The fight over how to best teach students math has two camps of thought, conservative and progressive, of which neither camp has a perfect solution. As with most methods they serve different learning styles and we need to have innovative mathematicians to solve the many problems that the future holds for civilization.
The Progressivist Form (1910-1940) is the pre-television, pre-calculator era and the stress for students was just to learn the basic operations and become fluent in them. As society changed and television and other industrial/commercial products flourished the era of New Math developed in the 1960's.
This type of thinking change was not in line with public perception or consumption and eventually was replaced with by the 1990's with National Standards. We have the era of television, calculators, and computers starting to influence all walks of the mathematical life.
A new standard needs to be formed to give way to a society that has grown by a factor of four
and has more highly educated citizens than ever before, that are more literate, more visual and more free thinking than ever before in history. This change in society moves forward driven by technology that is ever improving and is more visual than ever before and now has interactive television, wireless computer networks, and software that performs as sophisticatedly as we think and is moving faster than imagined into interactivity and modelling thoughts so that the level of mathematics for our futures must encourage more interactive free thinking students to move up the mathematical ladder to solve the challenging problems of the future.
Tuesday, September 29, 2009
Teacher Student Interviews
Students first with diverse ways of approaching and solving the same problem. If you follow my blog the scale problem had six unique solutions that were all different and approached the same problem with different theories which all produced the same answers numerically, however not all methods could be easily understood by other than their solution provider. So as there are many solutions there are many ways to learn how to solve a problem, and it is my goal as a future math teacher to provide as many ways as possible, as well as some visual representation of what the solution looks like.
Students in our interviews learned differently and it is up to the teacher to recognize this and teach the many learning styles that are in there classes so that all students and their ideas on how to solve problems are looked at and examined fairly.
Students in general feel there teachers cover the curriculum and many were able to improve their marks with the results on the final exam.
Involvement of the students seemed low in the math classes and visualization was minimal.
Teachers do the best they can when there is so much material to cover but by being open minded and willing to engage their students they may be surprised at how bright all their students really are and how much fun math can be.
As new teachers it is up to us to learn how each of your students learns and challenge them to expand their thinking.
Teachers work hard to plan lessons and try to help every child but with a large class this ideal case often is unattainable.
Students in our interviews learned differently and it is up to the teacher to recognize this and teach the many learning styles that are in there classes so that all students and their ideas on how to solve problems are looked at and examined fairly.
Students in general feel there teachers cover the curriculum and many were able to improve their marks with the results on the final exam.
Involvement of the students seemed low in the math classes and visualization was minimal.
Teachers do the best they can when there is so much material to cover but by being open minded and willing to engage their students they may be surprised at how bright all their students really are and how much fun math can be.
As new teachers it is up to us to learn how each of your students learns and challenge them to expand their thinking.
Teachers work hard to plan lessons and try to help every child but with a large class this ideal case often is unattainable.
Visual Image: Geometric Solution to the Scale Problem
Think of the operations that a weigh scale can do, that is +, -, and zero or balance. What mathematics principle does this remind you of? What pattern does it imply?
The scale is the principle of additive inverses. The range is in powers of 3.
Geometrically this is is a unit circle on the number line at zero (balance). The range of weights with a weight of one is two +-(-1)= 2. Fold this over at 1 and you get the next circle at three, which is the value of your next weight. Repeat, the range for 3-(-3)=6 fold over at three and you get 9, the value of your next weight. Repeat, the range for 9-(-9)=18 fold over at nine and you get 27, the value of your next weight. 1+3+9+27=40 and you have all the digits covered from 1 to forty.
Image is a numberline with circles about the origin with radius 1, 3, 9, and 27 and it can keep going.
The scale is the principle of additive inverses. The range is in powers of 3.
Geometrically this is is a unit circle on the number line at zero (balance). The range of weights with a weight of one is two +-(-1)= 2. Fold this over at 1 and you get the next circle at three, which is the value of your next weight. Repeat, the range for 3-(-3)=6 fold over at three and you get 9, the value of your next weight. Repeat, the range for 9-(-9)=18 fold over at nine and you get 27, the value of your next weight. 1+3+9+27=40 and you have all the digits covered from 1 to forty.
Image is a numberline with circles about the origin with radius 1, 3, 9, and 27 and it can keep going.
Thursday, September 24, 2009
Weights a Scale and Integral values 1 to 40
Weights a Scale and Integral values 1 to 40 in whatever units you need
The problem posed is to use only four weights to measure from 1 to 40 on a typical twin pan balance scale.
1. The first premise was to add weights to get all the values but with a quick addition this was not possible. How to get 1, by a weight of 1 or a difference in two weights? I chose to start with weight #1 being a weight of one, which leaves 39 numbers remaining. Factor as 13 x3.
2. The key premise was to realize that you could reach some numbers by adding to the bottom number and others by subtracting from the top number. Simply put, the weights could be put on both pans and the difference would allow you to reduce weight from the top number. Thus if the first three weights add up to 13 then the next weight minus 13 needed to be the next number 14. 27-13=14, and you can add the 13 to the 27 weight to get 40.
3. Checking how to get 2 as 3-1=2. Then the second weight must be 3 and the third weight is 1+3+w3=13 or w3 =9. 3+1=4, 9-4=5, 9-3=6, 9-3+1=7,9-1=8, 9, 9+1=10, 9+3-1=11, 9+3=12, 9+3+1=13. We can get all the numbers up to 13.
4. We need 14 which is 40-14=26 and divide by 2 = 13 which can be added or subtracted to get from 14 to 40.
5. This can continue on with n= weight number then;
Weight n = 3^(n-1)
n with a range of
weight 1 = 3^0 = 1 1
weight 1 = 3^1 = 3 1 - 4
weight 1 = 3^2 = 9 1 - 13
weight 1 = 3^3 = 27 1 - 40
weight 1 = 3^4 = 81 1 - 121
weight 1 = 3^5 = 243 1 - 365
and could continue.
The problem posed is to use only four weights to measure from 1 to 40 on a typical twin pan balance scale.
1. The first premise was to add weights to get all the values but with a quick addition this was not possible. How to get 1, by a weight of 1 or a difference in two weights? I chose to start with weight #1 being a weight of one, which leaves 39 numbers remaining. Factor as 13 x3.
2. The key premise was to realize that you could reach some numbers by adding to the bottom number and others by subtracting from the top number. Simply put, the weights could be put on both pans and the difference would allow you to reduce weight from the top number. Thus if the first three weights add up to 13 then the next weight minus 13 needed to be the next number 14. 27-13=14, and you can add the 13 to the 27 weight to get 40.
3. Checking how to get 2 as 3-1=2. Then the second weight must be 3 and the third weight is 1+3+w3=13 or w3 =9. 3+1=4, 9-4=5, 9-3=6, 9-3+1=7,9-1=8, 9, 9+1=10, 9+3-1=11, 9+3=12, 9+3+1=13. We can get all the numbers up to 13.
4. We need 14 which is 40-14=26 and divide by 2 = 13 which can be added or subtracted to get from 14 to 40.
5. This can continue on with n= weight number then;
Weight n = 3^(n-1)
n with a range of
weight 1 = 3^0 = 1 1
weight 1 = 3^1 = 3 1 - 4
weight 1 = 3^2 = 9 1 - 13
weight 1 = 3^3 = 27 1 - 40
weight 1 = 3^4 = 81 1 - 121
weight 1 = 3^5 = 243 1 - 365
and could continue.
Symbols and Meanings in School Mathematics
"Symbols and Meanings in School Mathematics" by David Pimm
Think, write, pair share.
Practice is necessary to review that you have understood the problem and have found a way at arriving at a solution. It is necessary to see that your understanding of the math problem and your solution work under many situations.
Practice makes perfect, or better yet; perfect practice makes perfect answers are cliches, that we need to test what we have read or learned to see that we understand how to use it to solve additional problems.
If the problems are all the same in a textbook in a section this is because each individual may require a different amount of practice to achieve understanding.
It is then up to the teacher to make increasingly difficult questions as the number within a section increases and harder in increasing sections. All questions should challenge the student to recall prior knowledge and to learn to use it.
Think, write, pair share.
Practice is necessary to review that you have understood the problem and have found a way at arriving at a solution. It is necessary to see that your understanding of the math problem and your solution work under many situations.
Practice makes perfect, or better yet; perfect practice makes perfect answers are cliches, that we need to test what we have read or learned to see that we understand how to use it to solve additional problems.
If the problems are all the same in a textbook in a section this is because each individual may require a different amount of practice to achieve understanding.
It is then up to the teacher to make increasingly difficult questions as the number within a section increases and harder in increasing sections. All questions should challenge the student to recall prior knowledge and to learn to use it.
Monday, September 21, 2009
Using Research to analyze, inform and assess changes in instruction
by Heather Robinson
I am sure that it is important to keep track of your own performance to see where improvements could be made to lesson plans, and to see how students learned, and if you could do something to make the lessons more engaging for students. Providing a variety of strategies for learning, and measuring student response will ultimately benefit both the student and the teacher. There is nothing more rewarding than to have your students succeed and therefore you as a teacher succeed. Knowing that you have helped a young person become a responsible adult capable of learning on their own is a most rewarding feeling.
Never underestimate your students provided you engage them in communicating their ideas of math and how they would solve problems. Students can come up with very clever solutions and in the process learn how to think critically.
Robinson has come up with a class management and instruction plan that limits pure lecture to 60 minutes per week. As a new teacher this time may need to be slightly longer as one gains experience and sees how students react to possible new methods.
It is important that the discussions be kept on track by asking questions. It is important for students to visualize math and feel how it works with interactive manipulatives.
by Heather Robinson
I am sure that it is important to keep track of your own performance to see where improvements could be made to lesson plans, and to see how students learned, and if you could do something to make the lessons more engaging for students. Providing a variety of strategies for learning, and measuring student response will ultimately benefit both the student and the teacher. There is nothing more rewarding than to have your students succeed and therefore you as a teacher succeed. Knowing that you have helped a young person become a responsible adult capable of learning on their own is a most rewarding feeling.
Never underestimate your students provided you engage them in communicating their ideas of math and how they would solve problems. Students can come up with very clever solutions and in the process learn how to think critically.
Robinson has come up with a class management and instruction plan that limits pure lecture to 60 minutes per week. As a new teacher this time may need to be slightly longer as one gains experience and sees how students react to possible new methods.
It is important that the discussions be kept on track by asking questions. It is important for students to visualize math and feel how it works with interactive manipulatives.
Two of my most memorable math teachers
Mr. Andrews started Math 11 by insulting the fellow who was sitting beside me and me by saying, with his hand covering his mouth, to another Math teacher Mr. Eisen, “Don’t these two in the front rows look like real winners”. Little did he know that the other fellow Chris and I would go on to complete Math 11 and Math 12 in the same year? This allowed us to leave his Math class two and a half months later and go on to do Math 12.
Mr. Andrews made so many mistakes that it was a pleasure to get out of his class into Mr. Norman-Martin’s Math 12 class. Needless to say Mr. Norman-Martin knew his math and produced two “A” students that went on to do first year university math in grade 12.
It is important as a teacher to never underestimate the learning skills of a student. Always show respect for any student should you want respect back in turn. The attitude difference in these two teachers was enormous. Mr Norman-Martin was inspiring; I hope that I can give my students that feeling when I start teaching.
Mr. Andrews started Math 11 by insulting the fellow who was sitting beside me and me by saying, with his hand covering his mouth, to another Math teacher Mr. Eisen, “Don’t these two in the front rows look like real winners”. Little did he know that the other fellow Chris and I would go on to complete Math 11 and Math 12 in the same year? This allowed us to leave his Math class two and a half months later and go on to do Math 12.
Mr. Andrews made so many mistakes that it was a pleasure to get out of his class into Mr. Norman-Martin’s Math 12 class. Needless to say Mr. Norman-Martin knew his math and produced two “A” students that went on to do first year university math in grade 12.
It is important as a teacher to never underestimate the learning skills of a student. Always show respect for any student should you want respect back in turn. The attitude difference in these two teachers was enormous. Mr Norman-Martin was inspiring; I hope that I can give my students that feeling when I start teaching.
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