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.
Sunday, September 20, 2009
Self Assessment: Microteaching – Hear a Knot See a Knot
Self Assessment: Microteaching – Hear a Knot See a Knot
I thought these things went well in my lesson:
Group size being small meant that things went quickly especially with the quick learners. Two of my three students were able to follow on the first instructions while one student needed several tries. This confirms an old statistic that your audience only hears 45% of audio the first time (this statistic needs to be confirmed and find research quote to back it up), meaning that it is normal to have to repeat the same instructions again.
Audio only gave some difficulty as expected, with the visual doing much better.
Everyone completed the task successfully!
If I were to teach the lesson again, I would work to improve it in these ways.
1. A better motivational introduction as to why we might need to learn to tie a knot. Such a story might be introduced with the question: Have you ever needed to tie a package and the string was too short? We will learn how to lengthen your string or rope so that it will hold together. The Short String Problem whereby the solution is to tie two or more strings together and test that they hold.
2. A better pre-test question to ensure that the group understands the difference between their right and left hands as without this knowledge it will be difficult to complete the knot correctly.
3. More clearly explain that we will be doing this exercise three times so that the student can evaluate how they best learn; auditory only, visual only (MIME), or both audio and visual.
4. Have a plan be to tie additional knot(s) for those who are very fast.
5. Make sure that each person pulls a hand width of string out when they grasp the ends to overlay on each other.
Here are some things that I reflected on based on my peer’s feedback:
1. Felt pressured from auditory only instructions. Improvement 1, 2, and 5.
2. More clearly explain the motivation for learning to do this task. Improvement 3.
3. I noticed that a hand width of string from the ends of the string made it easier and so this will be added to the instructions.
4. Have a Plan B, in this case additional knots that could be tied from written directions, to keep the fast learners busy.
5. Come up with a suitable test for the four methods written, auditory, visual, and combination of all of them.
There should be a combination to accommodate all learning styles.
I thought these things went well in my lesson:
Group size being small meant that things went quickly especially with the quick learners. Two of my three students were able to follow on the first instructions while one student needed several tries. This confirms an old statistic that your audience only hears 45% of audio the first time (this statistic needs to be confirmed and find research quote to back it up), meaning that it is normal to have to repeat the same instructions again.
Audio only gave some difficulty as expected, with the visual doing much better.
Everyone completed the task successfully!
If I were to teach the lesson again, I would work to improve it in these ways.
1. A better motivational introduction as to why we might need to learn to tie a knot. Such a story might be introduced with the question: Have you ever needed to tie a package and the string was too short? We will learn how to lengthen your string or rope so that it will hold together. The Short String Problem whereby the solution is to tie two or more strings together and test that they hold.
2. A better pre-test question to ensure that the group understands the difference between their right and left hands as without this knowledge it will be difficult to complete the knot correctly.
3. More clearly explain that we will be doing this exercise three times so that the student can evaluate how they best learn; auditory only, visual only (MIME), or both audio and visual.
4. Have a plan be to tie additional knot(s) for those who are very fast.
5. Make sure that each person pulls a hand width of string out when they grasp the ends to overlay on each other.
Here are some things that I reflected on based on my peer’s feedback:
1. Felt pressured from auditory only instructions. Improvement 1, 2, and 5.
2. More clearly explain the motivation for learning to do this task. Improvement 3.
3. I noticed that a hand width of string from the ends of the string made it easier and so this will be added to the instructions.
4. Have a Plan B, in this case additional knots that could be tied from written directions, to keep the fast learners busy.
5. Come up with a suitable test for the four methods written, auditory, visual, and combination of all of them.
There should be a combination to accommodate all learning styles.
Friday, September 18, 2009
Hear a Knot See a Knot
Ralph Baker 07434699 MaEd 314A Dr. Gerofsky
MAED314A
BOOPPPS lesson plan
1) BRIDGE: Tying a Reef Knot or will it be a Granny Knot
Test to see if anyone knows what a Granny Knot or Square Knot or Reef knot is?
2) Teaching OBJECTIVES:
Everyone should be able to tie a knot in a rope or string especially if they are going to be a sailor, or move and need to secure their belongings. By tying knots an individual and the group can bond together knowing they have a skill in common.
3) Learning OBJECTIVES: “SWBAT”
Students will be able to tie a Reef or Square knot and know that it is correct and not a Granny knot.
4) PRETEST: Have you ever tied a knot?
5) PARTICIPATORY This unit is to be carried out with their eyes closed first for all the instructions to find out if they respond well to auditory instructions and then it is to be done again with their eyes open.
6) POST-TEST Final results are easy to test. Students are provided with string, cord, and a picture of the final results for them to self evaluate.
7) SUMMARY A good exercise to see how the students learn, by hearing and seeing.
Problem is how to encounter special needs students.
Thursday, September 17, 2009
Relational and Instrumental Understanding
Relational and Instrumental Understanding
All quotes are from “Relational Understanding and Instrumental Understanding” by Richard R. Skemp.
Page 8, “These children need success to restore their self-confidence, and it can be argued that they can achieve this more quickly and easily in instrumental mathematics than in relational.”
I feel this is very true and that the instrumental method is a way of getting the quick start or kick start that is needed to capture student interest from a generation of students brought up in the “instant pudding” way of doing things. It is then up to the teacher to show that there can be limitations and that another way to find an understanding is to see that there is a deeper relational way to understand mathematics that can be carried over into everyday situations. It is up to the teacher to relate the mathematics to the relevant with examples of how this can be used in our future lives and jobs. This method is like climbing stairs, that is the instrumental gets us started, and the relational takes advantage of the momentum to keep interest and further enquiry as to how this can work for the student.
Page 11, All of these imply, as does the phrase ‘make a reasoned choice’, that he is able to consider the alternative goals of instrumental and relational understanding on their merits and in relation to a particular situation. To make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself.
A cycling of instrumental and relational can be beneficial if the instrumental eventually favours using the firmly grounded rules of mathematics more effectively. In our example of the dividing by a fraction, “invert and multiply”, I would use the rules of Identity to show how powerful it can be in solving this problem in a more relational way, that when learned will make solving many types of mathematical problems simpler. The instrumental may need to be used to get students started but they should be helped to see the path that relational understanding through the implementation of effective use of the Laws of Mathematics. There should be some dwell time to allow students to see that they can go deeper into long lasting understanding compared to the shorter term dissipative instrumental rules that may be forgotten. These can be memory jarring stepwise ways to remember the path from instrumental to relational understanding. I think that there could possibly be a third element that will make this teeter totter into a triad of understanding and that is the usefulness of the information to the learner. This means that the teacher must show that learning this information will have an effect on the potential occupations available to them, thus life and living goals can be achieved.
Page 14, And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’. I agree with this statement and find that it does not conform to my ideals of a good teacher, which is one that has the student learn how to learn so well that you are no longer needed for that material level anymore. The student becomes independent and the teacher is validated with a new student that needs the learning understanding to help them from instrumental to relational understanding and finally to understanding the value of knowing how to apply the knowledge they have learned and how to continue the process on their own.
Page 15, “but what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.” This statement needs to help the student to see that there are many more ways to look at the understanding of mathematics and these may be helped by new and still to come technology that allows students the time to view the visual interpretations of mathematics. By introducing the senses of visual, 2D and 3D as well as new interpretations of visual and audio, representations of what mathematics is and how to understand it and what it may be used for will expand the knowledge of mathematics in many new ways in the future.
Page 16, “One of these is the relationship between the goals of the teacher and those of the pupil. Another is the implications for a mathematical curriculum.
I agree that there needs to be some time devoted to the attainment of goals and career so that the teacher may show the pupil that a good foundation in mathematics learning will help in any future goals.
Relational and Instrumental Understanding
All quotes are from “Relational Understanding and Instrumental Understanding” by Richard R. Skemp.
Page 8, “These children need success to restore their self-confidence, and it can be argued that they can achieve this more quickly and easily in instrumental mathematics than in relational.”
I feel this is very true and that the instrumental method is a way of getting the quick start or kick start that is needed to capture student interest from a generation of students brought up in the “instant pudding” way of doing things. It is then up to the teacher to show that there can be limitations and that another way to find an understanding is to see that there is a deeper relational way to understand mathematics that can be carried over into everyday situations. It is up to the teacher to relate the mathematics to the relevant with examples of how this can be used in our future lives and jobs. This method is like climbing stairs, that is the instrumental gets us started, and the relational takes advantage of the momentum to keep interest and further enquiry as to how this can work for the student.
Page 11, All of these imply, as does the phrase ‘make a reasoned choice’, that he is able to consider the alternative goals of instrumental and relational understanding on their merits and in relation to a particular situation. To make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself.
A cycling of instrumental and relational can be beneficial if the instrumental eventually favours using the firmly grounded rules of mathematics more effectively. In our example of the dividing by a fraction, “invert and multiply”, I would use the rules of Identity to show how powerful it can be in solving this problem in a more relational way, that when learned will make solving many types of mathematical problems simpler. The instrumental may need to be used to get students started but they should be helped to see the path that relational understanding through the implementation of effective use of the Laws of Mathematics. There should be some dwell time to allow students to see that they can go deeper into long lasting understanding compared to the shorter term dissipative instrumental rules that may be forgotten. These can be memory jarring stepwise ways to remember the path from instrumental to relational understanding. I think that there could possibly be a third element that will make this teeter totter into a triad of understanding and that is the usefulness of the information to the learner. This means that the teacher must show that learning this information will have an effect on the potential occupations available to them, thus life and living goals can be achieved.
Page 14, And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’. I agree with this statement and find that it does not conform to my ideals of a good teacher, which is one that has the student learn how to learn so well that you are no longer needed for that material level anymore. The student becomes independent and the teacher is validated with a new student that needs the learning understanding to help them from instrumental to relational understanding and finally to understanding the value of knowing how to apply the knowledge they have learned and how to continue the process on their own.
Page 15, “but what constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.” This statement needs to help the student to see that there are many more ways to look at the understanding of mathematics and these may be helped by new and still to come technology that allows students the time to view the visual interpretations of mathematics. By introducing the senses of visual, 2D and 3D as well as new interpretations of visual and audio, representations of what mathematics is and how to understand it and what it may be used for will expand the knowledge of mathematics in many new ways in the future.
Page 16, “One of these is the relationship between the goals of the teacher and those of the pupil. Another is the implications for a mathematical curriculum.
I agree that there needs to be some time devoted to the attainment of goals and career so that the teacher may show the pupil that a good foundation in mathematics learning will help in any future goals.
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