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.
Thursday, October 8, 2009
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.
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