This article is about the three main movements in mathematics education in North America for the twentieth-century. The first movement was Progressivist reform (circa 1910-1940). It changed from “inculcation of meaningless memorized procedures” to “split between knowing and doing, or abstract and applied knowledge, proposing that students must engage in doing math as part of a reflective inquiry if they were to increase their intelligence and gain knowledge develop scientific and democratic thinkers.” Students were given the “challenge of doing and experimentation in math, accompanied by the sense-making activities of reflective practice” rather then “rigidly obedient rule-followers.” The second movement was the New Math reform of the 1960s. After USSR was beating the US in the space race, they changed the math curriculum for K-12, mainly focus on highly abstract math concepts. They wanted to familiarize students with the math basis for future careers as scientists, but many teachers had little knowledge with the new math topics, and they didn’t understand why new ways were being introduced. Therefore, many teachers, parents against the New Math since it trained students to be future scientist but not realize not all of them what to pursue that career. The third movement was the NCTM Standards-based Math Wars, from the 1990s to the present. This is a back-to-basics curriculum. They emphasized the development of flexible problem-solving skills and valued deep understanding and an ability to make math connections above calculation skills, although fluency in calculation was still considered an important goal.
In my opinion, I doubt that the reason for students in Asian countries have higher ranked in the world is because they have deeper conceptual understanding of mathematics. This is because I heard most of my Asian friends said they learned mathematic almost one hundred percent instrumentally before they came. Also, author said most people have negative attitude toward math. The reason might be that we didn’t have good math teacher at elementary school so seldom students understand the concepts, and students started to dislike math. Therefore, I think we definitely should have math specialized teacher at the elementary school.
Tuesday, September 29, 2009
Sunday, September 27, 2009
Summary to Interviews with Teacher and Students
In compliance with BCCT standards, “educators will engage in career-long learning”. Similarly, as teacher candidates and future educators we will continue to learn from our professors as well as from teacher and student interviews. In this interview, we have in our group of three collaboratively created a list of nine questions we most wanted to ask high school math teachers and students. From our interview with a math teacher and two high school students, we learned of the various resources and styles of teaching that we can make available to students to facilitate their learning in math.
Our first two interviewees were high school female students in grades nine and twelve respectively. We asked these students if they have troubles doing math, what would they do and why? The grade nine student said “if I have problem[s] doing math, I will go and ask my teacher” since she felt her teacher was approachable. The second student said that she will ask her friends because “discussions with her friends are good enough”. As teacher candidates then, we must recognize that our availability and approachability for students plays an important role in helping them learn math.
We also asked our two interviewees which area of math they find difficult and how their teachers can help them learn the topic better. One student has difficulties with volumes and angles and she prefers to learn “by sitting and taking notes” whereas the other student has difficulties with trigonometry and prefers to learn math “visually.” Thus, as teacher candidates we must keep in mind that each individual’s ways of learning is as different as the colors of the rainbow.
Our forth question was what can teachers do to motivate the students to learn math. In general, both students said it was hard to motivate them in a subject they disliked. However, they suggested that their teachers can try to interest them by making “the lesson a funner [group] activity.” Ironically, this ties into our last question, in which we asked the students how they would define a good math teacher. The grade nine student described “a good math teacher [as] someone [who] makes the class a little more fun and keeps everyone from NOT falling asleep.” On the other hand, the grade twelve student said a good math teacher “shoudn’t be monotone” and should make her “feel interested in what he/she teaches.”
In our interview with the high school math teacher however, we asked her what is the hardest thing to be a good math teacher? She replied “the most difficult part of being a math teacher is getting the students to get excited about Math.” What is interesting, however, is that she told us “some students say that they were once interested in Math” Therefore, as teachers it’s important that we try our best to teach math in fun and interesting ways.
This led into our second question inquiring any teaching advice(s) that our interviewee can give to teacher candidates. In her response, she told us we should avoid getting “disillusioned with teaching especially if the students are unmotivated.” In other words, it is important for teachers to overcome any teaching difficulties or harsh criticisms that she/he may encounter in the profession by approaching these problems optimistically and continuing to learn and develop professionally. For example, if a student is distracted, we shouldn’t take it personally and lose our initiative in teaching. Rather, we can refocus students back to the math lesson by “asking the distracted student to answer the question.” Distracted students also serve as a valuable sign in how well teachers are engaging their students in the class.
Thus, we asked the interviewee which is/are the most effective techniques she has used in teaching and why? She replied “I try to use real life examples…so that the seemingly abstract concepts can be more concrete.” This method however, “works only most of the time.” The reason for this as reflected by the interviewee is that she “wasn’t accommodating other learners especially those who have learning differences.” She also noted that “it was challenging to NOT teach the way she learned the concept.” Therefore, as teacher candidates, we should be flexible and adaptive in our teaching so that we are able to accommodate most if not all our students. Another effective teaching technique the interviewee had used all the time was “simplifying a complex concept using simpler examples.”
In conclusion, the interviews with the students helped us gain insight into their expectations from their math teachers and the methods that can be used to facilitate their learning in math. Generally, a great math teacher should be flexible and open-minded in his/her teaching to create a comforting classroom environment that involves plenty of excitement and fun group activities for students to engage in.
by Jenny, Maryanne, Candice
Our first two interviewees were high school female students in grades nine and twelve respectively. We asked these students if they have troubles doing math, what would they do and why? The grade nine student said “if I have problem[s] doing math, I will go and ask my teacher” since she felt her teacher was approachable. The second student said that she will ask her friends because “discussions with her friends are good enough”. As teacher candidates then, we must recognize that our availability and approachability for students plays an important role in helping them learn math.
We also asked our two interviewees which area of math they find difficult and how their teachers can help them learn the topic better. One student has difficulties with volumes and angles and she prefers to learn “by sitting and taking notes” whereas the other student has difficulties with trigonometry and prefers to learn math “visually.” Thus, as teacher candidates we must keep in mind that each individual’s ways of learning is as different as the colors of the rainbow.
Our forth question was what can teachers do to motivate the students to learn math. In general, both students said it was hard to motivate them in a subject they disliked. However, they suggested that their teachers can try to interest them by making “the lesson a funner [group] activity.” Ironically, this ties into our last question, in which we asked the students how they would define a good math teacher. The grade nine student described “a good math teacher [as] someone [who] makes the class a little more fun and keeps everyone from NOT falling asleep.” On the other hand, the grade twelve student said a good math teacher “shoudn’t be monotone” and should make her “feel interested in what he/she teaches.”
In our interview with the high school math teacher however, we asked her what is the hardest thing to be a good math teacher? She replied “the most difficult part of being a math teacher is getting the students to get excited about Math.” What is interesting, however, is that she told us “some students say that they were once interested in Math” Therefore, as teachers it’s important that we try our best to teach math in fun and interesting ways.
This led into our second question inquiring any teaching advice(s) that our interviewee can give to teacher candidates. In her response, she told us we should avoid getting “disillusioned with teaching especially if the students are unmotivated.” In other words, it is important for teachers to overcome any teaching difficulties or harsh criticisms that she/he may encounter in the profession by approaching these problems optimistically and continuing to learn and develop professionally. For example, if a student is distracted, we shouldn’t take it personally and lose our initiative in teaching. Rather, we can refocus students back to the math lesson by “asking the distracted student to answer the question.” Distracted students also serve as a valuable sign in how well teachers are engaging their students in the class.
Thus, we asked the interviewee which is/are the most effective techniques she has used in teaching and why? She replied “I try to use real life examples…so that the seemingly abstract concepts can be more concrete.” This method however, “works only most of the time.” The reason for this as reflected by the interviewee is that she “wasn’t accommodating other learners especially those who have learning differences.” She also noted that “it was challenging to NOT teach the way she learned the concept.” Therefore, as teacher candidates, we should be flexible and adaptive in our teaching so that we are able to accommodate most if not all our students. Another effective teaching technique the interviewee had used all the time was “simplifying a complex concept using simpler examples.”
In conclusion, the interviews with the students helped us gain insight into their expectations from their math teachers and the methods that can be used to facilitate their learning in math. Generally, a great math teacher should be flexible and open-minded in his/her teaching to create a comforting classroom environment that involves plenty of excitement and fun group activities for students to engage in.
by Jenny, Maryanne, Candice
Self-Reflection on Interview with Teacher and Students
In our interview with two female high school students, they both said that a good math teacher should make math classes more excited and more fun, which is not surprising since I agree that the most challenge part to teach math is to get the students interested and excited in math, and this corresponds with our teacher interviewee’s response. One way to make math more fun is group activities!
From teacher interviewee’s responses, I learn that we have to know various ways of solving same problems since every student learn and understand concepts differently, so we have to use different methods to teach them. From our interview with students, one like taking notes, yet one learn better by group activities, so it is really important for teachers to try different methods of teaching. To sum up, we have to flexible and open-minded.
Furthermore, interviewee said she ask teacher for help and the other one said she ask friends for help. So, I think it is really important to make myself approachable to students, and encourage them to come to me if they have questions; also, have some office hour or extra time outside the class.
From teacher interviewee’s responses, I learn that we have to know various ways of solving same problems since every student learn and understand concepts differently, so we have to use different methods to teach them. From our interview with students, one like taking notes, yet one learn better by group activities, so it is really important for teachers to try different methods of teaching. To sum up, we have to flexible and open-minded.
Furthermore, interviewee said she ask teacher for help and the other one said she ask friends for help. So, I think it is really important to make myself approachable to students, and encourage them to come to me if they have questions; also, have some office hour or extra time outside the class.
Wednesday, September 23, 2009
Response to Heather J. Robinson’s Article
“Using Research to Analyze, Inform, and Asses Changes in Instruction”
by Heather J. Robinson’s
I agree that she said “students learn best when they are actively engaged in the thinking about and doing mathematics.” Hence, I think teacher should give some time for group work and group discussion for every lecture and I think it is easier for students to talk to classmates rather than teachers. In addition, it is important to build an atmosphere “where students appreciate each other’s ideas and not afraid to be wrong in order to accomplish learning.” Hence, we have to encourages freedom of expression and value their own ideas, as well as respect and reward unusual or different ideas.
However, I’m doubtfully about how practical we can help students develop their critical thinking and problem solving skills since most of our curriculum topics are hard to teach them thinking critically, and plus group discussion takes a lot of time. Due to the topics/skills we need to teach students according to curriculum, it might be hard to give students a lot of time for discussion and teach them all the skills they have to know, sometimes it is just simply don’t have enough time, and ultimately and unfortunately people evaluate students by their grades/scores on the exam.
I have one more question about why her students score so high in class grade, but poorly in common final exam. Is it because she teaches total instrumentally, so students don’t understand it that they just simply memorize the formulas/rules, so they don’t know which one to apply when all kinds of questions are mixed together in final exam? Or is it because the questions on the final exam are just too different? Nevertheless, as a teacher, we are all life-long learners; we learn from experience, peers and students.
by Heather J. Robinson’s
I agree that she said “students learn best when they are actively engaged in the thinking about and doing mathematics.” Hence, I think teacher should give some time for group work and group discussion for every lecture and I think it is easier for students to talk to classmates rather than teachers. In addition, it is important to build an atmosphere “where students appreciate each other’s ideas and not afraid to be wrong in order to accomplish learning.” Hence, we have to encourages freedom of expression and value their own ideas, as well as respect and reward unusual or different ideas.
However, I’m doubtfully about how practical we can help students develop their critical thinking and problem solving skills since most of our curriculum topics are hard to teach them thinking critically, and plus group discussion takes a lot of time. Due to the topics/skills we need to teach students according to curriculum, it might be hard to give students a lot of time for discussion and teach them all the skills they have to know, sometimes it is just simply don’t have enough time, and ultimately and unfortunately people evaluate students by their grades/scores on the exam.
I have one more question about why her students score so high in class grade, but poorly in common final exam. Is it because she teaches total instrumentally, so students don’t understand it that they just simply memorize the formulas/rules, so they don’t know which one to apply when all kinds of questions are mixed together in final exam? Or is it because the questions on the final exam are just too different? Nevertheless, as a teacher, we are all life-long learners; we learn from experience, peers and students.
Two of my most memorable math teachers
One of my most memorable teachers is my junior math teacher, Mr. Hsu. He is really strict; I think he is the strictest teacher I ever had, but he is really reasonable. He has great knowledge in math. His thoughts are clear, and he is really good at explaining math concepts, he makes sure everyone understands them. He gives us a lot of homework since he says math need a lot of practice in order to be good at it. Most students don’t really like him at that time because everyone thinks he gives us too much homework, yet unbelievable he is the one who builds my cornerstone for math! I learn so much from him. I realize I have a strong math basic skill after I go into senior math class. He makes me realize that no pain, no gain. I have to admit that I don’t really like him that much at that time, but now I really thank him for what he taught me!
My second most memorable math teacher is Mrs. C. She is so passionate about math, and she knows the subject really well. Her writing is neat, and notes are really organized and clear. It’s surprised how often I refer back to her notes later on. During the class, she sometimes told us a lot of stories/experience she has, which made the class so much fun! Also, she is really approachable and always concerns about students. She is available after school if someone needs helps. She has such a great rapport with students!! Everyone just loves her!
Even thought the personalities are so different between these two wonderful teachers, but both of them have great knowledge in math and good at explaining the concepts! In order to be a great math teacher, we not only have to know our subject well, but also have to know how to explain clearly so students can understand.
Sunday, September 20, 2009
Reflection on Microteaching
Peer’s evaluation:
Strengths of my lesson:
It’s interesting, engaging, and fun to follow activity
Good time management
Clear explanation
Visual demonstration and hands-on activities
Areas need further work and development:
Could be a little louder voice (hard to hear in a noisy classroom)
Maybe give a little history of origami
Maybe give students chances to learn it on their own by looking at the handout
Self assessment:
Things went well in my lesson:
Time management was good.
I made sure everyone followed me step-by-step so no one was behind.
Everyone made their own origami by the end of lesson
If I were to teach again, I would work to improve it by:
More information about origami, such as the history of origami
If I got more time, I’ll have a fun activity at the end by using the origami they just made
Here are some things I reflected on based on my peers’ feedback:
I should work on voice, speak louder next time.
Most people think the learning object was clear, but some think it’s not too clear, so I could talk a bit more about origami before I start to teach them how to fold a motorboat.
Most people think it’s a good way to show the origami I made in order to catch people’s attention and interests. Also, most people think it's good that I check to see how everyone was progressing and provide a useful link for people who are interested in origami.
Strengths of my lesson:
It’s interesting, engaging, and fun to follow activity
Good time management
Clear explanation
Visual demonstration and hands-on activities
Areas need further work and development:
Could be a little louder voice (hard to hear in a noisy classroom)
Maybe give a little history of origami
Maybe give students chances to learn it on their own by looking at the handout
Self assessment:
Things went well in my lesson:
Time management was good.
I made sure everyone followed me step-by-step so no one was behind.
Everyone made their own origami by the end of lesson
If I were to teach again, I would work to improve it by:
More information about origami, such as the history of origami
If I got more time, I’ll have a fun activity at the end by using the origami they just made
Here are some things I reflected on based on my peers’ feedback:
I should work on voice, speak louder next time.
Most people think the learning object was clear, but some think it’s not too clear, so I could talk a bit more about origami before I start to teach them how to fold a motorboat.
Most people think it’s a good way to show the origami I made in order to catch people’s attention and interests. Also, most people think it's good that I check to see how everyone was progressing and provide a useful link for people who are interested in origami.
Friday, September 18, 2009
Lesson Plan for Microteaching
BOOPPPS lesson plan for making a motorboat origami !!
BRIDGE
We can have any kind of objects by folding a piece of paper.
Show students some origami I made.
Teaching OBJECTIVES
Get students interested in origami
Learning OBJECTIVES
Students will be able to fold a motorboat using a piece of paper.
PRETEST
Asking students “Do anyone know about origami?” and “Do anyone know how to fold anything using a piece of paper?”
PARTICIPATORY Activity Ideas
Students have a piece of paper and follow me step-by-step.
POST-TEST
Students have their origami motorboat and know how to folding a motorboat by themselves. I’ll provide a handout which has the steps in case they forget.
SUMMARY
Folding a boat is just an easy one of origami.
Give students a useful website of origami, so they have more resources if they are interested, and they could try harder origami by themselves.
Commentary on Skemp's article
In Skemp’s article, he defines the differences between instrumental understanding and relational understanding, and provides several examples to demonstrate the differences, which makes the readers easier to understand the concepts. He shows he believes relational understanding is better, yet he tries to understand why others use instrumental understanding. This shows he is reflective.
According to Skemp’s article, he says “there are two kinds of mathematical mis-matches which can occur.”(4) This is such a good point I never really thought about before. After I give it some thoughts, I believe it is inevitable, and a way to reduce this is to use both instrumental and relational understanding when teaching. This leads to how often and when to use which. For the concept of relational understanding is not too complex, teach relational first. After a while, teach instrumental for students who could not understand relational, so they would not feel they are beyond and still get a sense of confident. For instance, the concept of area that we can teach how to find the area of triangles, trapezoids, etc. relational first, and then instrumentally tell students those really convenient formulas for area. By doing this, we can satisfy both group of students. And this gets more elaborate when he says “I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’.”(6) A good teacher is flexible. People often agree that relational is better, but what if a teacher teach relational all the time, but none or seldom of his students can understand it. This would just make things worse. A good teacher will teach either relational or instrumental understanding or both depending on the students he has. Skemps also mentions that relational “is easier to remember.”(9) I agree, but not always. I have some rules I learned in elementary, and I still remember them now; those rules just stick in my mind. I agree that Skemps says “relational understanding of a particular topic is too difficult,”(11) and students just could not understand them at that stage. Relational understanding might just scared them away, and they would feel overwhelming and start to dislike math. He also mentions “a junior teacher in a school where all the other mathematics teaching is instrumental.”(11) This back to beginning where a good teacher should be responsible and know what is suitable for his students, relational or instrumental, since every teacher has different kinds of students every class, every term, and every year.
According to Skemp’s article, he says “there are two kinds of mathematical mis-matches which can occur.”(4) This is such a good point I never really thought about before. After I give it some thoughts, I believe it is inevitable, and a way to reduce this is to use both instrumental and relational understanding when teaching. This leads to how often and when to use which. For the concept of relational understanding is not too complex, teach relational first. After a while, teach instrumental for students who could not understand relational, so they would not feel they are beyond and still get a sense of confident. For instance, the concept of area that we can teach how to find the area of triangles, trapezoids, etc. relational first, and then instrumentally tell students those really convenient formulas for area. By doing this, we can satisfy both group of students. And this gets more elaborate when he says “I used to think that maths teachers were all teaching the same subject, some doing it better than others. I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’.”(6) A good teacher is flexible. People often agree that relational is better, but what if a teacher teach relational all the time, but none or seldom of his students can understand it. This would just make things worse. A good teacher will teach either relational or instrumental understanding or both depending on the students he has. Skemps also mentions that relational “is easier to remember.”(9) I agree, but not always. I have some rules I learned in elementary, and I still remember them now; those rules just stick in my mind. I agree that Skemps says “relational understanding of a particular topic is too difficult,”(11) and students just could not understand them at that stage. Relational understanding might just scared them away, and they would feel overwhelming and start to dislike math. He also mentions “a junior teacher in a school where all the other mathematics teaching is instrumental.”(11) This back to beginning where a good teacher should be responsible and know what is suitable for his students, relational or instrumental, since every teacher has different kinds of students every class, every term, and every year.
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