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.
Thursday, October 8, 2009
"What If Not..."
The "What If Not..." or "WIN" approach is an excellent tool that can be used to test the understanding of both the person creating the problem, which would typically be the teacher, and the person solving the problem, which would typically be the student. The reason I say "typically" is because students with a natural curiosity in mathematics will be tempted to push the boundaries of their knowledge and are likely to pose very similar questions to the teacher, so it is always helpful if the teacher has already reflected on similar questions themselves and are prepared to answer higher order questions from the students.
The strength of the WIN approach is the potential it has in identifying students that are adept in mathematics and seeking more of a challenge than the typical classroom environment offers. I believe the best application of the WIN approach would be as bonus questions on homework assignments and tests, where gifted students will be able to entertain themselves while the rest of the class catches up.
The weakness of the WIN approach is its ability to terrify students with a weak or moderate understanding in mathematics. A student that has a tentative understanding of a new concept may find that they are unable to grasp higher order questions until they have affirmed their own knowledge, which may not happen until after a unit is complete.
Unfortunately the WIN approach does not have any application in our micro teaching lesson on "Introduction to Algebra", since we only have 15 minutes to introduce the topic, and I believe the WIN approach is better left for a second or third class on the subject, when students are more comfortable with the material.
The strength of the WIN approach is the potential it has in identifying students that are adept in mathematics and seeking more of a challenge than the typical classroom environment offers. I believe the best application of the WIN approach would be as bonus questions on homework assignments and tests, where gifted students will be able to entertain themselves while the rest of the class catches up.
The weakness of the WIN approach is its ability to terrify students with a weak or moderate understanding in mathematics. A student that has a tentative understanding of a new concept may find that they are unable to grasp higher order questions until they have affirmed their own knowledge, which may not happen until after a unit is complete.
Unfortunately the WIN approach does not have any application in our micro teaching lesson on "Introduction to Algebra", since we only have 15 minutes to introduce the topic, and I believe the WIN approach is better left for a second or third class on the subject, when students are more comfortable with the material.
Sunday, October 4, 2009
Art of Problem Posing - Questions and Comments on Pgs 1-32.
Goal: Write 10 questions or comments to the author regarding pages 1-32.
(1) "We believe that is is necessary first to get "caught up in" (and sometimes even 'caught,' in the sense of 'trapped by') the activity in order to appreciate where we are headed." - Pg 2
I believe this is a fantastic idea for capturing the attention and imagination of the curious learner, but should we be concerned that we may frustrate the less interested students as well?
(2) "Coming to know something is not a 'spectator sport'... As Dewey asserted many years ago, and the constructivist school of thought has vigorously argued more recently, to claim that 'coming to know' is a participant sport is to require that we operate on and even modify the things we are trying to understand." - Pg 2
Thank you very much for clearly writing this statement. After reading this, I am finally beginning to understand the Brent Davis lectures, and I feel less stressed out over the first assignment.
(3) "It is worth appreciating, however, that not everything we experience comes as a problem. Imagine being given a situation in which no problem has been posed at all. One possibility of course is that we merely appreciate the situation, and do not attempt to act on it in any way... Another reasonable response for some situations, however,might be to generate a problem or to ask a question, not for the purpose of solving the original situation, but in order to uncover or to create a problem or problems that derive from the situation." - Pg 4-5
I find this statement acts as a reflection of my own life. There was a time when I questioned and thought critically about everything I experienced. From my questioning came a thirst for knowledge, but more questions arose than answers. Although I have already trained my mind to think critically and question everything, I found it wasn't until years later that I was simply able to appreciate the simplicity of certain situations, and let my mind relax. While I believe it is important to train the minds of students to question, it is equally important to train them how to be at peace. As for how to accomplish this second part, I am still unsure.
(4) "For example, exploration of the perimeters of various rectangles with area 24cm^2 by means of models and drawings, with data as recorded in [the table below], could lead to student recognition and formulation of such problems as the following: Is there a rectangle of minimum perimeter with the specified area? What are it's dimensions?" - Pg 9
This would also make a great launching activity for discussing which three-dimensional geometric shapes have the least surface area to volume ratio, and tie this into nature where soap bubbles minimize their surface area (and surface tension) by choosing a spherical shape. The student wouldn't need to know the complex mathematics and physics of the principle, just understand why this works.
(5) "x^2 + y^2 = z^2; What are some answers?" - Pg 12
Perhaps because of my physics background, I mentally solved for the f(x), f(y), and f(z) algebraically. Although I like the idea of an open ended mathematics question, I feel that further constraints are required before posing such a problem to the students... Half a page later, and I realize that this example was meant to illustrate that exact point. :P
(6) "x^2 + y^2 = z^2; What are some questions?" - Pg 14
My questions:
How do you solve for each individual variable?
Where is this formula used in mathematics?
When is this formula used in real life?
Again, I notice that my answers are different from those in the book. Because I view this formula as a tool, my first thought is "when do I use this tool?" I feel that if students can also see the relevance of this knowledge, they are more likely to be encouraged to learn it.
(7) "[Pythagorean Theorem] has to do with squares on the sides of a right triangle" - Pg 16
I am completely lost on this one. Is it a paper folding exercise?
(8) "Does the [Supreme Court Judges] handshake have an effect on subsequent cordiality? - Pg 20
I think this is a fantastic question to ask the students, but how can it be graded, since I do not know the answer myself?
(9) "If one angle of an isosceles triangle is twice another, is the shape of the triangle determined?" - Pg 21
From this question I immediately thought "Can you solve for every angle in the triangle?" Additionally, when I saw the figures below, I thought "If a triangle is made up of four identical isosceles triangles, what is the relationship between the angles of the larger triangle and the smaller ones?" This is a pretty self-evident question, but may help to make the struggling students in the class feel that they are developing a deeper understanding, and therefore open them up to further exploration in the subject.
(10) The geoboard has an appeal to the uninitiated student as well as to the sophisticated one... What would you do with it? - Pg 22
I would love to ask students how many squares can be made using the board. I recall a fellow that I did my physics undergrad with that had trouble with traditional homework questions, but excelled at puzzles like this. I think this would be a great opportunity to "level the playing field" in the classroom between the mathematically-inclined students and those that struggle with the subject.
(1) "We believe that is is necessary first to get "caught up in" (and sometimes even 'caught,' in the sense of 'trapped by') the activity in order to appreciate where we are headed." - Pg 2
I believe this is a fantastic idea for capturing the attention and imagination of the curious learner, but should we be concerned that we may frustrate the less interested students as well?
(2) "Coming to know something is not a 'spectator sport'... As Dewey asserted many years ago, and the constructivist school of thought has vigorously argued more recently, to claim that 'coming to know' is a participant sport is to require that we operate on and even modify the things we are trying to understand." - Pg 2
Thank you very much for clearly writing this statement. After reading this, I am finally beginning to understand the Brent Davis lectures, and I feel less stressed out over the first assignment.
(3) "It is worth appreciating, however, that not everything we experience comes as a problem. Imagine being given a situation in which no problem has been posed at all. One possibility of course is that we merely appreciate the situation, and do not attempt to act on it in any way... Another reasonable response for some situations, however,might be to generate a problem or to ask a question, not for the purpose of solving the original situation, but in order to uncover or to create a problem or problems that derive from the situation." - Pg 4-5
I find this statement acts as a reflection of my own life. There was a time when I questioned and thought critically about everything I experienced. From my questioning came a thirst for knowledge, but more questions arose than answers. Although I have already trained my mind to think critically and question everything, I found it wasn't until years later that I was simply able to appreciate the simplicity of certain situations, and let my mind relax. While I believe it is important to train the minds of students to question, it is equally important to train them how to be at peace. As for how to accomplish this second part, I am still unsure.
(4) "For example, exploration of the perimeters of various rectangles with area 24cm^2 by means of models and drawings, with data as recorded in [the table below], could lead to student recognition and formulation of such problems as the following: Is there a rectangle of minimum perimeter with the specified area? What are it's dimensions?" - Pg 9
This would also make a great launching activity for discussing which three-dimensional geometric shapes have the least surface area to volume ratio, and tie this into nature where soap bubbles minimize their surface area (and surface tension) by choosing a spherical shape. The student wouldn't need to know the complex mathematics and physics of the principle, just understand why this works.
(5) "x^2 + y^2 = z^2; What are some answers?" - Pg 12
Perhaps because of my physics background, I mentally solved for the f(x), f(y), and f(z) algebraically. Although I like the idea of an open ended mathematics question, I feel that further constraints are required before posing such a problem to the students... Half a page later, and I realize that this example was meant to illustrate that exact point. :P
(6) "x^2 + y^2 = z^2; What are some questions?" - Pg 14
My questions:
How do you solve for each individual variable?
Where is this formula used in mathematics?
When is this formula used in real life?
Again, I notice that my answers are different from those in the book. Because I view this formula as a tool, my first thought is "when do I use this tool?" I feel that if students can also see the relevance of this knowledge, they are more likely to be encouraged to learn it.
(7) "[Pythagorean Theorem] has to do with squares on the sides of a right triangle" - Pg 16
I am completely lost on this one. Is it a paper folding exercise?
(8) "Does the [Supreme Court Judges] handshake have an effect on subsequent cordiality? - Pg 20
I think this is a fantastic question to ask the students, but how can it be graded, since I do not know the answer myself?
(9) "If one angle of an isosceles triangle is twice another, is the shape of the triangle determined?" - Pg 21
From this question I immediately thought "Can you solve for every angle in the triangle?" Additionally, when I saw the figures below, I thought "If a triangle is made up of four identical isosceles triangles, what is the relationship between the angles of the larger triangle and the smaller ones?" This is a pretty self-evident question, but may help to make the struggling students in the class feel that they are developing a deeper understanding, and therefore open them up to further exploration in the subject.
(10) The geoboard has an appeal to the uninitiated student as well as to the sophisticated one... What would you do with it? - Pg 22
I would love to ask students how many squares can be made using the board. I recall a fellow that I did my physics undergrad with that had trouble with traditional homework questions, but excelled at puzzles like this. I think this would be a great opportunity to "level the playing field" in the classroom between the mathematically-inclined students and those that struggle with the subject.
Friday, October 2, 2009
Hypothetical Student Responses - 2019
Dear Mr. Bayer,
I would like to thank you for making mathematics fun for me. Although I decided not to continue on in math, I did find that your course was my favorite because of the laboratory "experiments" you included in the course. It really helped me to understand the concepts being taught, particularly algebra and trigonometry. I wasn't a fan of the word problems, but at least I could see how I could actually apply all the materials that I learned in class.
You really helped to make the subject "exciting" by being excited about teaching. I had a tendency to fall asleep in some of my other classes, but I was always able to stay awake in your class.
Mr. Bayer,
I wanted to let you know that I really did not enjoy your class. First, I thought that your "pop quizzes" were an extremely unfair form of punishment against the class for acting up and not focusing on work. Second, since when does Math need labs? If I wanted labs, I would take a science class. If I wanted a "sanity check", I would see a psychiatrist. You teach math... you shouldn't be doing anything other than punching numbers into a calculator. You suck.
Oh, and "Think-Pair-Share"? Why couldn't we do this on tests?
SUMMARY: My hopes and worries.
From my written work, I hope that I can convince most students that mathematics does not have to be dry, and I hope to accomplish this by incorporating simple physics concepts as a tool to teach mathematics, such as throwing a ball against a wall to measure velocity, a tool in introducing algebra.
My worry is that taking mathematics away from a strictly classroom instruction, I may end up alienating some students from mathematics. Also, I fear that the unspoken threat of pop quizzes as a classroom management tool may discourage some students from continuing on in mathematics.
I would like to thank you for making mathematics fun for me. Although I decided not to continue on in math, I did find that your course was my favorite because of the laboratory "experiments" you included in the course. It really helped me to understand the concepts being taught, particularly algebra and trigonometry. I wasn't a fan of the word problems, but at least I could see how I could actually apply all the materials that I learned in class.
You really helped to make the subject "exciting" by being excited about teaching. I had a tendency to fall asleep in some of my other classes, but I was always able to stay awake in your class.
Mr. Bayer,
I wanted to let you know that I really did not enjoy your class. First, I thought that your "pop quizzes" were an extremely unfair form of punishment against the class for acting up and not focusing on work. Second, since when does Math need labs? If I wanted labs, I would take a science class. If I wanted a "sanity check", I would see a psychiatrist. You teach math... you shouldn't be doing anything other than punching numbers into a calculator. You suck.
Oh, and "Think-Pair-Share"? Why couldn't we do this on tests?
SUMMARY: My hopes and worries.
From my written work, I hope that I can convince most students that mathematics does not have to be dry, and I hope to accomplish this by incorporating simple physics concepts as a tool to teach mathematics, such as throwing a ball against a wall to measure velocity, a tool in introducing algebra.
My worry is that taking mathematics away from a strictly classroom instruction, I may end up alienating some students from mathematics. Also, I fear that the unspoken threat of pop quizzes as a classroom management tool may discourage some students from continuing on in mathematics.
Wednesday, September 30, 2009
DAVE HEWITT VIDEO REFLECTION
I have a few thoughts on the video we watched today, both positive and negative, as well as an alternate method for introducing algebra to a class of students.
First the positive. I applaud Dave Hewitt for introducing a rather abstract concept in a manner that engages the entire class, and relating it to a familiar concept, the number line. I believe that "pure" mathematics instruction can be rather dry, and hard for a student that prefers to make associations to "real world" things to understand.
Unfortunately, I don't believe that this manner of teaching helps all students. In the video, there were a few students that answered incorrectly a few times originally, before correcting themselves. This may be because they weren't paying close enough attention the first time, or possibly that they are struggling with the concept but know that the class won't progress until they say the same answer as their classmates. There is no indication that they understood the subject material before moving on.
One realization that I did have while watching the Dave Hewitt video is that algebra can be introduced using the simple concept of kinematics. By asking the students questions like "If you are travelling 100kph for 2 hours, how far did you go?" and "If you are travelling at 50kph, and you have travelled 250km, how long have you been travelling?", the concept of formula manipulation can be introduced without actually writing anything. Once the students are able to answer a few questions, you ask them questions like "Is the speed you travel always equal to the distance divided by the time?" and "Is the time you travel always equal to the distance you travel divided by the speed?" and write up a few equations using full words, rather than symbols. Then let the students identify the similarity in all the equations. When they see the relation, introduce the symbols for velocity, distance, and time, and formally instruct them on how to manipulate an equation. After this concept is thoroughly understood, more difficult algebraic exercises can be done that don't necessarily have a "real world" counterpart.
First the positive. I applaud Dave Hewitt for introducing a rather abstract concept in a manner that engages the entire class, and relating it to a familiar concept, the number line. I believe that "pure" mathematics instruction can be rather dry, and hard for a student that prefers to make associations to "real world" things to understand.
Unfortunately, I don't believe that this manner of teaching helps all students. In the video, there were a few students that answered incorrectly a few times originally, before correcting themselves. This may be because they weren't paying close enough attention the first time, or possibly that they are struggling with the concept but know that the class won't progress until they say the same answer as their classmates. There is no indication that they understood the subject material before moving on.
One realization that I did have while watching the Dave Hewitt video is that algebra can be introduced using the simple concept of kinematics. By asking the students questions like "If you are travelling 100kph for 2 hours, how far did you go?" and "If you are travelling at 50kph, and you have travelled 250km, how long have you been travelling?", the concept of formula manipulation can be introduced without actually writing anything. Once the students are able to answer a few questions, you ask them questions like "Is the speed you travel always equal to the distance divided by the time?" and "Is the time you travel always equal to the distance you travel divided by the speed?" and write up a few equations using full words, rather than symbols. Then let the students identify the similarity in all the equations. When they see the relation, introduce the symbols for velocity, distance, and time, and formally instruct them on how to manipulate an equation. After this concept is thoroughly understood, more difficult algebraic exercises can be done that don't necessarily have a "real world" counterpart.
Friday, September 25, 2009
Summary and Response on "Battleground Schools"
Summary:
There exist two predominant stances on mathematics education that exist. The traditional classroom model, called the "Conservative Dichotomy", focuses on fluency and embraces the nature of the learner as a passive observer. A newer model has appeared, called the "Progressive Dichotomy", where understanding is the focus, and the student is encouraged to explore, relegating the teacher to a role of an orchestrator, preparing mathematical opportunities for the student to discover.
The Progressivist Reform (circa 1910-1940) sprang from criticism of a highly instrumentalized method of teaching. In an attempt to develop a more scientific and critical thinker, students were given a "programmed environment" where mathematical experimentation was encouraged.
The "New Math" Reform (1960's), led by the School Mathematics Study Group (SMSG), abruptly introduced university level mathematics into the grade school curriculum in an attempt to produce the next generations of scientists and astronauts. Teacher unfamiliarity of the subject material and media pressure ended the movement by the early 1970's.
A call for accountability began in the late 1970's, when a traditional curriculum and standardized testing were advocated by the United Kingdom and the United States. To prevent a possible implementation of national curricular standards, the National Council of Teachers of Mathematics (NCTM) developed their own set of standards, emphasizing the inclusion of new technology. While relational learning was preferred, fluency through instrumental learning was still viewed as important.
Currently the Third International Mathematics and Science Study (TIMSS) is at the center of the latest movement. After a poor ranking of American grade 8 math students and a video analysis suggesting that a higher ranking is related to a deep conceptual understanding, the TIMSS published recommended curriculum changes to grades K-8.
Response:
My opinion on the events that inspired this article can be accurately summarized by the following quote:
"If we had in this room a hundred teachers, good teachers from good schools, and asked them to define the word education, there would be very little general agreement." - William Glasser
I believe effective teachers are people who can organize their thoughts, and accurately convey ordered information to an attentive learner. There do exist teachers who are unable to order their thoughts, however, and any number of reforms on the educational system will not result in a better learning experience for the children. The problem lies not with the educational system, but the screening of the educators.
There exist two predominant stances on mathematics education that exist. The traditional classroom model, called the "Conservative Dichotomy", focuses on fluency and embraces the nature of the learner as a passive observer. A newer model has appeared, called the "Progressive Dichotomy", where understanding is the focus, and the student is encouraged to explore, relegating the teacher to a role of an orchestrator, preparing mathematical opportunities for the student to discover.
The Progressivist Reform (circa 1910-1940) sprang from criticism of a highly instrumentalized method of teaching. In an attempt to develop a more scientific and critical thinker, students were given a "programmed environment" where mathematical experimentation was encouraged.
The "New Math" Reform (1960's), led by the School Mathematics Study Group (SMSG), abruptly introduced university level mathematics into the grade school curriculum in an attempt to produce the next generations of scientists and astronauts. Teacher unfamiliarity of the subject material and media pressure ended the movement by the early 1970's.
A call for accountability began in the late 1970's, when a traditional curriculum and standardized testing were advocated by the United Kingdom and the United States. To prevent a possible implementation of national curricular standards, the National Council of Teachers of Mathematics (NCTM) developed their own set of standards, emphasizing the inclusion of new technology. While relational learning was preferred, fluency through instrumental learning was still viewed as important.
Currently the Third International Mathematics and Science Study (TIMSS) is at the center of the latest movement. After a poor ranking of American grade 8 math students and a video analysis suggesting that a higher ranking is related to a deep conceptual understanding, the TIMSS published recommended curriculum changes to grades K-8.
Response:
My opinion on the events that inspired this article can be accurately summarized by the following quote:
"If we had in this room a hundred teachers, good teachers from good schools, and asked them to define the word education, there would be very little general agreement." - William Glasser
I believe effective teachers are people who can organize their thoughts, and accurately convey ordered information to an attentive learner. There do exist teachers who are unable to order their thoughts, however, and any number of reforms on the educational system will not result in a better learning experience for the children. The problem lies not with the educational system, but the screening of the educators.
Questions/Answers - Personal Reflection
MAED314a-301
When starting this assignment, there were two questions that I knew I really wanted to ask. First, I wanted to know if there have been any changes to the "work at your own pace" mathematics system that was employed at Nechako Valley Secondary School, from where I graduated in 1997. Second, I was curious to see if "Think-Pair-Share" (TPS) actually worked in the math classroom.
When I started grade 8 math in 1992, Nechako Valley Secondary School had just implemented a revolutionary "work at your own pace" module system. Each student started grade 8 by writing tests that ascertained basic mathematics comprehension. The tests started out with single digit addition, and over the course of the first two weeks, tested skills that should have been taught in grades 1-7. In order to pass to the next section, a score of 8/10 was required. Very quickly the students gifted in mathematics started to outpace the ones who struggled in the subject.
When I first made contact with Mr. Jack French, who designed the "work at your own pace" module system, I was unsure as to if the program remained in its original form, if it had been modified, or if it had been wholly replaced by another system. While Mr. French wasn't as forthcoming in his direct written response, he did mention in the phone interview that there had been a stress on the system. Students that were struggling with grade 11 math were required to schedule a mathematics block in their grade 12 years to finish the required modules. Because a passing grade of 8/10 was required on all modules, these students were passing the course with 80%, but a year behind schedule. When Mr. French took a year sabbatical, and the other mathematics teacher supporting the system accepted a job as vice-principal in another school district, and the math program was replaced with a traditional classroom model.
The program now exists behind the scenes as an option for students that cannot fit a mathematics block into their schedule, or students that have other requirements forcing them to miss too much classroom time.
As a future mathematics teacher, I am very happy that I had the opportunity to talk with Mr. French, since I was considering a modified version of this same program for my own classroom. Because of this conversation, I am no longer considering this model.
My biggest surprise was the positive views of the "Think-Pair-Share" (TPS) technique. I have never seen this technique used in the secondary level, and had believed that it would quickly dissolve into a social opportunity for students. However, both students and teachers mentioned that the technique can be used for greater classroom management, which is opposite to my initial misconception. Additional benefits include more active student participation and deeper thought about the lesson being taught, less hesitance on sharing an answer, and increased creativity. This is definitely a technique that I will use in the future.
When starting this assignment, there were two questions that I knew I really wanted to ask. First, I wanted to know if there have been any changes to the "work at your own pace" mathematics system that was employed at Nechako Valley Secondary School, from where I graduated in 1997. Second, I was curious to see if "Think-Pair-Share" (TPS) actually worked in the math classroom.
When I started grade 8 math in 1992, Nechako Valley Secondary School had just implemented a revolutionary "work at your own pace" module system. Each student started grade 8 by writing tests that ascertained basic mathematics comprehension. The tests started out with single digit addition, and over the course of the first two weeks, tested skills that should have been taught in grades 1-7. In order to pass to the next section, a score of 8/10 was required. Very quickly the students gifted in mathematics started to outpace the ones who struggled in the subject.
When I first made contact with Mr. Jack French, who designed the "work at your own pace" module system, I was unsure as to if the program remained in its original form, if it had been modified, or if it had been wholly replaced by another system. While Mr. French wasn't as forthcoming in his direct written response, he did mention in the phone interview that there had been a stress on the system. Students that were struggling with grade 11 math were required to schedule a mathematics block in their grade 12 years to finish the required modules. Because a passing grade of 8/10 was required on all modules, these students were passing the course with 80%, but a year behind schedule. When Mr. French took a year sabbatical, and the other mathematics teacher supporting the system accepted a job as vice-principal in another school district, and the math program was replaced with a traditional classroom model.
The program now exists behind the scenes as an option for students that cannot fit a mathematics block into their schedule, or students that have other requirements forcing them to miss too much classroom time.
As a future mathematics teacher, I am very happy that I had the opportunity to talk with Mr. French, since I was considering a modified version of this same program for my own classroom. Because of this conversation, I am no longer considering this model.
My biggest surprise was the positive views of the "Think-Pair-Share" (TPS) technique. I have never seen this technique used in the secondary level, and had believed that it would quickly dissolve into a social opportunity for students. However, both students and teachers mentioned that the technique can be used for greater classroom management, which is opposite to my initial misconception. Additional benefits include more active student participation and deeper thought about the lesson being taught, less hesitance on sharing an answer, and increased creativity. This is definitely a technique that I will use in the future.
Question/Answer Group Reflection
MAED314a-301
(1) Why do you think we learn math in school, and why do you think math is an important or unimportant subject?
The general consensus is that Math is important to understand the world around us, and is a critical prerequisite to learning essential skills such as statistics, accounting, physics, poetry, etc. Additionally Math helps to develop critical thinking and problem solving skills.
(2a) For students: When learning a new material, do you tend to memorize the steps in solving a problem or do you try to understand the meaning and idea under each steps? and why do you do so?
Our students, who both struggled with Math, had differing opinions on Instrumental versus Relational understanding. The first student, who learned using a "work at your own pace" system, believed that the best method for him was 100% instrumental learning, since he viewed mathematics as an unnecessary learning exercise, and had no interest beyond passing the course. Our second student believed that if he had an Relational understanding, that it would be easier for him to develop (or "memorize") the Instrumental understanding.
(2b) For teachers: When teaching a new material, do you just work through a problem and showing all the steps in solving a problem or do you explain or try to explain the idea in each of the steps involved in solving a problem? and why do you do so?
Two of our teachers believe strongly that Relational teaching leads to a stronger understanding of the material than Instrumental teaching, which in turn allows the students to better apply their knowledge to a wider variety of problems. Our third teacher believes it is important to emphasize a variety of teaching techniques, both Relational and Instrumental, combined with classroom discussion and a final summary of the Relational concepts.
(3) What do you think of TPS (Think-Pair-Share) techniques where students break into groups of 2 to discuss the material before the class discusses it as a whole? How about groups of 5 students?
Two of our three teachers were willing to comment on TPS techniques, but for different reasons. Our first teacher believes that TPS can help both as a classroom management tool, as well as help reduce the possibility of public humiliation that occurs when a student answers a question wrong. Our second teacher supports TPS techniques because the students take a more active roll in their learning. She finds it inspires creativity in the students, improves their ability to communicate using math terms, and strengthens their understanding of the concepts.
Both students believe TPS techniques would help with classroom management. One student expressed concern regarding groups of 5, where not everyone may be actively involved.
(4) Administering tests can be an effective tool in determining class proficiency before moving on to new material. How frequently do you believe tests should be administered?
Our teachers as a whole believed that constant assessment is necessary, but did not believe that tests were essential to administer frequently. They all stressed that alternate methods of assessment were just as effective.
Both students explicitly stated that tests should be administered at the end of every unit, and suggested one test a week as a good pace. One of the students specified that he prefers tests to homework assignments as a means of assessment.
(5) What are your views on traditional class lecturing versus a system where students could work at their own pace and each individual could spend extra time on material they found difficult?
Our teachers agreed that a "work at your own pace" system would be nice, but is very difficult to implement due to time constraints and class management. Although not included in the written response, the one teacher, Mr. Jack French, did mention in a phone conversation that the administration and parents pushed heavily against this system due to students having to take a Grade 12 math block to finish their required Grade 11 mathematics, which by the program requirement, they eventually passed with 80% or better.
One student expressed concern that a "work at your own pace" system would result in the students getting behind, while the other student was concerned about teacher unavailability.
(1) Why do you think we learn math in school, and why do you think math is an important or unimportant subject?
The general consensus is that Math is important to understand the world around us, and is a critical prerequisite to learning essential skills such as statistics, accounting, physics, poetry, etc. Additionally Math helps to develop critical thinking and problem solving skills.
(2a) For students: When learning a new material, do you tend to memorize the steps in solving a problem or do you try to understand the meaning and idea under each steps? and why do you do so?
Our students, who both struggled with Math, had differing opinions on Instrumental versus Relational understanding. The first student, who learned using a "work at your own pace" system, believed that the best method for him was 100% instrumental learning, since he viewed mathematics as an unnecessary learning exercise, and had no interest beyond passing the course. Our second student believed that if he had an Relational understanding, that it would be easier for him to develop (or "memorize") the Instrumental understanding.
(2b) For teachers: When teaching a new material, do you just work through a problem and showing all the steps in solving a problem or do you explain or try to explain the idea in each of the steps involved in solving a problem? and why do you do so?
Two of our teachers believe strongly that Relational teaching leads to a stronger understanding of the material than Instrumental teaching, which in turn allows the students to better apply their knowledge to a wider variety of problems. Our third teacher believes it is important to emphasize a variety of teaching techniques, both Relational and Instrumental, combined with classroom discussion and a final summary of the Relational concepts.
(3) What do you think of TPS (Think-Pair-Share) techniques where students break into groups of 2 to discuss the material before the class discusses it as a whole? How about groups of 5 students?
Two of our three teachers were willing to comment on TPS techniques, but for different reasons. Our first teacher believes that TPS can help both as a classroom management tool, as well as help reduce the possibility of public humiliation that occurs when a student answers a question wrong. Our second teacher supports TPS techniques because the students take a more active roll in their learning. She finds it inspires creativity in the students, improves their ability to communicate using math terms, and strengthens their understanding of the concepts.
Both students believe TPS techniques would help with classroom management. One student expressed concern regarding groups of 5, where not everyone may be actively involved.
(4) Administering tests can be an effective tool in determining class proficiency before moving on to new material. How frequently do you believe tests should be administered?
Our teachers as a whole believed that constant assessment is necessary, but did not believe that tests were essential to administer frequently. They all stressed that alternate methods of assessment were just as effective.
Both students explicitly stated that tests should be administered at the end of every unit, and suggested one test a week as a good pace. One of the students specified that he prefers tests to homework assignments as a means of assessment.
(5) What are your views on traditional class lecturing versus a system where students could work at their own pace and each individual could spend extra time on material they found difficult?
Our teachers agreed that a "work at your own pace" system would be nice, but is very difficult to implement due to time constraints and class management. Although not included in the written response, the one teacher, Mr. Jack French, did mention in a phone conversation that the administration and parents pushed heavily against this system due to students having to take a Grade 12 math block to finish their required Grade 11 mathematics, which by the program requirement, they eventually passed with 80% or better.
One student expressed concern that a "work at your own pace" system would result in the students getting behind, while the other student was concerned about teacher unavailability.
Teacher/Student Questions and Answers
MAED 314A Assignment #1 Interview Responses
(1) Why do you think we learn math in school, and why do you think math is an important or unimportant subject?
Jack French (teacher): Mathematics is a critical subject for everyone to know, therefore to learn in school. Mathematics permits intelligent interpretation of many topics one reads about, permits one to learn topics that utilize mathematics as a base...statistics, accounting, physics, many other aspects of science to name just a few, and the study of mathematics develops thinking skills, which are transferable to all other aspects of life.
Ian Bayer (student): I think we learn it in school for better understanding of it. So if we use a calculator, we know why it gives us that answer. It's important up to a certain extent, and after that it should be optional. Complex fraction isn’t something you would run into unless you choose that path.
Carol Funk (teacher): Math is used in everyday lives regardless of what we do for a living. Although we may not use the exact Math that is studied, in particular in the academic classes, we learn how to problem solve, how to apply our knowledge to gain new knowledge. Math explains how our world works. Math teaches organization of our thoughts and how to explain our thinking.
Brandon Jentsch (student): We learn math to better learn how and why things, live, exist, and just function the way they do. It is very important because it helps us better understand the world existing around us.
Gabriele Gonzales (teacher): Math is an integral part of life: you use it in doing business (buying and selling), in keeping time, in planning, and you use it in more academic subjects like chemistry and physics. Even Poetry uses numbers (meters).
Taking math in school also teaches problem solving skills and analytical/ logical thinking.
(2) For students: When learning a new material, do you tend to memorize the steps in solving a problem or do you try to understand the meaning and idea under each steps? and why do you do so?
For teachers: When teaching a new material, do you just work through a problem and showing all the steps in solving a problem or do you explain or try to explain the idea in each of the steps involved in solving a problem? and why do you do so?
Jack French (teacher): If students understand why each step has been taken in solving a
problem they are more likely to be able to solve similar problems, and
eventually to solve many different types of problems, once their repertoire
of procedures is large enough. They will not remember steps they have
merely memorized.
Ian Bayer (student): The steps to get the answer. Uh... 'cause all I'm looking for is the answer. I'm not looking for understanding of the formula.
Carol Funk (teacher): The way we teach math has changed in the last couple of years. Rather than teaching to problem solve, we teach math through problem solving. Students solve problems using prior knowledge to learn new concepts. Multiple methods are emphasized and students communicate their methods to each other to understand that there is more than one method to solve any problem. Once students have shared how they approached a question I usually offer a formal summary of the concepts and clearly state what I expect to see when the students put their work on paper for future evaluation. I also stress the importance that students understand each step they use.
Note that today we are using manipulatives and models to explain new math concepts, something that was rarely used when you were in the junior grades.
Brandon Jentsch (student): I memorize the steps and why the steps are taken in solving a problem that way I better understand how it works so I can better memorize how to use that formula for solving the problem in the future.
Gabriele Gonzales (teacher): Explain ideas, the “why’s” so students can follow the thinking. This helps them understand and also teaches them logical thinking. Then summarize just going through the steps.
(3) What do you think of TPS (Think-Pair-Share) techniques where students break into groups of 2 to discuss the material before the class discusses it as a whole? How about groups of 5 students?
Jack French (teacher): Am not familiar with this procedure.
Ian Bayer (student): Yeah, I agree with the buddy system, sure. I don't know if I would want a group of 2 or 5, but it's better than a group of 20 students asking the teacher.
Carol Funk (teacher): We do this every day with the new curriculum and are trying to implement this into our senior classes. The size of the group depends on the difficulty of the task. Often we will start in pairs, then share with another pair, then share with the class. This gives students the opportunity to see alternative methods. It is amazing how creative students can be! By explaining the each other students improve on their ability to communicate using math terms and also strengthen their understanding of the concepts.
Brandon Jentsch (student): Groups of 2 would make sure everyone is doing something unlike groups of 5, but depending on class size 5 may be more appropriate, so my answer is 5 if you can make sure everyone is actively involved in working on the problem if not then my answer is 2.
Gabriele Gonzales (teacher): TPS make a student commit to an idea first, then takes away the possibility of public humiliation of being wrong by just comparing his answer to a partner. It allows him to defend his idea and forces both partners to think about their reasoning. It clarifies the concepts to both students before talking to the whole class. It would work with 5 students, too, if students can hold group discussions.
(4) Administering tests can be an effective tool in determining class proficiency before moving on to new material. How frequently do you believe tests should be administered?
Jack French (teacher): Teachers must receive fairly constant feedback from students in order
to monitor the effectiveness of the classroom dynamics for both the
teacher and individual students. If assignments are few, then tests
should be many. If many assignments are issued, then tests can be less
frequent.
Ian Bayer (student): That's a good question. Maybe at the end of every week or something. As often as the chapters move forward. I'm not a big homework guy. There is nobody at home to teach you, may as well be working at your own pace.
Carol Funk (teacher): Assessment is more than administering tests. A teacher must assess student understanding as they are first learning. This can be achieved by asking questions during the lesson, checking student work as they are working on the daily assignments, listening to students are they communicate their ideas to their peers…..
Students can assess how they are doing by checking their own work (assuming answers are provided). They can check with peers or the teacher to ensure they are on the right track.
I take in assignments daily to check progress and also give daily quizzes. Adjustments to lesson plans are made based on the results of the above. I also encourage students to let me know of any difficulties so I can deal with them before moving on.
By the time students get to the chapter test, they will have rehearsed several times on the assignments and quizzes.
Brandon Jentsch (student): Well probably weekly since stuff tends to be forgotten over weekends xp, but certainly after every chapter (basically after you finish 1 train of related thoughts and are about to move onto another is when you should test your class).
Gabriele Gonzales (teacher): Test at the end of the unit. Depending on the unit, intermittent quizzes are needed to check proficiency. However, they should not be the only means of assessment. Basically, they are mostly there to get grades.
(5) What are your views on traditional class lecturing versus a system where students could work at their own pace and each individual could spend extra time on material they found difficult?
Jack French (teacher): Individual learning is superior for the individual student in many
cases. Students learn how to learn effectively and can operate at a pace
that allows for effective learning. This school, for a time, had every
student learning this way. No students were enrolled in non-academic
courses.
Ian Bayer (student): I don't think there is a lot of teacher help at "work at your own pace". It's still 20 on 1. I think advanced students should take advanced math, and the rest should work as a class.
Carol Funk (teacher): In a perfect world it would be great to have students to work at their own pace and move on only when they have truly mastered prerequisite concepts. However we must deal with logistics such as large class sizes, time constraints (report cards, semestered classes…)
However with the use of models to help students develop better understanding, students who are not yet ready to move on to the use of algorithms can continue using models and manipulatives until they are ready, and in this way still be successful in problems.
Brandon Jentsch (student): Well while students being able to take their time and learn at their own pace is a nice idea, I have never had a teacher finish teaching us everything in a math book and I don’t think students have the time in the year to really learn at any slower pace than the teacher already has them going. If they need help they need to spend time after/before school with the teacher, get help from their parents, or hire a tutor.
Gabriele Gonzales(teacher): Difficult to implement in real life because most students who would take extra time on difficult material usually find MOST material difficult and would fall further and further behind. They would not complete the curriculum in time. Also may impede group work. However, some time and projects should be built in to allow for different speed of completion.
Tuesday, September 22, 2009
Robinson Article Response
ROBINSON ARTICLE RESPONSE
Goal: Write a 200 word response to the Robinson Article.
In the Heather J. Robinson's article, titled "Using Research to Analyze, Inform, and Assess Changes in Instruction", there were a number of points she made that I felt were especially important. The first was the opening statement:
"My vision for my classroom has always been one where students actively participate in the learning process by being engaged in enriching and meaningful learning activities that help make mathematics relevant and realistic - rather than an abstract "thing" out there to intimidate and use to separate the 'haves' from the 'have nots.'"
When I attended high school, my teachers made an effort to include "real world" word problems in our tests, which forced us to make the mental connection between word problems and mathematical concepts. I found greatly helped in my understanding, and encouraged me to minor in mathematics upon reaching high school.
A second statement of Heather J. Robinson's really struck a chord, as I thought back to my Grade 12 mathematics provincial exam:
"My students were able to perform mathematical operations with functions such as evaluating, adding, and multiplying functions, but I honestly was not sure that they ever understood what a function is and or how it might be applied."
Because our program at NVSS was a "work at your own pace" system, the students were self-taught, and usually we taught ourselves enough to pass the exam and progress on to the next topic, but no more. When I went to write the Grade 12 provincial exam, I found that there were a large number of questions that dealt with visual representations of logarithm and exponential functions, which was the one area that I was weakest. Because of this, I did very poorly on the exam, and spent the majority of the next summer relearning the material so that I could rewrite the exam.
I have mentioned several times in our MAED314a class that I support Instructional Learning with a focus on practical applications, however I believe that I should clarify that statement by explicitly stating that when students commit to learning higher level mathematics (Math 11 and 12), there should be more of a focus on relational understanding. However, I still support all students having a strong instrumental understanding, and relational understanding should be slowly introduced at successive grades.
Monday, September 21, 2009
Reflection on Memorable Math Teachers
Forward: Both the teachers I mention in this article are still active teachers at the high school and university levels, respectively. Because of this, I have decided that it is best to not record their last names.
Mrs. Lynn: Lynn taught my Grade 10 math class. I remember her mainly for her lack of understanding of probability, and insistence on not listening to an explanation on why another answer may be correct. Years later, after many probability and statistics classes, I still think back to this question:
You are a new kid in a school classroom made up of 10 boys and 10 girls. Initially you have an equal chance of meeting a boy or a girl. You have met 9 boys and 0 girls so far. What are the chances the next person you meet at random is a boy?
My answer was that there was a 1/11 chance, since there was only one boy in a class of eleven students. Lynn's answer was that there was a 50% chance. When I explained that the same logic could be applied to state that I have a 50% chance of winning the lottery, she agreed, stating that this was indeed the case.
Mr. Jim: Jim had a reputation as being a very difficult math instructor that was to be avoided at all costs. However, in order to complete a math minor, I needed a math class that fit into my schedule, and only Number Theory met this requirement. While I was hesitant about the course, Jim's enthusiasm about the subject material kept my attention, and this ended up being my favourite class that semester.
Reflection: In reflection, when I look back on these two separate instances, I had never made the connection until now how they were really flip sides of the same coin. Before now, when I thought of Lynn's misunderstanding of probability, I thought it was due to a lack of being able to properly grasp the subject material. Now that I look at these two instances "side-by-side", I realize that it was really due to Lynn's lack of interest of the subject material. From the interest would have developed curiosity to explore the material in further depth, and from curiosity would have come understanding.
On the flip side, Jim was an extremely curious person, who spent most of this free time finding new math problems or "hands-on" puzzle toys to challenge his students with. His curiosity led to his knowledge and passion for mathematics, which in turn encouraged his students.
I will be modelling my mathematics teaching career on Jim. Well, with the exception of the part where I instil fear in prospective students.
Saturday, September 19, 2009
How to Tie a Figure Eight Follow-Through Knot
Microteaching Self and Peer Assessment
The BOOPPPS Method: I found the BOOPPPS method (Bridge, Teacher Objectives, Student Objectives, Pretest, Participation, Posttest, Summary) to help keep my lesson organized. The bridge particularly was helpful, and is something that I think I might have overlooked otherwise, focusing strictly on the procedure and not the importance of the Figure Eight Follow-Through Knot.
Positive Relfections: My group found my presentation to be informative and relevant, which let me know that the bridge was very effective. They all mentioned that they really liked doing the participatory section was exceptionally “hands on”, which helped them learn.
My biggest surprise was that all of my students let me know that they really enjoyed the post-test. Rather than have them reproduce the knot that they had learned in the participatory activity, I had each of them recall one of the student objectives by posing a question to them:
“To create a Figure Eight Knot before looping through the carabineer, what steps need to be taken?”
“To create a Figure Eight Follow-Through Knot, which additional steps need to be taken?”
“ How do you safety check your knot?”
Truthfully, I had originally intended to have them do a “hands on” test, but when practicing my lesson a few days earlier, I realized that I needed to cut down on time, and made a conscious decision to change my post-test. I did not expect it to be the highlight of the lesson. I realize now that the post-test of the lesson does not need to put pressure to perform on my students, but simply reinforce the student objectives in a non-stressful way.
Negative Reflections: Two students mentioned that I talked to quickly, and my original instruction could have been done a little slower. I know this is a problem I have when I am nervous, and I will make every attempt to slow down in the future.
One student mentioned that the lesson would have been more effective if they had each been able to tie the knot several times, rather than each student taking turns tying the knot a single time. In reflection, I wish I had purchased another length of rope for the students to practice with.
Interesting Notes: Before I started the microteaching exercise, I did not believe that the pretest and posttest sections were a necessary part of every lesson. As I mentioned earlier, I was pleasantly surprised with how effective the posttest was. I am looking forward to seeing an occasion where the pretest section also surprises me with its effectiveness.
Thursday, September 17, 2009
How to Tie a Figure Eight Follow-Through Knot
Bridge: Rock climbing is an excellent workout that not only demands peak performance from the body, but from the mind as well. While your entire body is clinging to the surface of a rock face, and even your fingertips are getting tired, your mind must be judging where your next hold will be, and how to position your body so that you don’t lose your balance and fall off the wall.
Before even setting foot on the wall, however, you need to secure yourself to the rope using a Figure Eight Follow-Through Knot.
Teacher Objectives: Encourage excitement about rock-climbing through a simple knot exercise.
Student Objectives: Students will be able to tie a Figure Eight Knot and a Figure Eight Follow-Through Knot using the following steps:
(1) Create a Figure Eight Knot by making an alien head, tying a knot around it’s neck, and stabbing it in the eye.
(2) Loop the rope through the carabineer and trace the figure eight all the way through to create the Figure Eight Follow-Through Knot.
(3) Safety check by making sure that there are 5 pairs of parallel ropes.
Pretest: Ask students (1) which knots they are familiar with, (2) if they have ever gone rock climbing, and (3) can the describe the knot used to secure the climber to the rope.
Participation: Students will practice tying a Figure Eight Knot, then secure the rope to a carabineer using a Figure Eight Follow-Through Knot. Students can practice on each end of the rope, allowing two students to participate at once. The other students will help direct.
Posttest: Students will recall the steps required to make both a Figure Eight Knot and a Figure Eight Follow-Through Knot (see “Student Objectives”).
Summary: Before you ever go rock-climbing, you will need to be certified not only on how to tie a Figure Eight Follow-Through Knot, but also on how to properly wear a harness and some basic rock-climbing commands. However, not every instructor teaches the Figure Eight Follow-Through Knot using the “alien head” explanation, but I find that this is the best way to remember it. Even years later, I always think of stabbing the alien in the eye when making my knot.
Wednesday, September 16, 2009
Relational Understanding and Instrumental Understanding
Skemp takes the role of Devil’s Advocate to justify why teachers teach instrumental mathematics. His first justification is the only one I personally believe to be true, and therefore the one I would like to focus on:
“Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier.”
I will conclude this paper by taking another look at this justification and examining its incompleteness. In the meantime, I would like to examine Skemp’s justifications that teachers may be making against teaching relational mathematics.
Skemp mentions that one of the situational factors which contribute to a teacher consciously choosing instrumental teaching over relational teaching is the difficulty of assessing the student.
“From the marks he makes on paper, it is very hard to make valid inference about the mental processes by which a pupil has been led to make them… In a teaching situation, talking with the pupil is almost certainly the best way to find out; but in a class of over 30, it may be difficult to find the time.”
I do not believe it is classroom size that restricts teachers from judging a student’s understanding and assigning an appropriate grade by talking one-on-one. For parents of students with strictly instrumental mathematical knowledge, they will question why their child receives a failing grade despite passing all homework assignments and tests. As a teacher, justifying a subjective mathematical grade to a parent will be very difficult.
Skemp states another factor as being:
“The great psychological difficulty for teachers of accommodating (re-structuring) their existing and long-standing schemas, even for the minority who know they need to, want to do so, and have time for study.”
I’m going to tie this in with an earlier statement Skemp makes:
“Teaching for relational understanding may also involve more actual content. Earlier, an instrumental explanation was quoted leading to the statement ‘Circumference = pd’. For relational understanding, the idea of a proportion would have to be taught first (among others), and this would make it a much longer job than simply teaching the rules as given. But proportionality has such a wide range of other applications that it is worth teaching on these grounds also.”
I believe these statement to be unfair, putting unwarranted blame on individual teachers. Equations used in projectile motion, which in turn is used in both Physics and Math as an application of Algebra, cannot be derived without using Calculus. However, to require Grade 10 students to take Calculus as a prerequisite to Grade 11 Physics, or to postpone the teaching of a core science until second year university, would be a great injustice to science students, and to blame individual teachers, or even entire administrations, for not drastically altering the fundamental structure of Math/Science education at the high school and university level is simply preposterous.
Skemp partially addresses this concern with the following rationalization:
“An individual teacher might make a reasoned choice to teach for instrumental understanding [on the ground] that a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupil.”
To respond to this quote, I need to in turn quote Charles Vanden Eynden, author of Elementary Number Theory, to take Skemp’s statement to it’s extreme conclusion:
“Of course the integers are familiar to us from our earliest introduction to arithmetic. We have manipulated them thousands of times, and have formed an intuitive sense of many of their laws. This intuition carries some danger with it, because it may be hard to see the necessity for a proof to verify patterns that we have confirmed by instances over and over again since grade school. Although intuition can be of immense help in mathematics, no accumulation of special cases is sufficient to prove a general proposition.”
To truly understand a concept as simple as the laws governing integers, a number theory class is required to construct all the necessary proofs. Imagine a Grade 8 students reaction to finding out that they need to attend (and pass) a required Grade 8 proof class.
Again, Charles Vander Eynden says it best:
“Anxiety with respect to proofs makes success less likely in most cases. Therefore, the number of computational problems in [Elementary Number Theory] exceeds that of most number theory texts… Computational problems also lead to understanding of the theory behind them.”
I think the last part of this statement is particularly insightful. “Computational problems also lead to understanding of the theory behind them.” Skemp’s first justification stated “Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier.” However he overlooked the simple truth that instrumental teaching can lead to relational understanding.
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