This is a futuristic scenario regarding me as a teacher 10 years from now. At this point, I've been teaching for 10 years and have had an opportunity to teach around 2000 students. The text below describes two students. To the first student, I am their favourite teacher, and to the second student, I am a failure. At this time, it will be 2019 and I'll be 31.
Student #1:
Mr. Collins is my favourite teacher this year. ...and probably ever. I love coming to his classes because he always keeps me interested in what I'm learning and makes things fun. I have to think pretty hard because he poses lots of problems, but at least I know he'll give me the answer in the end if I can't figure it out myself. But most times I can, because Mr. Collins always shows us the 'why' behind math. I'm not the best student in the class, but I still get a chance to contribute and he always has time to hear what I have to say in class. I've chatted with him after class a couple times. Once 'cause I had to ask to extend a due date on one of my math assignments (although I'd finished most of it), but the others were to ask other questions about the concepts we were learning and to show him some of my ideas. I think our class is going to do really well on the provincial this year. And not just me, but all the kids will do well. Sometimes I feel it like a race to the answer, because when he poses the problem, we all know how to approach it. Mr. Collins would make a good professor one day and it'd be cool if he could be my professor in university. The only thing would be that he wouldn't look like a professor, because he's really young.
Student #2:
Mr. Collins is the worst teacher I have this year, but unfortunately, I have no choice 'cause he's the only teacher for our grade. I hate the fact that he always goes on these long-winded speeches about stuff that isn't even important. Sometimes I feel like he forgets we're even there! Plus, he always makes us feel stupid. He makes these ridiculously hard problems and then expects everyone to get them, even though there are only a couple people in the class that are actually passing!! Basically, if you want to pass this course, you have to put in extra time and ask questions outside of class. And I don't have that kinda time. I just want to be able to learn while in the class. Plus, I'm not really interested in all the extra stuff, stuff that isn't part of the textbook. He's just adding more work on top of the work we do out of the textbook and it's not even relevant.
He has questions about spaceships and stuff that's in the future and I don't get why it's so important. Also, Mr. Collins makes us get up in front of the class and solve problems on the board. Like, what's his deal? I don't have the answer, that why I'm learning!! Yet, we have to go up their by ourselves. Sure, the class is allowed to help a bit, but I don't get to go back to my seat until I've put something on the board. Whatever, once I fail, I'll just move on. I won't have him for another course again, 'cause I'll check ahead of time, or else just drop out if I do.
Friday, October 2, 2009
Wednesday, September 30, 2009
Reflection on MAED 314 in-class video
The video presented in class today was a real eye-opener. The grade 8 instruction felt like it was going at a snail pace and it made me remember how much different their learning is. Their speed, approach and objectives are so completetely different and it is something that I'm going to have to refamiliarize myself with if I'm going to be an effective teacher in their class. I can't go at my speed, or even close, or I will lose them for sure.
Some of our classmates commented on the slow presentation as boring, but that is just our minds adjusting to this style. We are used to fast paced information, and no time for repetition and practice. In my opinion, this was a great speed to teach at! The students certainly didn't seem bored! In fact, those students seemed completely absorbed in the process. The teacher was teaching to the slowest student and making sure everyone was on board. I was very impressed with that. In the space of 20min, he had successfully introduced the concept of algebra in an interactive, engaging and relational way. He had given the students the skills to perform algebra slowly and incrementally so that they all got it. He made the knowledge linear and broke it into small enough pieces.
I am very impressed. I plan on using some of his approaches in introducing concepts to my students in the future.
Some of our classmates commented on the slow presentation as boring, but that is just our minds adjusting to this style. We are used to fast paced information, and no time for repetition and practice. In my opinion, this was a great speed to teach at! The students certainly didn't seem bored! In fact, those students seemed completely absorbed in the process. The teacher was teaching to the slowest student and making sure everyone was on board. I was very impressed with that. In the space of 20min, he had successfully introduced the concept of algebra in an interactive, engaging and relational way. He had given the students the skills to perform algebra slowly and incrementally so that they all got it. He made the knowledge linear and broke it into small enough pieces.
I am very impressed. I plan on using some of his approaches in introducing concepts to my students in the future.
Monday, September 28, 2009
Battleground Schools (Summary & Reflection)
Summary:
This article addresses many of the problems that face the system of education of math curriculum in public schools. Primarily, there is a clash between progressive and conservative thinkers on the transmission of knowledge from teacher to student and what math skills should be the focus of instruction. Also, there is a recurring theme of prejudice, misconception and fear of mathematics that has a cripling effect as it propagates through the generations. It is not only socially acceptable to be mathematically illiterate, but there are students and teachers alike who pass through the education system without ever really understanding fundamental secondary math concepts. In this way, there are many math teachers who do not possess the skills to properly instruct their students in math and who are excused on the grounds that the textbook is considered 'teacher-proof'.
In the early 20th century (1910-1940), a Progressivist Reform sought to bring about the unification of knowledge and application. This meant using an inquiry-based approach to learning that had largely been ignored before. This included the facilitation and orchestration of student supported inquiries into learning. Then, in the 60s, spurred on by the launch of Russian spacecraft, the 'space race' became a national issue in America as an unmet demand for qualified and educated scientists was realized. This 'New Math' initiative saw the rewriting of mathematical structure to be based more on set theory. But many teachers struggled to adapt to the new changes and in the 70s, the popular media was denouncing it. Now (1990-present), there is a real reigning-in on teaching standards in an attempt to implement a 'back-t0-basics' approach that lessens the autonomy of teachers and holds them more accountable. The National Council of Teachers of Math (NCTM) produced a definitive set of standards in 2000.
Response:
This article had a lot that needed digesting. I can imagine that there would be incredible difficulties in moderating and regulating the instruction of any subject, but with math, it must be that much more difficult. Not only is it a subject that many people aren't fluent with, but because it is logically driven, discrepancies must be extremely difficult to resolve. If two teachers each believe their competing interpretations of the rules of math are correct, then there is little persuading them. Also, when it comes to the level of autonomy of teachers in the class room, I don't know where I stand. I know that it is essential that no detail of math instruction (either relational or instrumental) be left out, but at the same time, I don't want teachers acting as textbook paraphrasers either. That is why provincial exams are a good idea, because it holds the teachers within the bounds of the material covered therein. What I would like to see is an increase in the expectation of math students and teachers. If we can raise the standard (create a new trend of achivement), then all future generations can benefit.
This article addresses many of the problems that face the system of education of math curriculum in public schools. Primarily, there is a clash between progressive and conservative thinkers on the transmission of knowledge from teacher to student and what math skills should be the focus of instruction. Also, there is a recurring theme of prejudice, misconception and fear of mathematics that has a cripling effect as it propagates through the generations. It is not only socially acceptable to be mathematically illiterate, but there are students and teachers alike who pass through the education system without ever really understanding fundamental secondary math concepts. In this way, there are many math teachers who do not possess the skills to properly instruct their students in math and who are excused on the grounds that the textbook is considered 'teacher-proof'.
In the early 20th century (1910-1940), a Progressivist Reform sought to bring about the unification of knowledge and application. This meant using an inquiry-based approach to learning that had largely been ignored before. This included the facilitation and orchestration of student supported inquiries into learning. Then, in the 60s, spurred on by the launch of Russian spacecraft, the 'space race' became a national issue in America as an unmet demand for qualified and educated scientists was realized. This 'New Math' initiative saw the rewriting of mathematical structure to be based more on set theory. But many teachers struggled to adapt to the new changes and in the 70s, the popular media was denouncing it. Now (1990-present), there is a real reigning-in on teaching standards in an attempt to implement a 'back-t0-basics' approach that lessens the autonomy of teachers and holds them more accountable. The National Council of Teachers of Math (NCTM) produced a definitive set of standards in 2000.
Response:
This article had a lot that needed digesting. I can imagine that there would be incredible difficulties in moderating and regulating the instruction of any subject, but with math, it must be that much more difficult. Not only is it a subject that many people aren't fluent with, but because it is logically driven, discrepancies must be extremely difficult to resolve. If two teachers each believe their competing interpretations of the rules of math are correct, then there is little persuading them. Also, when it comes to the level of autonomy of teachers in the class room, I don't know where I stand. I know that it is essential that no detail of math instruction (either relational or instrumental) be left out, but at the same time, I don't want teachers acting as textbook paraphrasers either. That is why provincial exams are a good idea, because it holds the teachers within the bounds of the material covered therein. What I would like to see is an increase in the expectation of math students and teachers. If we can raise the standard (create a new trend of achivement), then all future generations can benefit.
Sunday, September 27, 2009
Re: Student/Teacher Interviews
Wow... what interesting responses! This has certainly given me a lot to think about. I like the fact that we managed to pose these questions with little overlap between student and teacher responses, so that we could learn as much as possible. The students (all from different schools, cities, backgrounds) gave good reviews of their teachers and that is encouraging to know that they are doing a good job. A couple obvious challenges still to overcome in the classroom and certainly traps that we will be looking out for during our practicum. The teacher gave some good advice and some nice ideas & guidelines for getting started in the classroom. Plus, it's nice to know that we have a resource to look to in the years to come; that there are teachers out there that are willing to help us as we step into the classroom. Now... off we go!
5 Burning Questions to Math Teachers and Math Students (Answered)
5 Burning Questions to Math Teachers and Math Students
Below are the answers we received from our willing student and teacher participants. The answers are candid, honest and anonymous:
Answers:
Teacher:
1. I am much busier at the full time job. Lesson preparation is the most time consuming thing. Lesson plans don't actually get written on the job. For class management, it is easier to be strict at first, then ease up later rather than the other way around. Seating plans can be very useful to break up chatting kids, although I don't use them that often.
2. Students will usually get 20-40 min. of seat work depending on the lesson. It is difficult to avoid a lot of talking and notes but I try to vary it with activities. Things like math bingo work well as a transition from notes to seat work.
3. In math, I don't use much... algebra tiles are stupid. Only when I get to geometry and surface area do I use solids to show concepts. I try to do many short lab-like activities where kids physically measure stuff with rulers or stopwatches. It is important to get them out of the seat once in a while.
4. I typically try to mimic the textbook, although for physics I don't even use the text and then I go with what is in the IRP's or whatever they'll encounter in university. You must keep notation simple. They will get lost at something so simple as f(x) = g(x) and then you lose them entirely.
5. Yes. It's difficult in practice when you are a new teacher and already have enough preparation and marking. Just be patient and offer you time outside of class. Peer tutors are useful if your school has them.
Students:
Student #1:
Grade 9
1. I'm pretty comfortable and math is one of my favourite subjects and I'm comfortable because it isn't super hard and I know how to sorta do it.
2. I remember most of it. It will have to do with my future a lot because it is how to pay taxes and how to know how much space there is in a room so I can place a rug or something in it.
3. Not really slowly, because they want it to get done but you can always come after school and ask for help on the stuff we just covered.
4. Not to much, because it was all really easy and in class we covered much harder stuff.
5. He would do the math for us, not explain and tell us to just do the work sheet.
Student #2:
Grade 12
1. I don't mind math once I grasp the concept, but I have to study hard to completely understand.
2. I remember most of last year's curriculum because most of it applies to this year's course. I don't see much use in this knowledge in the future because I'm not looking into any professions involving a lot of math.
3. Yes, my teacher went at a comfortable pace for me and he covered all the material needed for the exam.
4. I improved my grade with the final exam
5. What I didn't like about my previous teacher's instructing style was that he didn't really like going over the homework if we didn't understand.
Student #3:
Grade 9
1. I actually really enjoy math and I'm mostly comfortable using it because I find it sometimes fun and I understand it.
2. I don't remember everything but I remember a lot of it. A lot of it is important for the future because it was about taxes, fractions and other things that will be helpful for the future.
3. Yeah, she was very good and went over things if people didn't understand. She covered all the material very well.
4. The final exam helped my final grade a lot because it was easier than the work we did in class.
5. She was pretty good but she had a really quiet voice so I had to sit at the front. Other than that, she was a really good teacher.
Student #4:
Grade 11
1. I enjoy math to some extent, I usually don't understand it so I'm not comfortable with using it.
2. My brain likes to do this thing called erasing all memories of things I don't understand or like during the summer... so I don't remember much. I remember learning about certain things, but I don't remember how to do them. I see a use for some of the stuff, but I really don't see when I will need to do factoring and stuff.
3. My teacher went at a very good pace and covered all and more material then was required for our exam.
4. I have to say my exam wasn't even close to representing my course mark. I was getting a solid B (76%) iin the course and then proceeded to get a fabulous 57% on my exam.
5. My math teachers is one of the best around. I don't think there is anything I don't like about his teaching method. Though, he does like to just give us the basic rules of the problem and then we have to take what he showed us and use the same concept for the harder questions. This is one thing that I am still having difficulty with, but that is just me.
Student #5:
Grade 9
1. I really, really like math except last year the text book was all about drawing diagrams and I didn't learn anything so I wouldn't be comfortable using math everyday.
2. I only remember the equations, not the diagrams which would be helpful in the future.
3. My math teacher went way too slow so the whole class began to lose interest and when the finals rolled along, we weren't ready.
4. I did pretty well, it raised my mark a lot, but everyone else who was in our class scored lower than the average mark.
5. I didn't like how my teacher gave notes without explaining them.
Below are the answers we received from our willing student and teacher participants. The answers are candid, honest and anonymous:
Answers:
Teacher:
1. I am much busier at the full time job. Lesson preparation is the most time consuming thing. Lesson plans don't actually get written on the job. For class management, it is easier to be strict at first, then ease up later rather than the other way around. Seating plans can be very useful to break up chatting kids, although I don't use them that often.
2. Students will usually get 20-40 min. of seat work depending on the lesson. It is difficult to avoid a lot of talking and notes but I try to vary it with activities. Things like math bingo work well as a transition from notes to seat work.
3. In math, I don't use much... algebra tiles are stupid. Only when I get to geometry and surface area do I use solids to show concepts. I try to do many short lab-like activities where kids physically measure stuff with rulers or stopwatches. It is important to get them out of the seat once in a while.
4. I typically try to mimic the textbook, although for physics I don't even use the text and then I go with what is in the IRP's or whatever they'll encounter in university. You must keep notation simple. They will get lost at something so simple as f(x) = g(x) and then you lose them entirely.
5. Yes. It's difficult in practice when you are a new teacher and already have enough preparation and marking. Just be patient and offer you time outside of class. Peer tutors are useful if your school has them.
Students:
Student #1:
Grade 9
1. I'm pretty comfortable and math is one of my favourite subjects and I'm comfortable because it isn't super hard and I know how to sorta do it.
2. I remember most of it. It will have to do with my future a lot because it is how to pay taxes and how to know how much space there is in a room so I can place a rug or something in it.
3. Not really slowly, because they want it to get done but you can always come after school and ask for help on the stuff we just covered.
4. Not to much, because it was all really easy and in class we covered much harder stuff.
5. He would do the math for us, not explain and tell us to just do the work sheet.
Student #2:
Grade 12
1. I don't mind math once I grasp the concept, but I have to study hard to completely understand.
2. I remember most of last year's curriculum because most of it applies to this year's course. I don't see much use in this knowledge in the future because I'm not looking into any professions involving a lot of math.
3. Yes, my teacher went at a comfortable pace for me and he covered all the material needed for the exam.
4. I improved my grade with the final exam
5. What I didn't like about my previous teacher's instructing style was that he didn't really like going over the homework if we didn't understand.
Student #3:
Grade 9
1. I actually really enjoy math and I'm mostly comfortable using it because I find it sometimes fun and I understand it.
2. I don't remember everything but I remember a lot of it. A lot of it is important for the future because it was about taxes, fractions and other things that will be helpful for the future.
3. Yeah, she was very good and went over things if people didn't understand. She covered all the material very well.
4. The final exam helped my final grade a lot because it was easier than the work we did in class.
5. She was pretty good but she had a really quiet voice so I had to sit at the front. Other than that, she was a really good teacher.
Student #4:
Grade 11
1. I enjoy math to some extent, I usually don't understand it so I'm not comfortable with using it.
2. My brain likes to do this thing called erasing all memories of things I don't understand or like during the summer... so I don't remember much. I remember learning about certain things, but I don't remember how to do them. I see a use for some of the stuff, but I really don't see when I will need to do factoring and stuff.
3. My teacher went at a very good pace and covered all and more material then was required for our exam.
4. I have to say my exam wasn't even close to representing my course mark. I was getting a solid B (76%) iin the course and then proceeded to get a fabulous 57% on my exam.
5. My math teachers is one of the best around. I don't think there is anything I don't like about his teaching method. Though, he does like to just give us the basic rules of the problem and then we have to take what he showed us and use the same concept for the harder questions. This is one thing that I am still having difficulty with, but that is just me.
Student #5:
Grade 9
1. I really, really like math except last year the text book was all about drawing diagrams and I didn't learn anything so I wouldn't be comfortable using math everyday.
2. I only remember the equations, not the diagrams which would be helpful in the future.
3. My math teacher went way too slow so the whole class began to lose interest and when the finals rolled along, we weren't ready.
4. I did pretty well, it raised my mark a lot, but everyone else who was in our class scored lower than the average mark.
5. I didn't like how my teacher gave notes without explaining them.
MAED 314 'Timed-write' exercise 2
On Understanding and Fluency
"What makes good math teaching?
What method of approach should be considered optimal? Should we teach students to treat homework questions as unique problems or as derivatives of each other? What truly is the purpose of repetition and what should be its focus? I have found that repetition (as a derivative approach) is a good way to train the mind to be conditioned to the instrumental style of problem solving. Sort of as a way to ingrain the knowledge and give the students a certain 'muscle-memory' for following algorithms. I like this method, as, in the same way that relational learning is built iteratively from fundamental ideas, so must the instrumental tools as well. This is how it is achieved. You can't give someone tools and not teach them how to use them. Tackling new problems every time is often bewildering and futile without proper guidance, experience & knowledge. "
"What makes good math teaching?
What method of approach should be considered optimal? Should we teach students to treat homework questions as unique problems or as derivatives of each other? What truly is the purpose of repetition and what should be its focus? I have found that repetition (as a derivative approach) is a good way to train the mind to be conditioned to the instrumental style of problem solving. Sort of as a way to ingrain the knowledge and give the students a certain 'muscle-memory' for following algorithms. I like this method, as, in the same way that relational learning is built iteratively from fundamental ideas, so must the instrumental tools as well. This is how it is achieved. You can't give someone tools and not teach them how to use them. Tackling new problems every time is often bewildering and futile without proper guidance, experience & knowledge. "
Wednesday, September 23, 2009
Reflection... how I learn...
This was a 'timed-write' exercise for my EDST 314 class that dealt with the subject of learning and I was pleased with what I wrote at the time, so I'd like to share it with you. These moments of clarity, where endless streams of consciousness prevail, there is opportunity for learning and growth. And I was lucky to be writing when I encountered this one:
"I often see problems or solutions as an aggregate of ideas. There is some process, whether scientific or artistic, for distilling and refining concepts in some way. Like fitting a curve to data or finding a theme in a body of work, there is truth hidden in the details. As students, we do this process of analysis in the greatest space possible to provid e ourselves a maximum amount of meaning.
In education, I see this process as most common among students. Each contributor to the students' learning (beit teacher, peer, mentor, etc.) provides "some part of it"; some part of an accumulative data set of perspectives. Contained within each, is that grain of substance that must be isolated and appreciated. The meaning of life, in part, is devised in this way also, as a summation of a global definition of this meaning.
We search continuously for more ideas and opinions that can clarify our world. Like a compilation of 'trace paper' sketches all totalled to produce a final picture. In this way, as individuals, we see only a part of it, but as a collective academic and learned society, we continue to approach the entire thing. But like the drawing of a fractal image, each layer of detail only attends to a small piece of new information or structure. And like a fractal, this process continues indefinitely on an infinitesimal scale. Perhaps the meaning of life is to continue this trial for as long as we believe it serves us. Or perhaps, we will stumble upon a discrete end; a finishing step. We will have to wait for the unification of our beliefs and understanding for that to occur. Like a mathematician conceding defeat at the hands of a quantized world. But we are grateful for that opportunity of discovery, our chance to be wrong. "
This brief piece of writing was done in about 5 minutes and so there was little preparation, proof reading or lifting of the pen from the page. Gleam from this what you will (if anything) and use it as a jumping-off point in your own thinking.
"I often see problems or solutions as an aggregate of ideas. There is some process, whether scientific or artistic, for distilling and refining concepts in some way. Like fitting a curve to data or finding a theme in a body of work, there is truth hidden in the details. As students, we do this process of analysis in the greatest space possible to provid e ourselves a maximum amount of meaning.
In education, I see this process as most common among students. Each contributor to the students' learning (beit teacher, peer, mentor, etc.) provides "some part of it"; some part of an accumulative data set of perspectives. Contained within each, is that grain of substance that must be isolated and appreciated. The meaning of life, in part, is devised in this way also, as a summation of a global definition of this meaning.
We search continuously for more ideas and opinions that can clarify our world. Like a compilation of 'trace paper' sketches all totalled to produce a final picture. In this way, as individuals, we see only a part of it, but as a collective academic and learned society, we continue to approach the entire thing. But like the drawing of a fractal image, each layer of detail only attends to a small piece of new information or structure. And like a fractal, this process continues indefinitely on an infinitesimal scale. Perhaps the meaning of life is to continue this trial for as long as we believe it serves us. Or perhaps, we will stumble upon a discrete end; a finishing step. We will have to wait for the unification of our beliefs and understanding for that to occur. Like a mathematician conceding defeat at the hands of a quantized world. But we are grateful for that opportunity of discovery, our chance to be wrong. "
This brief piece of writing was done in about 5 minutes and so there was little preparation, proof reading or lifting of the pen from the page. Gleam from this what you will (if anything) and use it as a jumping-off point in your own thinking.
Weigh-scale Problem (New Solution)
In math class today, we were challenged with a classic problem about measuring weights using an old-fashioned set of scales. While the correct solution was not immediately evident to me, an alternative solution presented itself. I take the time now to show you the original solution, my solution and how I found it.
Problem: A weigh-scale has four 'test weights' that are used to help balance the scales and measure the weight of unknown objects. If these scales can correctly measure the weight of any object whose weight is a whole number from 1 to 40 (1kg, 2kg,...39kg, 40kg.), how much do each of the four 'test weights' weigh?
Solution:
1kg - 3kg - 9kg - 27kg
This is because, the difference between weights is a range twice the amount of the former weight, allowing the any number in between to be reached from either bound.
Proof:
Below is a table values of weights. The first number shows the weight of the unknown object and the second and third sets of numbers show the balancing of numbers.
1: 1 = 1
2: 2 + 1 = 3
3: 3 = 3
4: 4 = 1 + 3
5: 5 + 1 + 3 = 9
6: 6 + 3 = 9
7: 7 + 3 = 1 + 9
8: 8 + 1 = 9
9: 9 = 9
10: 10 = 9 + 1
11: 11 + 1 = 3 + 9
12: 12 = 3 + 9
13: 13 = 1 + 3 + 9
14: 14 + 1 + 3 + 9 = 27
15: 15 + 3 + 9 = 27
16: 16 + 3 + 9 = 1 + 27
17: 17 + 1 + 9 = 27
18: 18 + 9 = 27
19: 19 + 9 = 1 + 27
20: 20 + 1 + 9 = 3 + 27
...
40: 40 = 1 + 3 + 9 + 27
The interesting thing about this solution is that you can carry this process on indefinitely. In the same way that you can reach all numbers with just 1,2,4,8,16, ... etc. you can reach them with 1,3,9,27,81, ... etc. The trick to this puzzle seems so mundane when presented in this fashion, so let's bring in a new idea!
My Solution:
2kg - 6kg - 18kg - 54kg
You'll notice now that my numbers are quite big. In fact, rather coincidentally, they are exactly double. But what is funny, is that I discovered this solution and remained oblivious to the original solution, until some one showed it to me. So let me show you why this one works. Note, this solution works so well, that you can go farther than 40kg, you can go to 80kg!!
Proof:
Reaching the even numbers with my solution is identical to the original solution, although I didn't see that at the time. However, without the number '1', reaching the odd numbers is difficult. Now, two 'weighings' are required. But despite that, reaching the numbers is just as fast except now we can reach 80 with just four 'test weights'!
1: 1 < 2
1 > 0
2: 2 = 2
3: 3 + 2 < 6
3 > 2
4: 4 + 2 = 6
5: 5 < 6
5 + 2 > 6
6: 6 = 6
7: 7 < 2 + 6
7 > 6
8: 8 = 2 + 6
9: 9 + 2 + 6 < 18
9 > 2 + 6
10: 10 + 2 + 6 = 18
11: 11 + 6 < 18
11 + 2 + 6 > 18
12: 12 + 6 = 18
13: 13 + 6 < 2 + 18
13 + 6 > 18
14: 14 + 6 = 2 + 18
15: 15 + 2 < 18
15 + 6 > 2 + 18
16: 16 + 2 = 18
17: 17 < 18
17 + 2 > 18
18: 18 = 18
19: 19 < 2 + 18
19 > 18
20: 20 = 2 + 18
...
27: 27 + 2 + 6 + 18 < 54
27 > 2 + 6 + 18
...
35: 35 + 18 < 54
35 + 2 + 18 > 54
36: 36 + 18 = 54
37: 37 + 18 < 2 + 54
37 + 18 > 54
...
71: 71 < 18 + 54
71 + 2 > 18 + 54
...
79: 79 < 2 + 6 + 18 + 54
79 > 6 + 18 + 54
80: 80 = 2 + 6 + 18 + 54
Just to name a few!
I really like how pretty this looks. It's very easy to do, once you see the pattern play out over enough numbers. Now, I'll give you a little explanation behind how I landed on these numbers.
When I first saw the problem, I knew that the four numbers (weights) should probably add up to 40 (or at least close to). That way, when the biggest number came up, these four numbers could still reach it. But I was struggling to identify (only) four numbers that were large enough to reach 40, but small enough to hit each little number as it came up. Believe it or not, my first quess came along the lines of 1 - 4 - 9 - 20. It seemed to me that there was adequate space between the numbers, but that they could still reach in between. So seemed a good place to start, save for my initial constraint that the four numbers had to add up to 40. These added up to only 34. This lead me to believe that '1' was out! No way could I have four numbers and have '1' as one of them. So I next looked at 2 as my lowest number. This immediately prompted the question, well how would you weight '1'? Hmm... well, I suppose '1' is less that '2' (my lowest number), and since these are weigh-scales, we'd easily be able to determine that '1>0'. Thus, began my rationale behind using inequalities. After '1' was solved, '2' was easy. It was '2=2'! But '3' was hard. If my second weight was '5', then '3+2=5'. But by that reaoning, '6+5+2=13' and '14+13+5+2=34'. So, my four numbers would be 2 - 5 - 13 - 34. Does this work? Actually, it does!! But it can only reach 54 (not 80) and more importantly, before I had even got to working out all four numbers, I had already thought of changing it.
If I can get away with 1<2, then I can get away with 2+3<6 and from that I can do anything. 2+6=8 and so, the next number we could have to deal with is 9. And again, I can get away with 2+6+9<18, so I did so. And from this, I produced way of figuring out the next number. 2+6+18=26, so 2+6+18+27<54 was the next step in the logic. And from that I produced my guiding numbers. And they are 2 - 6 - 18 - 54. And with these, I can tackle all of them to 80. Not 2+6+18+54=80. So, if I only had three weights, I could reach 26. i.e. 2+6+18=26.
So, if you are faced with this problem in the future, you have the skills to overcome it. And what if you are presented with 1-40,000 instead of 1-40?? Can it be done? Is this the most efficient way? What are the implications? Below I have set a question dealing with just that:
How many weights (#) are necessary to reach 40,000kg? How high would those weights reach (kg)?
ANSWER MY QUIZ
(side bar of page)
Hope you enjoyed!
Problem: A weigh-scale has four 'test weights' that are used to help balance the scales and measure the weight of unknown objects. If these scales can correctly measure the weight of any object whose weight is a whole number from 1 to 40 (1kg, 2kg,...39kg, 40kg.), how much do each of the four 'test weights' weigh?
Solution:
1kg - 3kg - 9kg - 27kg
This is because, the difference between weights is a range twice the amount of the former weight, allowing the any number in between to be reached from either bound.
Proof:
Below is a table values of weights. The first number shows the weight of the unknown object and the second and third sets of numbers show the balancing of numbers.
1: 1 = 1
2: 2 + 1 = 3
3: 3 = 3
4: 4 = 1 + 3
5: 5 + 1 + 3 = 9
6: 6 + 3 = 9
7: 7 + 3 = 1 + 9
8: 8 + 1 = 9
9: 9 = 9
10: 10 = 9 + 1
11: 11 + 1 = 3 + 9
12: 12 = 3 + 9
13: 13 = 1 + 3 + 9
14: 14 + 1 + 3 + 9 = 27
15: 15 + 3 + 9 = 27
16: 16 + 3 + 9 = 1 + 27
17: 17 + 1 + 9 = 27
18: 18 + 9 = 27
19: 19 + 9 = 1 + 27
20: 20 + 1 + 9 = 3 + 27
...
40: 40 = 1 + 3 + 9 + 27
The interesting thing about this solution is that you can carry this process on indefinitely. In the same way that you can reach all numbers with just 1,2,4,8,16, ... etc. you can reach them with 1,3,9,27,81, ... etc. The trick to this puzzle seems so mundane when presented in this fashion, so let's bring in a new idea!
My Solution:
2kg - 6kg - 18kg - 54kg
You'll notice now that my numbers are quite big. In fact, rather coincidentally, they are exactly double. But what is funny, is that I discovered this solution and remained oblivious to the original solution, until some one showed it to me. So let me show you why this one works. Note, this solution works so well, that you can go farther than 40kg, you can go to 80kg!!
Proof:
Reaching the even numbers with my solution is identical to the original solution, although I didn't see that at the time. However, without the number '1', reaching the odd numbers is difficult. Now, two 'weighings' are required. But despite that, reaching the numbers is just as fast except now we can reach 80 with just four 'test weights'!
1: 1 < 2
1 > 0
2: 2 = 2
3: 3 + 2 < 6
3 > 2
4: 4 + 2 = 6
5: 5 < 6
5 + 2 > 6
6: 6 = 6
7: 7 < 2 + 6
7 > 6
8: 8 = 2 + 6
9: 9 + 2 + 6 < 18
9 > 2 + 6
10: 10 + 2 + 6 = 18
11: 11 + 6 < 18
11 + 2 + 6 > 18
12: 12 + 6 = 18
13: 13 + 6 < 2 + 18
13 + 6 > 18
14: 14 + 6 = 2 + 18
15: 15 + 2 < 18
15 + 6 > 2 + 18
16: 16 + 2 = 18
17: 17 < 18
17 + 2 > 18
18: 18 = 18
19: 19 < 2 + 18
19 > 18
20: 20 = 2 + 18
...
27: 27 + 2 + 6 + 18 < 54
27 > 2 + 6 + 18
...
35: 35 + 18 < 54
35 + 2 + 18 > 54
36: 36 + 18 = 54
37: 37 + 18 < 2 + 54
37 + 18 > 54
...
71: 71 < 18 + 54
71 + 2 > 18 + 54
...
79: 79 < 2 + 6 + 18 + 54
79 > 6 + 18 + 54
80: 80 = 2 + 6 + 18 + 54
Just to name a few!
I really like how pretty this looks. It's very easy to do, once you see the pattern play out over enough numbers. Now, I'll give you a little explanation behind how I landed on these numbers.
When I first saw the problem, I knew that the four numbers (weights) should probably add up to 40 (or at least close to). That way, when the biggest number came up, these four numbers could still reach it. But I was struggling to identify (only) four numbers that were large enough to reach 40, but small enough to hit each little number as it came up. Believe it or not, my first quess came along the lines of 1 - 4 - 9 - 20. It seemed to me that there was adequate space between the numbers, but that they could still reach in between. So seemed a good place to start, save for my initial constraint that the four numbers had to add up to 40. These added up to only 34. This lead me to believe that '1' was out! No way could I have four numbers and have '1' as one of them. So I next looked at 2 as my lowest number. This immediately prompted the question, well how would you weight '1'? Hmm... well, I suppose '1' is less that '2' (my lowest number), and since these are weigh-scales, we'd easily be able to determine that '1>0'. Thus, began my rationale behind using inequalities. After '1' was solved, '2' was easy. It was '2=2'! But '3' was hard. If my second weight was '5', then '3+2=5'. But by that reaoning, '6+5+2=13' and '14+13+5+2=34'. So, my four numbers would be 2 - 5 - 13 - 34. Does this work? Actually, it does!! But it can only reach 54 (not 80) and more importantly, before I had even got to working out all four numbers, I had already thought of changing it.
If I can get away with 1<2, then I can get away with 2+3<6 and from that I can do anything. 2+6=8 and so, the next number we could have to deal with is 9. And again, I can get away with 2+6+9<18, so I did so. And from this, I produced way of figuring out the next number. 2+6+18=26, so 2+6+18+27<54 was the next step in the logic. And from that I produced my guiding numbers. And they are 2 - 6 - 18 - 54. And with these, I can tackle all of them to 80. Not 2+6+18+54=80. So, if I only had three weights, I could reach 26. i.e. 2+6+18=26.
So, if you are faced with this problem in the future, you have the skills to overcome it. And what if you are presented with 1-40,000 instead of 1-40?? Can it be done? Is this the most efficient way? What are the implications? Below I have set a question dealing with just that:
How many weights (#) are necessary to reach 40,000kg? How high would those weights reach (kg)?
ANSWER MY QUIZ
(side bar of page)
Hope you enjoyed!
Tuesday, September 22, 2009
Response: Article by Heather J. Robinson
'Using Research to Analyze, Inform, and Assess changes in Instruction'
by Heather J. Robinson
This article struck me particularly because I see myself, primarily, as a 'lecture-based' teacher. Perhaps this spawns from having a lecture-style teacher for my grade 12 year. But, having the right coping and learning skills, I thrived regardless and in fact was probably aided by the large quantity of information and demonstration. The participation factor wasn't necessary for me to succeed, where as it may be for many others. So, I'm looking to try and be more effective than that at reaching all my students.
I am curious as to why the issue of high failure rates doesn't seem to have been addressed by the school. Did they not know this was happening? Also, what school has 20 math teachers?!? Even for 2,500 students. My high school had 3,000 students and had a french teacher teaching math along with a small handful of math teachers. And there was no 'Advanced Placement Statistics' or 'Money Management' courses... there was just 'Math'! Is it possible that this school and its teachers were simply teaching material that was too advanced for the students? Or were there some real challenges in engaging the students, without sacrificing the essential time spent on theory and lecture.
It even seemed that there might be a fundamental challenge in learning instrumental techniques (as in the case of the Quizzes). There is clearly a piece of the puzzle missing if the students can't even solve a simplification problem instrumentally.
In my future classroom, I plan on supplying more practice, more assessment and more clarity of information, to best avoid some of these learning crises.
by Heather J. Robinson
This article struck me particularly because I see myself, primarily, as a 'lecture-based' teacher. Perhaps this spawns from having a lecture-style teacher for my grade 12 year. But, having the right coping and learning skills, I thrived regardless and in fact was probably aided by the large quantity of information and demonstration. The participation factor wasn't necessary for me to succeed, where as it may be for many others. So, I'm looking to try and be more effective than that at reaching all my students.
I am curious as to why the issue of high failure rates doesn't seem to have been addressed by the school. Did they not know this was happening? Also, what school has 20 math teachers?!? Even for 2,500 students. My high school had 3,000 students and had a french teacher teaching math along with a small handful of math teachers. And there was no 'Advanced Placement Statistics' or 'Money Management' courses... there was just 'Math'! Is it possible that this school and its teachers were simply teaching material that was too advanced for the students? Or were there some real challenges in engaging the students, without sacrificing the essential time spent on theory and lecture.
It even seemed that there might be a fundamental challenge in learning instrumental techniques (as in the case of the Quizzes). There is clearly a piece of the puzzle missing if the students can't even solve a simplification problem instrumentally.
In my future classroom, I plan on supplying more practice, more assessment and more clarity of information, to best avoid some of these learning crises.
MAED 314 'Timed-write' exercise
6 min. Timed-Write:
Prompt: "2 most memorable math teachers"
"My most memorable math teacher was in grade 12. He was a young guy much like I will be in my first year. I believe he taught us in his first year, although he didn't dare reveal that to us. He was very friendly, considerate and pensive. He had a great answer for most questions and on questions where he didn't, he would ponder it and return triumphantly the next class with the answer/solution. I admired his love of math (and physics) and appreciated his relaxed attitude/philosophy on everything. He also had great demonstrations, I learned a lot.
My second most memorable math teacher was in grade 11. No surprise there. She taught me/trained me to write out all my steps in a solution. Up until grade 11, I used to solve problems only in my head. She helped me show my work... something essential for university! Also, she had us do yoga at the beginning of each math class, which at the very least helped us focus but probably gave us a better perspective also. Very cool!"
In looking back on this 'timed-write', I'm amazed that I didn't chose to focus on a math teacher that I hadn't enjoyed. Although, having succeeded at math, I suppose that I would have survived even bad math classes without much trouble. This exercise has helped remind me of how these math classes had been conducted and I know that as a math teacher, I will incorporate some of their styles whether I know I using them or not. But I feel confident that these teachers have prepared me and given me a model to follow in my teaching, and I'm excited!
Prompt: "2 most memorable math teachers"
"My most memorable math teacher was in grade 12. He was a young guy much like I will be in my first year. I believe he taught us in his first year, although he didn't dare reveal that to us. He was very friendly, considerate and pensive. He had a great answer for most questions and on questions where he didn't, he would ponder it and return triumphantly the next class with the answer/solution. I admired his love of math (and physics) and appreciated his relaxed attitude/philosophy on everything. He also had great demonstrations, I learned a lot.
My second most memorable math teacher was in grade 11. No surprise there. She taught me/trained me to write out all my steps in a solution. Up until grade 11, I used to solve problems only in my head. She helped me show my work... something essential for university! Also, she had us do yoga at the beginning of each math class, which at the very least helped us focus but probably gave us a better perspective also. Very cool!"
In looking back on this 'timed-write', I'm amazed that I didn't chose to focus on a math teacher that I hadn't enjoyed. Although, having succeeded at math, I suppose that I would have survived even bad math classes without much trouble. This exercise has helped remind me of how these math classes had been conducted and I know that as a math teacher, I will incorporate some of their styles whether I know I using them or not. But I feel confident that these teachers have prepared me and given me a model to follow in my teaching, and I'm excited!
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