
I have a proof that two equals one.
It's mathematical.
No, it's mathtastical.
I use it to challenge the students.
They'll check the algebra,
and question the logic.
But the secret should be obvious,
Division by zero.
Reflections
Timed writing and poetry as a pedagogical exercise helped me to reflect on how difficult mathematics may be for some children, since poetry and "free writing" are some of my weakest skills. Aside from putting me outside of my comfort zone, I do not see this as an effective tool for teaching mathematics or sciences, but I am open to further interpretation of this exercise by others.
Wednesday, October 14, 2009
Reflection on Kinemalgebratics!
MAED 314a-301 INTRODUCTION TO ALGEBRA THOUGH KINEMATICS (aka "KINEMALGEBRATICS")
I would initially like to take a moment and thank Amelia and Erwin for supervising the ball-rolling activity. Although I was managing the other group in their derivations of the kinematics equations and building their concept of algebra, I could hear how much fun university students were having with a simple activity, and it was overwhelmingly reflected in their reviews and comments. I take great pride in knowing that this mathematics activity could easily be the highlight of any grade 8 students day.
Also, many thanks to Erwin for his fantastic sign. That small detail was the highlight of my day. Not that today was a bad day... just otherwise uneventful. :P
As a mathematician, it would hurt me not to include a statistical analysis of my peer feedback, so I'll just get that out of the way now. Each topic is listed by category (Clarity, Active learning, and Connected mathematical ideas) and simply the assessment item number, to save space. I've included the average (AVG) and the standard deviation (STDEV), as well as my own personal assessment (MIKE).
CLARITY 1 >> AVG 4.36 >> STDEV 0.48 >> MIKE 4.00
CLARITY 2 >> AVG 4.68 >> STDEV 0.44 >> MIKE 4.00
CLARITY 3 >> AVG 4.80 >> STDEV 0.40 >> MIKE 4.00
ACTIVE 1 >> AVG 4.82 >> STDEV 0.39 >> MIKE 5.00
ACTIVE 2 >> AVG 4.73 >> STDEV 0.45 >> MIKE 5.00
ACTIVE 3 >> AVG 4.50 >> STDEV 0.67 >> MIKE 4.00
CONNECT 1 >> AVG 4.36 >> STDEV 0.48 >> MIKE 1.00
CONNECT 2 >> AVG 4.45 >> STDEV 0.66 >> MIKE 3.00
My personal assessment reflects how well I think our group did on planning and executing this activity. I believe that every group member really helped to make this a fantastic group project and a great learning experience for the "students" in the classroom.
There should be two items that stand out immediately to the casual observer. First, I scored our group 1/5 for CONNECT 1 (The instructors offered activities that connected to other areas of math). I would like to explicitly state that although this is a "low" mark, I do not believe it represents a failure of any sort, but simply a narrow focus on a specific 15 minute lesson.
Second, I scored our group 3/5 for CONNECT 2 (The lesson connected to other areas of life & culture). I believe that given the 15 minutes, this is a sufficient undertaking. However, if we had been given 30 minutes, I would have liked to introduce more complicated algebra using kinematics (d = d0 + v*t). Simply put, these low scores are not a failure on the part of the group, but instead a reflection of the time constraint.
Next, I would like to address some of the more prevalent "constructive criticism" comments that were listed on the peer reviews. The first comment deals with coordination of the two groups. Unfortunately I ran a half minute behind on the first group, and the second group had to wait until we finished. Although this does not seem like very long, I'm sure it would be an eternity for a grade 8 student. If we were doing an activity like this in a class with multiple instructors, I think having an official "time keeper" to give 2min and 1min warnings would help everyone (read: Mike B.) to pace their lessons appropriately.
The second "constructive criticism" was that we should have better outlined our objectives to the class before beginning. I am still on the fence about this comment, since although I know that many students, myself included, prefer an ordered and clearly laid out lesson structure, I do believe that the world "Algebra" might invoke terror in young students, and I really do like the idea of teaching it implicitly without actually naming the technique until the students are comfortable with the material. Any thoughts on this matter?
I would initially like to take a moment and thank Amelia and Erwin for supervising the ball-rolling activity. Although I was managing the other group in their derivations of the kinematics equations and building their concept of algebra, I could hear how much fun university students were having with a simple activity, and it was overwhelmingly reflected in their reviews and comments. I take great pride in knowing that this mathematics activity could easily be the highlight of any grade 8 students day.
Also, many thanks to Erwin for his fantastic sign. That small detail was the highlight of my day. Not that today was a bad day... just otherwise uneventful. :P
As a mathematician, it would hurt me not to include a statistical analysis of my peer feedback, so I'll just get that out of the way now. Each topic is listed by category (Clarity, Active learning, and Connected mathematical ideas) and simply the assessment item number, to save space. I've included the average (AVG) and the standard deviation (STDEV), as well as my own personal assessment (MIKE).
CLARITY 1 >> AVG 4.36 >> STDEV 0.48 >> MIKE 4.00
CLARITY 2 >> AVG 4.68 >> STDEV 0.44 >> MIKE 4.00
CLARITY 3 >> AVG 4.80 >> STDEV 0.40 >> MIKE 4.00
ACTIVE 1 >> AVG 4.82 >> STDEV 0.39 >> MIKE 5.00
ACTIVE 2 >> AVG 4.73 >> STDEV 0.45 >> MIKE 5.00
ACTIVE 3 >> AVG 4.50 >> STDEV 0.67 >> MIKE 4.00
CONNECT 1 >> AVG 4.36 >> STDEV 0.48 >> MIKE 1.00
CONNECT 2 >> AVG 4.45 >> STDEV 0.66 >> MIKE 3.00
My personal assessment reflects how well I think our group did on planning and executing this activity. I believe that every group member really helped to make this a fantastic group project and a great learning experience for the "students" in the classroom.
There should be two items that stand out immediately to the casual observer. First, I scored our group 1/5 for CONNECT 1 (The instructors offered activities that connected to other areas of math). I would like to explicitly state that although this is a "low" mark, I do not believe it represents a failure of any sort, but simply a narrow focus on a specific 15 minute lesson.
Second, I scored our group 3/5 for CONNECT 2 (The lesson connected to other areas of life & culture). I believe that given the 15 minutes, this is a sufficient undertaking. However, if we had been given 30 minutes, I would have liked to introduce more complicated algebra using kinematics (d = d0 + v*t). Simply put, these low scores are not a failure on the part of the group, but instead a reflection of the time constraint.
Next, I would like to address some of the more prevalent "constructive criticism" comments that were listed on the peer reviews. The first comment deals with coordination of the two groups. Unfortunately I ran a half minute behind on the first group, and the second group had to wait until we finished. Although this does not seem like very long, I'm sure it would be an eternity for a grade 8 student. If we were doing an activity like this in a class with multiple instructors, I think having an official "time keeper" to give 2min and 1min warnings would help everyone (read: Mike B.) to pace their lessons appropriately.
The second "constructive criticism" was that we should have better outlined our objectives to the class before beginning. I am still on the fence about this comment, since although I know that many students, myself included, prefer an ordered and clearly laid out lesson structure, I do believe that the world "Algebra" might invoke terror in young students, and I really do like the idea of teaching it implicitly without actually naming the technique until the students are comfortable with the material. Any thoughts on this matter?
Tuesday, October 13, 2009
Introduction to Algebra using Kinematics
MAED 314a-301
INTRODUCTION TO ALGEBRA
BRIDGE: Today in Math we are going to be doing an activity involving a ball and measurements!
TEACHER OBJECTIVES: To teach algebra implicitly using a Kinematics Lab.
STUDENT OBJECTIVES: Basic understanding of algebra and the ability to apply it to real world situations involving simple kinematics (v=d/t).
PRE-TEST: Intentionally none done, as we don't want to "tip our hat".
PARTICIPATION: Two tables.
1)The first table is a "Kinematics Shout-Out", where the students derive the basic Kinematics equations and put them in algebraic notation. We will be using the equation v=d/t and solving for the variables v, d, t using repetition.
2)The second table is a "Mini-Lab" with a ball rolling a set distance against a wall. The students will measure the distance from a line to the wall, and measure the time it takes the ball to get to the wall. This will then later be applied to the equation learned previously.
Required Materials:
* Masking tape
* Measuring tape
* Stop watches
* Balls
POST-TEST: Students will use their measurements and their new equations to calculate the speeds that the balls were rolling using v=d/t
SUMMARY: Surprise... you just learned algebra!
Write down the equation you learned and how you could apply this to another real life situation for next class.
INTRODUCTION TO ALGEBRA
BRIDGE: Today in Math we are going to be doing an activity involving a ball and measurements!
TEACHER OBJECTIVES: To teach algebra implicitly using a Kinematics Lab.
STUDENT OBJECTIVES: Basic understanding of algebra and the ability to apply it to real world situations involving simple kinematics (v=d/t).
PRE-TEST: Intentionally none done, as we don't want to "tip our hat".
PARTICIPATION: Two tables.
1)The first table is a "Kinematics Shout-Out", where the students derive the basic Kinematics equations and put them in algebraic notation. We will be using the equation v=d/t and solving for the variables v, d, t using repetition.
2)The second table is a "Mini-Lab" with a ball rolling a set distance against a wall. The students will measure the distance from a line to the wall, and measure the time it takes the ball to get to the wall. This will then later be applied to the equation learned previously.
Required Materials:
* Masking tape
* Measuring tape
* Stop watches
* Balls
POST-TEST: Students will use their measurements and their new equations to calculate the speeds that the balls were rolling using v=d/t
SUMMARY: Surprise... you just learned algebra!
Write down the equation you learned and how you could apply this to another real life situation for next class.
Sunday, October 11, 2009
MAED314
CITIZENSHIP EDUCATION
MICHAEL BAYER
"If we think about it, we begin to realize that there is much in our society which has been quantified - the gross national product, the DOW index, unemployment rates, the weather forecast, the smog index, the quality of a hockey player's game performance, a student's understanding of literature, and intelligence itself, with the Intelligence Quotient."
I think this is a really important statement. I believe that teachers should encourage students interest in mathematics by relating new concepts to familiar uses of those concepts, whether they be basic kinematics or statistical analysis. Every new concept should immediately have an application presented. I believe that by making mathematics more "friendly" to the general student, I will foster interest in the subject.
"Mathematics is one of the greatest cultural and intellectual achievements of humankind." - NCTM, 2000
Unfortunately the preceding statement makes mathematics seem to be out of the grasp of the general student, and implies that only the most adept student will ever progress in mathematics, which will in turn make them the elite of humankind. I feel this is a very dangerous statement to make, which may lead to students becoming discouraged in even attempting to learn the subject.
"Teaching students to identify and pose problems, to explain themselves in terms others can understand and to question the invisible structures of mathematics is key to developing informed, active and critical citizens."
I agree completely with this statement. As a mathematics teacher, I believe an interesting form of assessment would be to have students work together in groups to develop a list of questions based on the material already covered, submit this list to the teacher for approval, and in turn have the questions returned to the students in the form of a group homework assignment. Groups would work together to create questions difficult enough that other groups were unable to answer them, which in turn would foster deeper understanding from the problem posers and develop their social "professional" interactions and problem-solving skills.
CITIZENSHIP EDUCATION
MICHAEL BAYER
"If we think about it, we begin to realize that there is much in our society which has been quantified - the gross national product, the DOW index, unemployment rates, the weather forecast, the smog index, the quality of a hockey player's game performance, a student's understanding of literature, and intelligence itself, with the Intelligence Quotient."
I think this is a really important statement. I believe that teachers should encourage students interest in mathematics by relating new concepts to familiar uses of those concepts, whether they be basic kinematics or statistical analysis. Every new concept should immediately have an application presented. I believe that by making mathematics more "friendly" to the general student, I will foster interest in the subject.
"Mathematics is one of the greatest cultural and intellectual achievements of humankind." - NCTM, 2000
Unfortunately the preceding statement makes mathematics seem to be out of the grasp of the general student, and implies that only the most adept student will ever progress in mathematics, which will in turn make them the elite of humankind. I feel this is a very dangerous statement to make, which may lead to students becoming discouraged in even attempting to learn the subject.
"Teaching students to identify and pose problems, to explain themselves in terms others can understand and to question the invisible structures of mathematics is key to developing informed, active and critical citizens."
I agree completely with this statement. As a mathematics teacher, I believe an interesting form of assessment would be to have students work together in groups to develop a list of questions based on the material already covered, submit this list to the teacher for approval, and in turn have the questions returned to the students in the form of a group homework assignment. Groups would work together to create questions difficult enough that other groups were unable to answer them, which in turn would foster deeper understanding from the problem posers and develop their social "professional" interactions and problem-solving skills.
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