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.
Wednesday, September 30, 2009
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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