Presented here, is a brief history of the conflicts surrounding mathematics education throughout the 20th Century. Starting with the Progressivist movement in the early part of the century, followed by New Math in the 1960s and finishing up with the NCTM Standards reforms that are currently being debated we see the development of math curriculum and the controversy it has created.
I find all of this debate on the way math “should” be taught very confusing. We have no real experience and are therefore relying on readings and lectures to show us the way but, there is so much information out there. My mathematics education began to fall apart when my lack of true fundamental understanding of the subject became apparent in honours courses I was taking at UBC. No longer could I just practice and get better, I actually had to think differently and I wasn’t prepared for that. So, I tend to think that we should be working on the students thinking processes. My feeling is that once these are developed properly, the rest should fall in to place. I know how I feel, but how do I substantiate that feeling? What proof is there that I am right? The arguments in the article are interesting, but they are all based on the assumption that progressive is better. Are there statistics to back this up?
Wednesday, September 30, 2009
Longer School Days
I wonder what other candidates think about this http://news.yahoo.com/s/ap/20090927/ap_on_re_us/us_more_school ?
Monday, September 28, 2009
Questions Reflection
There seems to be a real disconnect between what the teachers are trying to teach, and what students think they need to learn. How then do we make students appreciate math once it is not as obvious how the knowledge will apply to their lives? Ilija really stressed the importance of developing student's thinking processes. So do we say to the students that we are not really teaching them math, but developing their thinking processes? It seems that motivation to learn is definitely going to be the hardest thing for us to nurture. I know that I have always struggled to motivate my own education, especially when I felt like I was having a hard time understanding the ideas, and I like math so I can only imagine how difficult it is to motivate someone who doesn't enjoy math to begin with. I hope to gain some insight in to this once I start my practicum.
Questions For Math Teachers and Students Summary
Questions Regarding Mathematics in Secondary Schools
For students:
1. What do you like about other subjects that you don't find in math?
2. Do you think the math you learn will help you in your life and how so?
3. Do you think the grades you get in math reflect your knowledge or understanding?
4. Give reasons math should or shouldn't be learned if a student "knows" they won't use it in their future?
5. Which classes that you are taking currently do you think twill be most useful in your life?
Rate math from 1 to 10.
For teachers:
1. What do you wish someone told you when you were starting out the profession?
2. What are your views on technology in the classroom as a teaching tool?
3. As course learning outcomes shrink, how does this benefit/detract from the class?
4. What methods of evaluation do you find works best?(quizzes, tests, homework, participation)
5. Do you add depth to the course material with extra information like introduction to complex or fundamental ideas? Why or why not?
We interviewed three high school students, all through email. They all found math to be an okay subject but none of them really loved it. Their negative views stemmed from their belief that math class is uncreative and doesn’t allow opportunities for group collaboration, experiments or other less solitary work. Again all three agreed that math is useful and should be learned up to about grade 10, at which point they thought math became useless for everyday life.
The idea of "floundering" in the first few years was apparent. The "how" in asking a colleague was not hard it was the "what?" That is, with all these ideas of classroom management, content, and good rapport swirling around, it was hard to form the best question for that situation. It was clear a mentor is the most valuable source because one cannot find this information in books.
Technology in the classroom was thought of as a good supplement to the course, but all three teachers agreed it was not a good tool to be used in place of teaching. They said that using technology too often or before students fully understand the concept sometimes caused extra confusion for students. One teacher said although his students loved some of the technology/software he used to use in his lessons, he realized they weren’t actually learning anything from it - even though they were having fun - so he has since stopped using it.
The curriculum is shrinking and weakening. Learning outcomes are fewer with less of a level required for mastery in math. This affects us on a global scale as countries continue to "innovate" each year and within a few years their respective curricula has doubled when ours' has decreased. We are trying to "pass" our students and let them graduate from high school. The solution is for a "stream-lined" approach. Most students are not ready to take principles of math 10 since the grade eight and nine curriculum focuses on general math and is not connected with grade 10 and beyond. Enrichment in grade eight and nine is a band-aid fix. The real solution comes in the stream-lining of math from grade one.
All three teachers were very interested in the question about student assessment. All three use a similar method of assessment: constant formative assessment followed by final summative assessment. Using constant feedback the teachers track the students understanding of the material and give bonus marks for quality group work, inquiries that help to further the lesson, and to students willing to present their work to the class. None of the teachers see any reason to punitively mark homework, which they view as content the students are in the process of learning. At the end of each unit students are required to perform on a unit exam, this is the graded portion of the class. The teachers stressed that any bonus marks are given for performance, and not participation. One teacher also uses “math journals” to keep track of her student’s progress. She has them write in their journals during the last two minutes of class every day and then reads over them to test how well they understand the lesson.
The instances where extending or introducing complex ideas can be used as an introductory tool. Of course it is not the main focus but these concepts that go beyond the curriculum can be addressed with research projects on the history of mathematics or when a new topic is being taught. The latter idea is a good transition into a new lesson because it can utilize concepts from physics or other sciences to give a taste of how the concept fits in with the scope of the lesson.
For students:
1. What do you like about other subjects that you don't find in math?
2. Do you think the math you learn will help you in your life and how so?
3. Do you think the grades you get in math reflect your knowledge or understanding?
4. Give reasons math should or shouldn't be learned if a student "knows" they won't use it in their future?
5. Which classes that you are taking currently do you think twill be most useful in your life?
Rate math from 1 to 10.
For teachers:
1. What do you wish someone told you when you were starting out the profession?
2. What are your views on technology in the classroom as a teaching tool?
3. As course learning outcomes shrink, how does this benefit/detract from the class?
4. What methods of evaluation do you find works best?(quizzes, tests, homework, participation)
5. Do you add depth to the course material with extra information like introduction to complex or fundamental ideas? Why or why not?
We interviewed three high school students, all through email. They all found math to be an okay subject but none of them really loved it. Their negative views stemmed from their belief that math class is uncreative and doesn’t allow opportunities for group collaboration, experiments or other less solitary work. Again all three agreed that math is useful and should be learned up to about grade 10, at which point they thought math became useless for everyday life.
The idea of "floundering" in the first few years was apparent. The "how" in asking a colleague was not hard it was the "what?" That is, with all these ideas of classroom management, content, and good rapport swirling around, it was hard to form the best question for that situation. It was clear a mentor is the most valuable source because one cannot find this information in books.
Technology in the classroom was thought of as a good supplement to the course, but all three teachers agreed it was not a good tool to be used in place of teaching. They said that using technology too often or before students fully understand the concept sometimes caused extra confusion for students. One teacher said although his students loved some of the technology/software he used to use in his lessons, he realized they weren’t actually learning anything from it - even though they were having fun - so he has since stopped using it.
The curriculum is shrinking and weakening. Learning outcomes are fewer with less of a level required for mastery in math. This affects us on a global scale as countries continue to "innovate" each year and within a few years their respective curricula has doubled when ours' has decreased. We are trying to "pass" our students and let them graduate from high school. The solution is for a "stream-lined" approach. Most students are not ready to take principles of math 10 since the grade eight and nine curriculum focuses on general math and is not connected with grade 10 and beyond. Enrichment in grade eight and nine is a band-aid fix. The real solution comes in the stream-lining of math from grade one.
All three teachers were very interested in the question about student assessment. All three use a similar method of assessment: constant formative assessment followed by final summative assessment. Using constant feedback the teachers track the students understanding of the material and give bonus marks for quality group work, inquiries that help to further the lesson, and to students willing to present their work to the class. None of the teachers see any reason to punitively mark homework, which they view as content the students are in the process of learning. At the end of each unit students are required to perform on a unit exam, this is the graded portion of the class. The teachers stressed that any bonus marks are given for performance, and not participation. One teacher also uses “math journals” to keep track of her student’s progress. She has them write in their journals during the last two minutes of class every day and then reads over them to test how well they understand the lesson.
The instances where extending or introducing complex ideas can be used as an introductory tool. Of course it is not the main focus but these concepts that go beyond the curriculum can be addressed with research projects on the history of mathematics or when a new topic is being taught. The latter idea is a good transition into a new lesson because it can utilize concepts from physics or other sciences to give a taste of how the concept fits in with the scope of the lesson.
Wednesday, September 23, 2009
Robinson Reflection
I sense that our thinking on math education is being directed slightly. I, for one, am desperate to learn how to teach math at a deeper and more enriched level (relationally if you will). I never learned mathematics in this fashion and so have no clear idea of how I would transfer the ideas of a discussion oriented class in to a mathematics classroom. I feel like the best way to learn something is to at least get the impression that you taught yourself. That is how I’ve felt in most of my arts courses. Through discussions and projects I was allowed to come to my own conclusions, which were usually in line with the course objectives. I have only had one math course where discourse was encouraged and a class project was part of the course work. I did my project on Knot Theory, a topic that is rarely covered in any undergraduate coursework but could be understood easily by secondary students. The learning advantages were clear, I learned more about Knot Theory in the two weeks I worked on the project than I did about non-Euclidean geometry throughout the course. It is ideas like this that I hope to be able to apply to my teaching. I don’t want to send a bunch of computers (in the old-timey sense) out of my classroom, I would rather send out a bunch of thinkers.
Quick Write Reflection
I think 17 year old me would think it is crazy that I am currently pursuing a career in education. I did not like high school. I barely went to high school. Why then, am I doing this? I want to be better than what my high school offered me. For my inspiration I will mainly have to draw on University Professors. That is fine. I will also look to my peers for inspiration. That is also fine. Students are inspiring as well and the reason I am here. I guess you could say my lack of memorable high school teachers inspired me in a way.
Quick Write
I find it hard to do these. I had two high school math teachers and neither were particularly memorable.
Two memorable teachers I had post high school were: Peter Danenhower at Langara College, and Jozsef Solymosi at UBC.
Peteris a really positive dud who always was really good at breaking down really complex ideas for the class. The reason he's so memorable for me is he really knew his stuff. If a student questioned something that he'd done, Peter was never afraid to digress.
Jozsef made his class exciting with lots of discussion. He was also very demanding. Homework was not a result of the lecture, but designed to enhance the lectures.
Two memorable teachers I had post high school were: Peter Danenhower at Langara College, and Jozsef Solymosi at UBC.
Peteris a really positive dud who always was really good at breaking down really complex ideas for the class. The reason he's so memorable for me is he really knew his stuff. If a student questioned something that he'd done, Peter was never afraid to digress.
Jozsef made his class exciting with lots of discussion. He was also very demanding. Homework was not a result of the lecture, but designed to enhance the lectures.
Sunday, September 20, 2009
Reflections on Microteaching Lesson
Overall, I thought my lesson went extremely well. I had not really figured out how I was going to start the lesson so the beginning was a bit rocky. I wanted to reveal the true intent of the lesson after the quiz so that I could get a bit more of a wow reaction. Like I said, this made the beginning a bit rough.
Throughout the lesson I relied a bit too heavily on my passion for the material, I substituted this for actual teaching style at times. This is not to say that I did not have a plan or a clear idea of what my objectives were. If I were trying to teach a subject where my passion was not as obvious I think that this could be a problem.
The technique I used that I think the learners liked was my build up to the learning outcomes. I didn’t reveal right away what I was expecting from them that way, when they took my introductory quiz, they would not have any preconceived ideas of how I wanted them to answer. This gave me a truer knowledge of their understanding of the material.
One thing I learned from the experience is that it is very important to know your audience and have a good idea of where they are coming from. I think having a good idea of who I was teaching to made it much easier to teach the lesson.
Hmm. . . .it seems that I am being fairly negative here, but I enjoyed the exercise and felt like my lesson went really well. I look forward to doing more in the future.
Throughout the lesson I relied a bit too heavily on my passion for the material, I substituted this for actual teaching style at times. This is not to say that I did not have a plan or a clear idea of what my objectives were. If I were trying to teach a subject where my passion was not as obvious I think that this could be a problem.
The technique I used that I think the learners liked was my build up to the learning outcomes. I didn’t reveal right away what I was expecting from them that way, when they took my introductory quiz, they would not have any preconceived ideas of how I wanted them to answer. This gave me a truer knowledge of their understanding of the material.
One thing I learned from the experience is that it is very important to know your audience and have a good idea of where they are coming from. I think having a good idea of who I was teaching to made it much easier to teach the lesson.
Hmm. . . .it seems that I am being fairly negative here, but I enjoyed the exercise and felt like my lesson went really well. I look forward to doing more in the future.
Summary of Microteaching Feedback
The feedback was generally the same. People enjoyed the use of my “quiz” to introduce the general idea of the lesson. Everybody really enjoyed the activity. I came off a little disorganized but made up for it in enthusiasm. My disorganization may have muddled the learning objectives for the lesson. In the end, my enthusiasm for the subject made up for the shortcomings.
Thursday, September 17, 2009
How to play guitar, with a primer on how to make some musicalish noise
BOOPPPS!!!
Bridge: Who wants to be a rock star? Well, not exactly a rock star. Who wants to make music?
Teaching Objective: Make music creation accessible to everybody.
Learning Objective: Students will be able to take a guitar and use it however they want, to create music. Students will also (hopefully) re-examine the boundaries of traditional music.
Pre-test: Short quiz.
Participatory Activity: Making music as a group.
Post-test: Question period. Revisit quiz. Student impressions.
Summary: Music is whatever you want it to be. I used a guitar as a simple example because it is a very flexible instrument that is used very rigidly. Jim Reid of the Jesus and Mary Chain never used to tune his guitar. Ricky Wilson of the B-52’s only used 4 strings on his guitar (it wasn’t a bass guitar). Keith Richards often played in different tunings with only 5 strings. Point is, do what works for you. Anyone can make some form of music.
Caleb’s Intro To Guitar Quiz:
How many strings are on a guitar?
What notes do you tune the strings to?
What makes a minor chord different from a major chord?
Bridge: Who wants to be a rock star? Well, not exactly a rock star. Who wants to make music?
Teaching Objective: Make music creation accessible to everybody.
Learning Objective: Students will be able to take a guitar and use it however they want, to create music. Students will also (hopefully) re-examine the boundaries of traditional music.
Pre-test: Short quiz.
Participatory Activity: Making music as a group.
Post-test: Question period. Revisit quiz. Student impressions.
Summary: Music is whatever you want it to be. I used a guitar as a simple example because it is a very flexible instrument that is used very rigidly. Jim Reid of the Jesus and Mary Chain never used to tune his guitar. Ricky Wilson of the B-52’s only used 4 strings on his guitar (it wasn’t a bass guitar). Keith Richards often played in different tunings with only 5 strings. Point is, do what works for you. Anyone can make some form of music.
Caleb’s Intro To Guitar Quiz:
How many strings are on a guitar?
What notes do you tune the strings to?
What makes a minor chord different from a major chord?
Wednesday, September 16, 2009
Thoughts on Relational and Instrumental Understanding
In Richard R. Skemp’s article he gives examples of several faux amis, each one capable of causing confusion. Skemp proposes that understanding and mathematics are both faux amis and that this could be one problem facing mathematics education. Is an instrumental understanding of any subject, truly understanding? In history, does one truly understand the ratification of Canada’s Constitution just by memorizing the date and using a mnemonic device to remember the names of those involved? No, one must contextualize the event in order to understand its place in history. Math should also receive the contextualization it deserves.
In Skemp’s defence of instrumental understanding he writes: “If what is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily” (8). Is this really a defence? When in life, outside of school, will we ever be faced with a page of problems just waiting to be solved? Practically never, it is a situation that is unique to the school experience, and thus a skill that is of very little practicality.
Continuing Skemp’s (lacklustre) defence of instrumental understanding: “The pupils just won’t want to know all the careful groundwork [the teacher] gives in preparation for whatever is to be learnt next. All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest” (4). Why then, would anybody just ask for the answer? Can we motivate students to learn in another way? According to Skemp, yes: “Relational knowledge can be effective as a goal in itself” (10). I find it odd that in other subjects this is taken for granted. In English, and most other arts classes, we write essays or engage in discussions to show that we have some insight in to the subject at hand. What would an equivalent test of relational knowledge be for math?
Perhaps, one problem is that most people view math instrumentally. Skemp suggests a solution for this problem: “One [point] is whether the term ‘mathematics’ ought not to be used for relational mathematics only” (16). I would tend to agree with him, with one reservation: who is to say that we get the term? I agree that there should be some sort of differentiation between instrumental and relational mathematics but, if the common usage of the word mathematics means instrumental mathematics then it is an uphill battle to make people even see that there is such a thing as relational mathematics.
People want to learn “useful math.” Skemp says of instrumental mathematics: Pupils “will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects, and whose lack has recently been the subjects of complaints by teachers of science, employers and others” (6). I would argue that it is instrumental mathematics that is creating these shortcomings. Employers and science teachers want math skills that are adaptable to new situations. It is not useful to only be able to solve a set of problems on a sheet of paper. It is useful to know how to fit these problems in to one’s schema. Thus, “useful math” is actually relational.
I spent a lot of time after reading this article testing my understanding of fundamental concepts in math. It is often the simplest concepts that can be the most difficult to explain, especially to explain properly but, explaining them is what we do. So, why not do it properly?
References
R.R. Skemp: Relational Understanding and Instrumental Understanding (1976)
In Skemp’s defence of instrumental understanding he writes: “If what is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily” (8). Is this really a defence? When in life, outside of school, will we ever be faced with a page of problems just waiting to be solved? Practically never, it is a situation that is unique to the school experience, and thus a skill that is of very little practicality.
Continuing Skemp’s (lacklustre) defence of instrumental understanding: “The pupils just won’t want to know all the careful groundwork [the teacher] gives in preparation for whatever is to be learnt next. All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest” (4). Why then, would anybody just ask for the answer? Can we motivate students to learn in another way? According to Skemp, yes: “Relational knowledge can be effective as a goal in itself” (10). I find it odd that in other subjects this is taken for granted. In English, and most other arts classes, we write essays or engage in discussions to show that we have some insight in to the subject at hand. What would an equivalent test of relational knowledge be for math?
Perhaps, one problem is that most people view math instrumentally. Skemp suggests a solution for this problem: “One [point] is whether the term ‘mathematics’ ought not to be used for relational mathematics only” (16). I would tend to agree with him, with one reservation: who is to say that we get the term? I agree that there should be some sort of differentiation between instrumental and relational mathematics but, if the common usage of the word mathematics means instrumental mathematics then it is an uphill battle to make people even see that there is such a thing as relational mathematics.
People want to learn “useful math.” Skemp says of instrumental mathematics: Pupils “will at least acquire proficiency in a number of mathem-atical techniques which will be of use to them in other subjects, and whose lack has recently been the subjects of complaints by teachers of science, employers and others” (6). I would argue that it is instrumental mathematics that is creating these shortcomings. Employers and science teachers want math skills that are adaptable to new situations. It is not useful to only be able to solve a set of problems on a sheet of paper. It is useful to know how to fit these problems in to one’s schema. Thus, “useful math” is actually relational.
I spent a lot of time after reading this article testing my understanding of fundamental concepts in math. It is often the simplest concepts that can be the most difficult to explain, especially to explain properly but, explaining them is what we do. So, why not do it properly?
References
R.R. Skemp: Relational Understanding and Instrumental Understanding (1976)
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