Gilbert, M.J., Coomes, J., (2010). What Mathematics Do High School Teachers Need to Know? Mathematics Teacher, 103, 418-423.
People assume that if you can do, you can teach. Gilbert and Coomes claim this is a fallacy, and that teaching mathematics effectively requires much more mathematical knowledge than is necessary to merely solve the problem. They list and explain several types of knowledge that a teacher needs in addition to Common Content Knowledge – the ability to solve a problem.
A Teacher must be able to do more than simply mark “right” or “wrong”, as Students will approach a mathematical problem in varied ways and analyzing the methods used by students and allows teachers to assess student learning. This requires that teachers understand and are able to interpret each of these solutions and determine how the student was thinking so that they can best help the student to improve.
For example, it is important for a teacher to recognize common errors and to realize why a student might make these errors. This allows the teacher to provide the student with guidance on which part of their concept model is incorrect, so that they can adapt it accordingly. To increase understanding, perhaps a teacher can teach with these possible errors in mind, outlining why a student should avoid certain techniques and so on. Teachers must also recognize how each approach relates to the other tactics that students use, to help the students develop connections, and allowing the teacher to determine when the students are ready for new material. Additionally, they must understand how these solutions connect to instructional goals, so that they can help their students to achieve them.
I think this is actually one of the main reasons why so many students struggle. I haven’t worked in a high school, but in working with elementary school teachers in their mathematics, I find it is often assumed that a teacher can teach mathematics because they can do the mathematics. But many teachers are simply on a hunt for correct answers – if correct answers are being produced by the majority of students, the teacher feels they have done a good job and their work is done. However, in my observations, it is never this simple. Some students manage to extract a correct answer from dubious methods and are never able to correct their misconceptions because their lack of understanding is rewarded with a pass. Others think deeply about the material but make some trivial calculation error. Their thought processes are correct but they abandon them after being told they are doing it wrong. Others’ responses show that the students have a serious misconception that needs to be addressed, and as such, telling a student to go back and check their work will not be very constructive.
Students always initially try to understand the material. They take the information provided, build a mental model, and apply it as best they can to the problem. Then they are told, no, that answer is not right, but they are confused as to why. Gradually they come to the conclusion that math just doesn’t make sense. Few children can survive years of this treatment and still be willing to actually think mathematically. If a teacher only understands how they would do it themselves, then they will not be able to teach much to the students who don’t think exactly the same as them…. i.e all students.
Wednesday, March 24, 2010
Friday, March 19, 2010
Readers: Have you ever found this to be an issue? I'm curious....
Gordon, S. P. (2006) Placement Tests: The Shaky Bridge Connecting School and College Mathematics. Mathematics Teacher, 100, 174-178.
The main point of this article concerns the transition between high school and college mathematics. Most incoming freshman are placed in a class determined by their results on placement tests. Gordon argues that this approach is extremely unsuccessful and contributes to the unpopularity of .He believes that since the changes implemented by NCTM’s Curriculum and Evaluation Standards for School Mathematics, Colleges no longer understand what students have been taught in their schools. With the increased emphasis on conceptual understanding, the greater use of technology, and the reduced emphasis on algebraic manipulation and procedural proficiency, freshmen are equipped with a very different set of skill, skills that many would argue to be preferable in a college setting. However, these Placement tests – used at almost every college for more than 20 years- test the degree to which students have mastered basic algebraic manipulation, and little else. Students are often placed far below where they are conceptually, and the colleges complain about how students are placing lower and thus how unprepared freshman are. On the other hand, some students from more traditional high schools do very well on the tests and are surprised to find that the course requires much higher levels of thinking and reasoning then they are able. Gordon suggests combining placement tests with other indicators of ability, such as ACT or SAT scores, portfolios of work, and semesters of math taken.
The more I look at this article, the more I feel that the author is making a big deal out of a small issue. There are no clear statistics to show that so many students are being misplaced. Gordon uses only anecdotal evidence and the vague data collected by angry high school teachers who would have liked to see their students placed higher. I would like to see, at the very least, quotes from the students themselves showing their negative feelings about their placement, preferably some statistics about the portion of students who feel misplaced. The students should be the judges based on their own knowledge of their own skills.
Secondly, from my own experience, I do not feel that students will be able to succeed in college mathematics without the algebraic manipulation skill that Gordon so casually dismisses. If students cannot, for example, quickly factor polynomials then they will have a tough road ahead in college mathematics. If students do not have these skills, perhaps it is better for them to take remedial or other math classes so that they are more prepared.
Gordon also claims that students perform poorly on the test because their understanding is more conceptual than procedural. From our reading in class, particularly the comparison between the 2 schools Amber Hill and Phoenix Park, I am convinced that if students have a solid conceptual understanding of mathematical concepts, then they should not have problems with these questions, in fact, they should score just as high or higher on the tests.
The main point of this article concerns the transition between high school and college mathematics. Most incoming freshman are placed in a class determined by their results on placement tests. Gordon argues that this approach is extremely unsuccessful and contributes to the unpopularity of .He believes that since the changes implemented by NCTM’s Curriculum and Evaluation Standards for School Mathematics, Colleges no longer understand what students have been taught in their schools. With the increased emphasis on conceptual understanding, the greater use of technology, and the reduced emphasis on algebraic manipulation and procedural proficiency, freshmen are equipped with a very different set of skill, skills that many would argue to be preferable in a college setting. However, these Placement tests – used at almost every college for more than 20 years- test the degree to which students have mastered basic algebraic manipulation, and little else. Students are often placed far below where they are conceptually, and the colleges complain about how students are placing lower and thus how unprepared freshman are. On the other hand, some students from more traditional high schools do very well on the tests and are surprised to find that the course requires much higher levels of thinking and reasoning then they are able. Gordon suggests combining placement tests with other indicators of ability, such as ACT or SAT scores, portfolios of work, and semesters of math taken.
The more I look at this article, the more I feel that the author is making a big deal out of a small issue. There are no clear statistics to show that so many students are being misplaced. Gordon uses only anecdotal evidence and the vague data collected by angry high school teachers who would have liked to see their students placed higher. I would like to see, at the very least, quotes from the students themselves showing their negative feelings about their placement, preferably some statistics about the portion of students who feel misplaced. The students should be the judges based on their own knowledge of their own skills.
Secondly, from my own experience, I do not feel that students will be able to succeed in college mathematics without the algebraic manipulation skill that Gordon so casually dismisses. If students cannot, for example, quickly factor polynomials then they will have a tough road ahead in college mathematics. If students do not have these skills, perhaps it is better for them to take remedial or other math classes so that they are more prepared.
Gordon also claims that students perform poorly on the test because their understanding is more conceptual than procedural. From our reading in class, particularly the comparison between the 2 schools Amber Hill and Phoenix Park, I am convinced that if students have a solid conceptual understanding of mathematical concepts, then they should not have problems with these questions, in fact, they should score just as high or higher on the tests.
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