Upon finishing Lockhart’s A Mathematician’s Lament, he discusses his main issue with how mathematics is taught in schools. Math is no longer something which is presented to students as a topic to be curious about, but rather a set of rules that they should blindly accept and are drilled into them. In math classes today rules are presented, such as with absolute value inequalities you write down two equations one in which you switch the inequality and negate the numerical value, but it is never explained why they are doing this or to what purpose or even how this procedure came to be. Lockhart argues for math to be brought back to how mathematicians discovered mathematics, through their natural curiosity. The way math classes are structured today, students no longer have that natural curiosity and quite frankly are not really expected to have that curiosity. I am currently taking Combinatorics with Professor Storm, in which there really is no structure to the class. Students come up with the theorems that they present based off of questions that they thought of. At first this type of class was difficult for me to adapt to since I had never experienced a type of class run like this. He does guide the discussion, but the theorems presented all come from students’ ideas. At times there is down time in which students continue to work on the theorems they are currently trying to prove and he will circulate the classroom, giving some advice, but never a straight answer. This is exactly what Lockhart is calling for in math classrooms throughout children’s education starting with grade school. This would be difficult to implement in a classroom without collaboration from other teachers, since by the time students would enter my middle school or high school classroom they would have already learned to expect not to have to think about the process they are taking to do a math problem and to expect that the teacher would present everything to them. In order for this to truly have an effect on students, mathematics should be taught in this manner from the beginning of their schooling. Rather than just developing a lesson according to Lockhart, it would be better to construct the entire year course in this manner. Also, if number theory was presented in this manner all of the “rules” that are drilled into students to memorize, would be discovered by them, making them more memorable and more meaningful. This would make mathematics more interesting to students, with them being driven to act like real mathematicians.
You are very lucky to be experiencing that type of learning with Prof Storm. The difficulty will be in makin this happen in a public setting with younger students who se base level is obviously not as foundation ally solid. Additionally the new pressures from admin, colleagues, and falling back on what is comfortable make it hard... BUT THAT DOESN'T MEAN YOU STOP TRYING
you are very right. THe way it used to be, students were forced to take the math classes and hated every minute of it. They hated it because they did not know how to apply it to real life. They did not ask questions as to why this was important for them to know. They would memorize a problem just for a test. Lockhart says that teaching math is like an art form. we each teach in our own ways and try to make it interesting for everyone. With students understanding where equations and problems come from they will be more inclined to not only memorize it for a test, but remember it for life.
I completely agree Chris. A major problem with the mathematics curriculum is that it's not relatable to kids. They get bored and lose interest in class because they don't see the importance of it or how interesting math can be when you are challenged to think in abstract ways.
As a student in Combinatorics with Professor Storm, I know exactly what Nicole means. I was never taught math by coming up with my own questions and finding the answers to them, which is why I was confused and unsure of the class at the beginning. I did not know what to expect or if I would even be able to handle working like that. However, I have to say that I am now enjoying it. I am able to broaden my thinking and can focus on what interests me within number theory. Thus, finding the answers to the questions I pose is a much more satisfying experience. Through my experiences in this class I can connect with what Paul Lockhart says about students being bored by math. They are hardly ever given the opportunity to think outside of the box. Everything is set out in a very formulaic manner and all the students have to do is follow a set of procedures. Therefore, students are not engaging with the material and are missing out on a lot of exciting material that may harness their enthusiasm for math rather than increase their apathy. For example, Lockhart mentions the story of Mr. C and Mrs. A to discuss area and circumference of a circle and how it makes no sense for a teacher to go through this story when the teacher could tell the story of how this came to be. They could even discuss the various people throughout hundreds of year who have tried to approximate pi. Giving students this kind of background may grab their attention and cause them to look deeper into mathematics. In addition, this shows how math classes focus too much on the “how” and usually ignore the “why.” To illustrate, students are always taught what the quadratic formula is and how to use, but are never told why it works. Isn’t why going to be the inevitable question? How can students internalize the quadratic formula, use it, and understand it, when they have no idea why it works (other than because teachers say so) and where it came from? I just learned how the quadratic formula came to be this semester and now, years later I know why it works. I now have a greater appreciation for it and the mathematics behind, which makes it a shame when students do not get to see the math behind the scenes. Students are not able to inquire because they are always told how to solve a problem and are not expected to question why that solution works. Thus, when Lockhart explains that the way mathematics is taught in schools needs to be changed, I cannot help but to agree with him. Teachers can provide students with the basic tools they need to come up with their own questions and inquiry about how to solve them. Even if teachers posed the questions, students would be more active participants in their math education and would get more out of it. However, this may shock students, like it did me on my first day of Combinatorics, which is why I agree with Nicole. It has to happen throughout the school years.
"Math is no longer something which is presented to students as a topic to be curious about, but rather a set of rules that they should blindly accept and are drilled into them." I completely agree with what Nicole said. It is like learning the quadratic formula. Most of us learned the formula through song and it just stuck with us for well, forever. But we never see it anymore and when we did, we were just plugging in numbers to the formula. we never really learned what it was or why it's like that. In the book, Lockhart was talking to another teacher about paint by numbers. He said "Why do colleges care if you can fill in numbered regions with the corresponding color?" and the reply was, "Oh, well, you know, it shows clear and logical thinking." Well duh. Anyone can fill in a numbered region with the right color. You don't have to be an art major to do that. It does not show clear and logical thinking in the same way that solving a math problem does. But colleges don't care in the least bit if you can color in the lines..they want people who think outside the box. But what we were "taught" before college seems irrelevant now because the things that we learned that should have guided us here, no longer apply...the quadratic formula. Math is supposed to be filled with discovering new things...not memorizing and regurgitating and forgetting.
I agree that one of the problems we face when going to college is that we were taught to "color in the lines", where in college we are expect to "think outside the box". We face this dilemma in not understanding what it truly is to think outside the box because we were never introduced to it. We go to college not really prepared to do college work.
It seems like Professor Storm is really trying to adapt this theory. I took Proofs with him and while it wasn't as student centered to the extent that Nicole and Mariacristina described he still required the class to run lessons. We were given problems to solve with very little guidance, but the point was for us to work it out on our own and to try and fail and try again. I think I learned more by being given the chance to solve a problem and get it wrong. When I realized I didn't have the right answer I could learn from my mistakes and make changes to my techniques and try again. I think this kind of learning gets students more interested in math. I understand what Lockhart is saying about math class being so boring. Handing out formulas and rules for students to follow is no way to learn. I especially liked his description of geometric proofs, which were a nightmare for me in high school for the exact reasons he described. Memorizing all these statements and changing one or two words to fit every other proof was tedious. If students were given the chance to look at a triangle and make their own observations about the sides and the angles, I think it would spark their interest more. Unfortunately today geometric proofs in high school are taught so strictly that if you do not write the exact statement the teacher provided in class, you'll get the answer wrong. Students are writing the right ideas, but not in the exact words as the teacher and getting the answer wrong. There is something seriously wrong with that. Math is supposed to allow students to discover ideas and principles about things that they observe on their own. Instead their being handed a list of theorems, steps and rules they must follow to solve a problem, which seriously inhibits their curiosity and interest in mathematics.
My experience with geometry in high school was exactly as Kayla and Lockhart described it as, a nightmare. I did not discover the properties of triangles or quadrilaterals out of my own curiosity. Instead, I was given the list of properties and told exactly how to prove the theorems. All I had to do was regurgitate them back up. In high school even proofs are presented as just more rules for students to follow. My geometry proofs in high school were framed exactly how Lockhart has them written in his example, strictly as statement reason proofs, always starting off with the first statement restating the information that was given. This proof technique is not at all helpful for students to learn the real flavor of proofs, which is discovering the proof on your own using your own technique. So why are we limiting high school students to learn this version of mathematics when it bastardizes it so badly?
I agree with Kayla, Nicole, and Lockhart—high school geometry is not enjoyable at all. Statement and reason proofs are way too formulaic and again only require memorization. Students just remember how the first proof they did went and continue using that standard while occasionally switching out definitions. The opportunity for creative is completely removed from the equation. Students are not able to play around with a problem or think about how to prove it because they are told how to do so. Like Nicole said, they are then not able to develop their own proof techniques or even their own style. Because they students become so far removed from the problem they do not form meaningful or lasting connections with the material and thus are unable to understanding the thought processes behind what they are doing. Students are simply giving the teacher what is expected and like Kayla said teachers like students to stick to the “script.” How does that benefit a student in any way?
It's really sad to see that we've had similar experiences. There's so much pressure to memorize a list of statements just to have students regurgitate the information on a test. The problem with this is that the information never becomes knowledge so students store it temporarily and forget it when they've passed the tests. Students are definitely not being inspired to learn when they're being taught in this manner.
As Kayla said, all we are told to do in a math class is to memorize information only to spit it back out and then go onto the next class, forgetting what we "learned" the year before. The problem is that we didn't learn much of anything. We learned how to memorize information and apply it over and over again, however we didn't learn anything about that information.
I also took Proofs with Kayla and I'm taking Combinatorics with Nicole, Mariacristina, and Debbie, which are both taught by Prof. Storm. I completely agree with everything that has been said. Proofs was almost an introduction to the teaching style that Storm is using in Combinatorics. Like Kayla said, we were given theorems and then had to figure out how to go about proving them. Since, I was not introduced to this style of teaching before I didn't like it at first because I didn't know what to do. Once I started Combinatorics, my mind was like in shock by the student-based learning. At first I wanted to drop the class because I felt that my brain didn't work the way Storm was asking us to, but that was only because I never knew that it could work that way. As I started to understand what the course was about and what we were being asked to do, I started understanding and learning that my brain was able to work in ways that I wasn't aware of. Granted I liked being able to just plug in numbers in high school because it was organized and clear and it just all made sense to me, but this style of student-based learning is even more satisfying once you get the hang of it. It often frustrates me when Storm doesn't give a straight answer as to wether you are right about something, as Nicole said, however once the proof is finished and I find out that it was right, I feel like I have accomplished something. As Lockhart has expressed, students would be so much more interested in math if it wasn't taught like list of directions, because they would get more gratification out of using their minds and expanding on their own curiosity. Of course there will be students who will want to be lazy and not want to use their brains, but critical thinking is a huge part of education in general and it is of main importance for all students to develop this skill. As Mariacristina said, I agree that a good way to create structure in this style of teaching would be to pose beginning questions or give definitions and let the students' minds wander from there to see what they come up with. We can incorporate this learning into our mathematics curriculum and still have structure.
Michelle, we were in the same group so I completely understand. Proofs was so hard for me at first, but eventually I liked that we were allowed to discover the proof on our own and then share what we found with the class. It was definitely more rewarding to finish a proof and know that I put the pieces together on my own without someone giving me the steps to do it.
I agree with everyone's "thinking outside the box" idea. In high school, all you're required to do is memorize formulas and steps to a problem. Being that we are all math students, this was probably easy for us. We are going to teach math because we enjoy it. But most high school students don't enjoy it. So they won't be interested in just following the directions and memorizing seemingly useless information. If students were taught to think outside the box at an earlier age, it could make students more involved and interested in the class. It would also help prepare students for college and future jobs. That is what the whole purpose of high school seems to be according to the SLOs and Common Core Standards, so one way to help that is to make students inquire into mathematics. I feel we as students weren't sufficiently prepared for the difficult upper level math courses because we have been trained to believe that math is just an advanced version of paint by numbers. We are able to adapt and succeed now in college, but if this teaching was introduced at an earlier age, all students would be able to become better thinkers.
I am also in Combinatorics with Nicole, Mariacristina, Debbie and Michelle and just like they stated, in the beginning it was intimidating. Now that we have been in the class for some time, it makes more sense (most of the time) and is helping me come up with my own questions and being able to prove them. I enjoy the freedom that the class has in the sense that I can make up my own problems, present them to the class and even if it isn’t completely right, I have the rest of the class there to help me fix my mistakes. We are learning from each other and have a professor to guide us. If only high school Geometry had been similar to this course and not a nightmare as Nicole and Kayla mentioned. Every example Lockhart had throughout the book is exactly how I learned the material all through high school. If Geometrical proofs were to change in high school, then students will be more into the course and not just trying to pass a test, and they would be better prepared to take a Proofs course in college if necessary. Those geometrical proofs were basically the only proofs I knew before taking the course in college and once you are told that you will no longer be doing your proofs in that same way it takes a bit of time to be able to change your techniques.
Another point Lockhart made towards the end of the book was that “you don’t start with definitions, you start with problems.” I couldn’t agree more with this statement. It is easier to be working on a problem and have to stop to check the definition than to have a definition without knowing where the word will come up. Definitions, in my opinion are better in context. A list of definitions is useless if you have nowhere to apply the words. What would a classroom of 25 students taking geometry do if they were given a question on the first day of class or first day of a new topic? With the way math is taught today, I am sure most of them would just sit there with blank papers waiting for the teacher to go over the problem on the board instead of trying to come with their own ideas.
Lockhart’s Lament really spoke to me because I feel like it is my lament as well. So many of the problems he points to in math education have been things that have frustrated me as a future educator, especially when I look back on my own career as a student. I was taught math through memorization of formulas and applying those formulas to different practice problems I was given. I always liked math, because to me, it was a game. If you knew the rules and when they were applicable, you could win every time. I was never taught to see the beauty of math as an art form. I think I was able to see this beauty, and that is why I have always been interested, but I really wish I was taught to appreciate the art and the process of creating it. I feel cheated in a way. I never realized it was a creative process, or that I could find another way to solve a problem that might be just as good as the way it was done in the textbook. I really love the parallels Lockhart draws to art and music because first, the metaphors clearly show how misguided our math instruction is in school. Math is an art, and should be taught as such. I also love the metaphor because I have a passion for art as well. I love that every artist’s style is unique, and I really like Lockhart’s suggestion that every mathematician has a unique style as well. Math is not a boring system of numbers, it is a form of expression and art, and an outlet for creativity. We need to teach our students to be artists. To be creative. To think critically and to build their own style and find their own answers to questions. It just seems so absurd that we ask students a question and tell them exactly how to get to the answer. It takes all the excitement out of it. It removes all the true inquiry and learning. I would love to teach math by teaching my students to be artists. To be honest, I’m afraid I don’t know how to do it myself. I was never taught this way, and as Lockhart says, it is a lot of work. But I do believe that the only way to really teach math is to foster creativity, and, as Lockhart puts it, to have an honest intellectual relationship with our students.
I thought "A Mathematician's Lament" brought up some really great ideas of how education, more specifically, math education should be represented in schools. Personally, I think that the teacher that is able to use all these ideas in his/her instruction would be a 'superstar' educator. The curriculum that is given to teachers doesn't easily give the time for any changes. The fact that students have to learn a set of ideas and be able to apply them in time for their next test limits the teacher's ability to change the way it can be taught. I think it's unfortunate that it has to be this way, but I don't really see any other way around it. Although I think the ideas in Lockhart's book help math to be more of a creative subject, new teachers can only try their best to incorporate his methods into their daily instruction. In graduate school, I feel like I've only been learning that the way today's teachers educate is wrong. I find this to be very frustrating and it almost makes me wish I changed my profession. How are the teachers that I had in high school instructing the wrong way if I was able to graduate and had always been on honor roll? If I was able to "get it", why can't other students? Also, I feel like I only hear what the wrong way to teach is; what are some examples of the "right" way to teach? Finally, why is all the pressure on the graduate students who are preparing to become teachers in the future? Shouldn't administrators and principals be putting this pressure on the current teachers? As a graduate student, I sit in classrooms and observe the way the curriculum is currently being taught. This becomes the model for me. How can I learn how to teach better if I don't have a role model to follow? Though I'm not against the ideas in Lockhart's book, I honestly find that it would be a very difficult task to try and incorporate them into everyday lessons.
The key idea of Paul Lockhart’s “A Mathematician’s Lament” is that there has to be a fundamental change on how we teach mathematics in schools today. Lockhart believes that math is really an art and that society has made math into a boring tool to be used by science and technology. He thinks our education system destroys a student’s natural curiosity towards math. Lockhart proposes that students should be allowed to pose their own math problems and that they should be allowed to create their own explanations and proofs to these problems. This book helps me to better understand teaching and learning through inquiry because I was able to reflect on my own math experiences as a student in high school. Maybe if my own initial math learning experiences had not been so rote and dedicated to formula memorization, I would have pursued a study of mathematics at an earlier stage in my life.
Upon finishing Lockhart’s A Mathematician’s Lament, he discusses his main issue with how mathematics is taught in schools. Math is no longer something which is presented to students as a topic to be curious about, but rather a set of rules that they should blindly accept and are drilled into them. In math classes today rules are presented, such as with absolute value inequalities you write down two equations one in which you switch the inequality and negate the numerical value, but it is never explained why they are doing this or to what purpose or even how this procedure came to be. Lockhart argues for math to be brought back to how mathematicians discovered mathematics, through their natural curiosity. The way math classes are structured today, students no longer have that natural curiosity and quite frankly are not really expected to have that curiosity. I am currently taking Combinatorics with Professor Storm, in which there really is no structure to the class. Students come up with the theorems that they present based off of questions that they thought of. At first this type of class was difficult for me to adapt to since I had never experienced a type of class run like this. He does guide the discussion, but the theorems presented all come from students’ ideas. At times there is down time in which students continue to work on the theorems they are currently trying to prove and he will circulate the classroom, giving some advice, but never a straight answer. This is exactly what Lockhart is calling for in math classrooms throughout children’s education starting with grade school. This would be difficult to implement in a classroom without collaboration from other teachers, since by the time students would enter my middle school or high school classroom they would have already learned to expect not to have to think about the process they are taking to do a math problem and to expect that the teacher would present everything to them. In order for this to truly have an effect on students, mathematics should be taught in this manner from the beginning of their schooling. Rather than just developing a lesson according to Lockhart, it would be better to construct the entire year course in this manner. Also, if number theory was presented in this manner all of the “rules” that are drilled into students to memorize, would be discovered by them, making them more memorable and more meaningful. This would make mathematics more interesting to students, with them being driven to act like real mathematicians.
ReplyDeleteYou are very lucky to be experiencing that type of learning with Prof Storm. The difficulty will be in makin this happen in a public setting with younger students who se base level is obviously not as foundation ally solid. Additionally the new pressures from admin, colleagues, and falling back on what is comfortable make it hard... BUT THAT DOESN'T MEAN YOU STOP TRYING
Deleteyou are very right. THe way it used to be, students were forced to take the math classes and hated every minute of it. They hated it because they did not know how to apply it to real life. They did not ask questions as to why this was important for them to know. They would memorize a problem just for a test. Lockhart says that teaching math is like an art form. we each teach in our own ways and try to make it interesting for everyone. With students understanding where equations and problems come from they will be more inclined to not only memorize it for a test, but remember it for life.
DeleteI completely agree Chris. A major problem with the mathematics curriculum is that it's not relatable to kids. They get bored and lose interest in class because they don't see the importance of it or how interesting math can be when you are challenged to think in abstract ways.
DeleteAs a student in Combinatorics with Professor Storm, I know exactly what Nicole means. I was never taught math by coming up with my own questions and finding the answers to them, which is why I was confused and unsure of the class at the beginning. I did not know what to expect or if I would even be able to handle working like that. However, I have to say that I am now enjoying it. I am able to broaden my thinking and can focus on what interests me within number theory. Thus, finding the answers to the questions I pose is a much more satisfying experience. Through my experiences in this class I can connect with what Paul Lockhart says about students being bored by math. They are hardly ever given the opportunity to think outside of the box. Everything is set out in a very formulaic manner and all the students have to do is follow a set of procedures. Therefore, students are not engaging with the material and are missing out on a lot of exciting material that may harness their enthusiasm for math rather than increase their apathy. For example, Lockhart mentions the story of Mr. C and Mrs. A to discuss area and circumference of a circle and how it makes no sense for a teacher to go through this story when the teacher could tell the story of how this came to be. They could even discuss the various people throughout hundreds of year who have tried to approximate pi. Giving students this kind of background may grab their attention and cause them to look deeper into mathematics. In addition, this shows how math classes focus too much on the “how” and usually ignore the “why.” To illustrate, students are always taught what the quadratic formula is and how to use, but are never told why it works. Isn’t why going to be the inevitable question? How can students internalize the quadratic formula, use it, and understand it, when they have no idea why it works (other than because teachers say so) and where it came from? I just learned how the quadratic formula came to be this semester and now, years later I know why it works. I now have a greater appreciation for it and the mathematics behind, which makes it a shame when students do not get to see the math behind the scenes. Students are not able to inquire because they are always told how to solve a problem and are not expected to question why that solution works. Thus, when Lockhart explains that the way mathematics is taught in schools needs to be changed, I cannot help but to agree with him. Teachers can provide students with the basic tools they need to come up with their own questions and inquiry about how to solve them. Even if teachers posed the questions, students would be more active participants in their math education and would get more out of it. However, this may shock students, like it did me on my first day of Combinatorics, which is why I agree with Nicole. It has to happen throughout the school years.
ReplyDeleteBetween both if you I am thrilled and would gladly looks to hire your thoughts in my school should the opportunity present itself!
Delete"Math is no longer something which is presented to students as a topic to be curious about, but rather a set of rules that they should blindly accept and are drilled into them." I completely agree with what Nicole said. It is like learning the quadratic formula. Most of us learned the formula through song and it just stuck with us for well, forever. But we never see it anymore and when we did, we were just plugging in numbers to the formula. we never really learned what it was or why it's like that. In the book, Lockhart was talking to another teacher about paint by numbers. He said "Why do colleges care if you can fill in numbered regions with the corresponding color?" and the reply was, "Oh, well, you know, it shows clear and logical thinking." Well duh. Anyone can fill in a numbered region with the right color. You don't have to be an art major to do that. It does not show clear and logical thinking in the same way that solving a math problem does. But colleges don't care in the least bit if you can color in the lines..they want people who think outside the box. But what we were "taught" before college seems irrelevant now because the things that we learned that should have guided us here, no longer apply...the quadratic formula. Math is supposed to be filled with discovering new things...not memorizing and regurgitating and forgetting.
ReplyDeleteTrue, but at what level does old facts and raw information need to be just memorized!
DeleteI agree that one of the problems we face when going to college is that we were taught to "color in the lines", where in college we are expect to "think outside the box". We face this dilemma in not understanding what it truly is to think outside the box because we were never introduced to it. We go to college not really prepared to do college work.
DeleteIt seems like Professor Storm is really trying to adapt this theory. I took Proofs with him and while it wasn't as student centered to the extent that Nicole and Mariacristina described he still required the class to run lessons. We were given problems to solve with very little guidance, but the point was for us to work it out on our own and to try and fail and try again. I think I learned more by being given the chance to solve a problem and get it wrong. When I realized I didn't have the right answer I could learn from my mistakes and make changes to my techniques and try again. I think this kind of learning gets students more interested in math. I understand what Lockhart is saying about math class being so boring. Handing out formulas and rules for students to follow is no way to learn. I especially liked his description of geometric proofs, which were a nightmare for me in high school for the exact reasons he described. Memorizing all these statements and changing one or two words to fit every other proof was tedious. If students were given the chance to look at a triangle and make their own observations about the sides and the angles, I think it would spark their interest more. Unfortunately today geometric proofs in high school are taught so strictly that if you do not write the exact statement the teacher provided in class, you'll get the answer wrong. Students are writing the right ideas, but not in the exact words as the teacher and getting the answer wrong. There is something seriously wrong with that. Math is supposed to allow students to discover ideas and principles about things that they observe on their own. Instead their being handed a list of theorems, steps and rules they must follow to solve a problem, which seriously inhibits their curiosity and interest in mathematics.
ReplyDeleteMy experience with geometry in high school was exactly as Kayla and Lockhart described it as, a nightmare. I did not discover the properties of triangles or quadrilaterals out of my own curiosity. Instead, I was given the list of properties and told exactly how to prove the theorems. All I had to do was regurgitate them back up. In high school even proofs are presented as just more rules for students to follow. My geometry proofs in high school were framed exactly how Lockhart has them written in his example, strictly as statement reason proofs, always starting off with the first statement restating the information that was given. This proof technique is not at all helpful for students to learn the real flavor of proofs, which is discovering the proof on your own using your own technique. So why are we limiting high school students to learn this version of mathematics when it bastardizes it so badly?
ReplyDeleteI agree with Kayla, Nicole, and Lockhart—high school geometry is not enjoyable at all. Statement and reason proofs are way too formulaic and again only require memorization. Students just remember how the first proof they did went and continue using that standard while occasionally switching out definitions. The opportunity for creative is completely removed from the equation. Students are not able to play around with a problem or think about how to prove it because they are told how to do so. Like Nicole said, they are then not able to develop their own proof techniques or even their own style. Because they students become so far removed from the problem they do not form meaningful or lasting connections with the material and thus are unable to understanding the thought processes behind what they are doing. Students are simply giving the teacher what is expected and like Kayla said teachers like students to stick to the “script.” How does that benefit a student in any way?
DeleteIt's really sad to see that we've had similar experiences. There's so much pressure to memorize a list of statements just to have students regurgitate the information on a test. The problem with this is that the information never becomes knowledge so students store it temporarily and forget it when they've passed the tests. Students are definitely not being inspired to learn when they're being taught in this manner.
DeleteAs Kayla said, all we are told to do in a math class is to memorize information only to spit it back out and then go onto the next class, forgetting what we "learned" the year before. The problem is that we didn't learn much of anything. We learned how to memorize information and apply it over and over again, however we didn't learn anything about that information.
DeleteI also took Proofs with Kayla and I'm taking Combinatorics with Nicole, Mariacristina, and Debbie, which are both taught by Prof. Storm. I completely agree with everything that has been said. Proofs was almost an introduction to the teaching style that Storm is using in Combinatorics. Like Kayla said, we were given theorems and then had to figure out how to go about proving them. Since, I was not introduced to this style of teaching before I didn't like it at first because I didn't know what to do. Once I started Combinatorics, my mind was like in shock by the student-based learning. At first I wanted to drop the class because I felt that my brain didn't work the way Storm was asking us to, but that was only because I never knew that it could work that way. As I started to understand what the course was about and what we were being asked to do, I started understanding and learning that my brain was able to work in ways that I wasn't aware of. Granted I liked being able to just plug in numbers in high school because it was organized and clear and it just all made sense to me, but this style of student-based learning is even more satisfying once you get the hang of it. It often frustrates me when Storm doesn't give a straight answer as to wether you are right about something, as Nicole said, however once the proof is finished and I find out that it was right, I feel like I have accomplished something. As Lockhart has expressed, students would be so much more interested in math if it wasn't taught like list of directions, because they would get more gratification out of using their minds and expanding on their own curiosity. Of course there will be students who will want to be lazy and not want to use their brains, but critical thinking is a huge part of education in general and it is of main importance for all students to develop this skill. As Mariacristina said, I agree that a good way to create structure in this style of teaching would be to pose beginning questions or give definitions and let the students' minds wander from there to see what they come up with. We can incorporate this learning into our mathematics curriculum and still have structure.
ReplyDeleteMichelle, we were in the same group so I completely understand. Proofs was so hard for me at first, but eventually I liked that we were allowed to discover the proof on our own and then share what we found with the class. It was definitely more rewarding to finish a proof and know that I put the pieces together on my own without someone giving me the steps to do it.
DeleteYou appear to have had the perfect class at the perfect time of your career... Just continue those understandings into your future employment!
ReplyDeleteI agree with everyone's "thinking outside the box" idea. In high school, all you're required to do is memorize formulas and steps to a problem. Being that we are all math students, this was probably easy for us. We are going to teach math because we enjoy it. But most high school students don't enjoy it. So they won't be interested in just following the directions and memorizing seemingly useless information. If students were taught to think outside the box at an earlier age, it could make students more involved and interested in the class. It would also help prepare students for college and future jobs. That is what the whole purpose of high school seems to be according to the SLOs and Common Core Standards, so one way to help that is to make students inquire into mathematics. I feel we as students weren't sufficiently prepared for the difficult upper level math courses because we have been trained to believe that math is just an advanced version of paint by numbers. We are able to adapt and succeed now in college, but if this teaching was introduced at an earlier age, all students would be able to become better thinkers.
ReplyDeleteI am also in Combinatorics with Nicole, Mariacristina, Debbie and Michelle and just like they stated, in the beginning it was intimidating. Now that we have been in the class for some time, it makes more sense (most of the time) and is helping me come up with my own questions and being able to prove them. I enjoy the freedom that the class has in the sense that I can make up my own problems, present them to the class and even if it isn’t completely right, I have the rest of the class there to help me fix my mistakes. We are learning from each other and have a professor to guide us. If only high school Geometry had been similar to this course and not a nightmare as Nicole and Kayla mentioned. Every example Lockhart had throughout the book is exactly how I learned the material all through high school. If Geometrical proofs were to change in high school, then students will be more into the course and not just trying to pass a test, and they would be better prepared to take a Proofs course in college if necessary. Those geometrical proofs were basically the only proofs I knew before taking the course in college and once you are told that you will no longer be doing your proofs in that same way it takes a bit of time to be able to change your techniques.
ReplyDeleteAnother point Lockhart made towards the end of the book was that “you don’t start with definitions, you start with problems.” I couldn’t agree more with this statement. It is easier to be working on a problem and have to stop to check the definition than to have a definition without knowing where the word will come up. Definitions, in my opinion are better in context. A list of definitions is useless if you have nowhere to apply the words. What would a classroom of 25 students taking geometry do if they were given a question on the first day of class or first day of a new topic? With the way math is taught today, I am sure most of them would just sit there with blank papers waiting for the teacher to go over the problem on the board instead of trying to come with their own ideas.
Lockhart’s Lament really spoke to me because I feel like it is my lament as well. So many of the problems he points to in math education have been things that have frustrated me as a future educator, especially when I look back on my own career as a student. I was taught math through memorization of formulas and applying those formulas to different practice problems I was given. I always liked math, because to me, it was a game. If you knew the rules and when they were applicable, you could win every time. I was never taught to see the beauty of math as an art form. I think I was able to see this beauty, and that is why I have always been interested, but I really wish I was taught to appreciate the art and the process of creating it. I feel cheated in a way. I never realized it was a creative process, or that I could find another way to solve a problem that might be just as good as the way it was done in the textbook.
ReplyDeleteI really love the parallels Lockhart draws to art and music because first, the metaphors clearly show how misguided our math instruction is in school. Math is an art, and should be taught as such. I also love the metaphor because I have a passion for art as well. I love that every artist’s style is unique, and I really like Lockhart’s suggestion that every mathematician has a unique style as well. Math is not a boring system of numbers, it is a form of expression and art, and an outlet for creativity. We need to teach our students to be artists. To be creative. To think critically and to build their own style and find their own answers to questions. It just seems so absurd that we ask students a question and tell them exactly how to get to the answer. It takes all the excitement out of it. It removes all the true inquiry and learning.
I would love to teach math by teaching my students to be artists. To be honest, I’m afraid I don’t know how to do it myself. I was never taught this way, and as Lockhart says, it is a lot of work. But I do believe that the only way to really teach math is to foster creativity, and, as Lockhart puts it, to have an honest intellectual relationship with our students.
I thought "A Mathematician's Lament" brought up some really great ideas of how education, more specifically, math education should be represented in schools. Personally, I think that the teacher that is able to use all these ideas in his/her instruction would be a 'superstar' educator. The curriculum that is given to teachers doesn't easily give the time for any changes. The fact that students have to learn a set of ideas and be able to apply them in time for their next test limits the teacher's ability to change the way it can be taught. I think it's unfortunate that it has to be this way, but I don't really see any other way around it. Although I think the ideas in Lockhart's book help math to be more of a creative subject, new teachers can only try their best to incorporate his methods into their daily instruction. In graduate school, I feel like I've only been learning that the way today's teachers educate is wrong. I find this to be very frustrating and it almost makes me wish I changed my profession. How are the teachers that I had in high school instructing the wrong way if I was able to graduate and had always been on honor roll? If I was able to "get it", why can't other students? Also, I feel like I only hear what the wrong way to teach is; what are some examples of the "right" way to teach? Finally, why is all the pressure on the graduate students who are preparing to become teachers in the future? Shouldn't administrators and principals be putting this pressure on the current teachers? As a graduate student, I sit in classrooms and observe the way the curriculum is currently being taught. This becomes the model for me. How can I learn how to teach better if I don't have a role model to follow? Though I'm not against the ideas in Lockhart's book, I honestly find that it would be a very difficult task to try and incorporate them into everyday lessons.
ReplyDeleteYeny Santiago October 12, 2012
ReplyDeleteThe key idea of Paul Lockhart’s “A Mathematician’s Lament” is that there has to be a fundamental change on how we teach mathematics in schools today. Lockhart believes that math is really an art and that society has made math into a boring tool to be used by science and technology. He thinks our education system destroys a student’s natural curiosity towards math. Lockhart proposes that students should be allowed to pose their own math problems and that they should be allowed to create their own explanations and proofs to these problems.
This book helps me to better understand teaching and learning through inquiry because I was able to reflect on my own math experiences as a student in high school. Maybe if my own initial math learning experiences had not been so rote and dedicated to formula memorization, I would have pursued a study of mathematics at an earlier stage in my life.