tag:blogger.com,1999:blog-8725813156088961385.post6697112603040748353..comments2014-06-06T02:25:18.619-07:00Comments on Dialogue in Mathematics Research: The Pedagogical Content Knowledge in Mathematics: Pre-Service Primary Mathematics Teachers' Perspectives in TurkeyPatricia Dickensonhttps://plus.google.com/101987906598017674170noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-8725813156088961385.post-80294892443587201152011-07-27T15:54:17.950-07:002011-07-27T15:54:17.950-07:00This was an excellent article introducing the impo...This was an excellent article introducing the importance of mathematical content knowledge and pedagogical knowledge as a teacher. When these two concepts intertwine, a dynamic teacher rises. By having the teacher be able to fully understand the concept, they are able to fully introduce it to the student in ways that it helps them ask the correct question so they can fully grasp the concept on their own. I feel that is the overall goal of the “dynamic” combination of these two concepts. And I truly feel that is what is required for a new brand of teachers to have in order to be able to introduce the common concepts of mathematics that students continue to struggle with. <br />I agree with Noemi, that understanding the student’s thought process is the most challenging and rewarding aspects of teaching math. The reason is because when you are able to understand their thought process, one is able to present the material in ways they will find a connection with. By creating this relationship, the students begin to understand the concept in their own terms and begin to make connections on to the new ideas being introduced. <br />Also the idea of extinguishing the negative aspects of WRONG answers was interesting. Instead of just saying the answer is wrong and begin to explain the answer; one can use that misconception to allow the students to figure out (on their own) why the answer is wrong and thus come up with the solution. That process of questioning in order to guide the student to the correct answer is something that I find to be quite successful and interesting while tutoring. This would definitely be something I would like to incorporate in to my own form of teaching.Juan C.https://www.blogger.com/profile/17922602786024933828noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-54954130688217031232011-07-27T14:00:35.605-07:002011-07-27T14:00:35.605-07:00One idea that this paper articulated well was that...One idea that this paper articulated well was that in addition to guiding students how to construct meaningful understanding of the material, an excellent teacher IDENTIFIES fundamental misunderstandings in a students' knowledge and then INTERPRETS those misconceptions.<br /><br />Understanding a student's thought process is, in my opinion, one of the most challenging and rewarding aspects of teaching math. Sometimes, looking at the student's work alone may not be enough to understand his or her thought process (especially if they have arrived at the correct numerical answer) so, as the paper explains, it is important to ask questions. These questions should not ask the student do explain the task – rather, to describe their thought process. From this, a good teacher will interpret the student's misconception, and will develop a method to address it.<br /><br />I recognize the importance of asking, not telling, to allow students the opportunity to develop their own understanding. Equally important and necessary is the use of intentional questions that are responsive to students' needs and will guide them to their gap in understanding or inconsistency in their reasoning.<br /><br />As I've developed as a in my teaching and tutoring, I've begun to ask my students more questions, but asking more concise and constructive questions is something that I would like to improve on. I look forward to learning more strategies to do this.<br /><br />NaomiNaomi Branzhttps://www.blogger.com/profile/03089382851242561299noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-38804511552799652922011-07-27T00:46:05.787-07:002011-07-27T00:46:05.787-07:00I found this article very engaging. It really put ...I found this article very engaging. It really put into perspective about how some teachers are in the public education system. It shows how desperately the education system needs teachers who are capable of math and also capable of learning how to teach. I agree with Rick because this article does a good job of creating a balance between teaching students and understanding math.<br /><br />Some of the responses to the 4 questions seemed so outrageous and unheard of, probably due to our background in math. I believe since we are proficient in mathematics as teachers we should make the effort to learn about how to teach and how to make the subject easy to learn. <br /><br />Teacher preparation should focus on training those that are proficient in mathematics so we can balance out the content knowledge with how to teach. Teacher preparation should also focus on different representations. Seeing as many representations as we could helps us see how other students learn. It also opens our eyes to problems that students may encounter and common issues that students have. <br /><br />Overall, this was a great article to read because it helps me build up a little more confidence with becoming a teacher and it challenges me to see how some students think.<br /><br />ChrisAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-65625062897587508002011-07-27T00:23:44.823-07:002011-07-27T00:23:44.823-07:00And by Juan I meant Jesus. lol whoops. Goodnight :...And by Juan I meant Jesus. lol whoops. Goodnight :) <br />-AlsaciaAlsaciahttps://www.blogger.com/profile/03953485953782916931noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-29950927284804064662011-07-27T00:08:03.791-07:002011-07-27T00:08:03.791-07:00I agree that this article does a great job in stre...I agree that this article does a great job in stressing the importance that effective teachers have in-depth knowledge of both pedagogy and mathematics content to teach students for conceptual understanding. Because there is substantial evidence to support this claim, I think it should be the main goal of Teacher Preparation programs in preparing teachers for their role in the classroom. <br /><br />The results of this analysis were shocking & disappointing, but not surprising. It seems to be a pretty common thing that teachers in the classrooms don't fully understand the content they're teaching and therefore cannot teach it effectively to students. As Melissa said, when you simply state facts to students, you aren't teaching them anything. They need to discover procedures on their own in order to truly understand the meanings behind them. Additionally, it is important that teachers thoroughly understand these procedure/concepts in order to help students make these discoveries (and also to understand their possible conceptions/misconceptions).<br /><br />I really liked the idea of looking at samples of student work and analyzing their steps to see where they are coming from (it's like a teaching/learning technique). There are so many possible reasons for their solutions and I think it's important to have practice in this technique so that when we experience it in the classroom we have some idea at how to approach it. <br /><br />In response to Juan's question: I feel that it doesn't matter which level math classes you teach, there should always be a balance between content and pedagogical knowledge. There is "more to know" if you're looking at math as sequential (in the sense that you need to know algebra for calculus), but overall I think that teachers of all subjects should understand their content enough to be able to teach it effectively.Unknownhttps://www.blogger.com/profile/03953485953782916931noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-4841757020974649462011-07-26T22:24:10.474-07:002011-07-26T22:24:10.474-07:00Melissa Wallace
Teacher Preparation programs shou...Melissa Wallace<br /><br />Teacher Preparation programs should prepare teachers by providing them problems that develop their conceptual understanding of mathematics. It's not difficult for anyone to understand and teach the procedure of a mathematical concept, for example, multiplying fractions. One can tell students, "this is how you do it. Multiply the numerator and denominator". However, this has no meaning to students. Teacher preparation programs should provide problems, similar to the one in the article, for teachers to practice analyzing students' misconceptions about math, and provide teachers with opportunities to develop different methods in order to develop students' content knowledge. The example on page 11, with dividing the squares into fourths, and then fifths to represent, 1/4 * 1/5 is one method teachers should use to explain the concept of fraction multiplication.<br /><br />I really enjoyed reading this article. As I was reading through the four problems that were given to see the teacher's interpretation of students' misconceptions or mathematical knowledge, it made me think of potential methods,or ways of teaching this concept in order for students to develop conceptual understaing, and procedural fluency. I was surprised and disappointed by the results. The fact that out of 50 teachers, none of them are level 3, excellent, and how out of those 50, 8 teachers were considered insufficient. FOr problem 2, less than half of the teachers understood the student's misconception, which concerns me that many of the math teachers are actually unprepared to be teaching math because they do not have the knowledge of both pedagogy nor mathematics content knowledge.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-13078266323059152362011-07-26T22:22:27.752-07:002011-07-26T22:22:27.752-07:00I agree with Rick and Christy that the article sho...I agree with Rick and Christy that the article showcased the necessity of having both mathematical content knowledge and pedagogical knowledge is important. Its very interesting to continue to read and hear that being a good mathematician is not good enough to be a good teacher. Its important for the Teacher Preparation Programs to provide a wide spectrum of examples and opportunities to learn how to bridge content and pedagogy. The student teaching will be a great opportunity.<br /><br />The article indeed showed the necessity of having the mathematical background to help students learn with understanding. The results was not surprising to me, but it does worry me and makes me think about the balance of content and pedagogy. This particular study makes me think about TIMMS and how they compared the cultural context of learning. I would like to see if similar studies were conducted in other countries and compare the results. <br /><br />I would like to open the following questions for discussion with the group: Does the grade level you teach matter when you consider the balance between mathematical content and pedagogical content? Do you need to have more mathematical content as you begin to teach a course more sophisticated? For example, teaching pre-algebra in comparison to teaching Calculus.Jesus Diazhttps://www.blogger.com/profile/04479452633647097820noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-13966843500661300822011-07-26T20:09:43.855-07:002011-07-26T20:09:43.855-07:00I agree with Rick, I think the article did a great...I agree with Rick, I think the article did a great job of stressing the balance of pedagogical content knowledge and the mathematics knowledge. I think that both of these things are important for great teachers in math. I think that good Teacher Preparation Programs would have classes in both teaching Mathematics in an effective way and in teaching the pedagogical content knowledge. I am excited to learn more about teaching mathematics and observing math teachers to see what kind of explanations they give their students. <br /><br />I thought the article was very interesting and a little scary knowing that these teachers were uninformed about certain math concepts. I hope that as a future math teacher, I will be more knowledgeable and able to understand why my students are misunderstanding certain concepts.<br /><br />I am excited to learn more about teaching math and ready to apply some of the already learned pedagogical content knowledge. <br /><br />~Christina (Christy)Christy Austinhttps://www.blogger.com/profile/15158862907009601708noreply@blogger.comtag:blogger.com,1999:blog-8725813156088961385.post-18442737007584013102011-07-26T19:56:40.164-07:002011-07-26T19:56:40.164-07:00I thought this article did a good job of stressing...I thought this article did a good job of stressing the importance of balance in the education of math teachers.<br /><br />Of particular interest was the discussion of assessment knowledge. Teacher canadites' lack of assessment knowledge impedes their ability to accuratly assess students' knowledge. For example, an answer may be wrong, but the teacher still gives a high grade. I'm curious to discuss assessmet knowledge more.<br /><br />~rickAnonymousnoreply@blogger.com