Approaching zero,
Entering the dark.
To the blackest black,
We dive and divide.
Improbable,
But not impossible.
To exist at all,
Throughout space.
.
.
.
During the free write, the first word was 'divide'.
One into another. Over another. Fraction. Dum doodle dee. There is a line with a dot above and bellow. Or you could see a backslash. A divider! Two sides to something, ............ Cut a cucumber or an apple into 2, 3, 4 pieces. Oh it's a miracle. Big number over smaller number could be bigger or smaller. Divide those people into groups.
The second word was 'zero'.
The name of a character in Code Geass... What is 0? Is it nothing? Is it very very small, approaching nothing? What do you get when you divide by 0? Calculators say undefined. But what is a number divided by something so incredibly small? Something so big... Infinity. That's right, Infinity. Yeah, capitalize that! Where does 0 exist? Where in the Universe, Galaxy, Solar System, Star, Planet, Continent, Household, Pocket?
Would I use this in a 9, 10, 11 classroom? Depends on the class and how I feel they learn. Writing a poem about zero is interesting, but is it useful? When doing the free write it wasn't as if I had a revelation about zero. I didn't come to any new conclusions. I just wrote what I already knew with the knowledge of limits as x approaches 0. This is what would probably happen if my high school students were given the same assignment. They would write what they knew already without learning anything new or coming to an understanding about the topic.
Monday, October 19, 2009
Thursday, October 15, 2009
Math lesson reflections
Our lesson on logarithms did not go as planned. Our original plan was as follows,
1) Ralph do a short history as a hook
2) Write up some of the rules on logarithms
3) Derive multiplication and division laws for logarithms
4) Have the students do two quick questions to review the logarithm rule blog(x) = log(x^b)
5) Have an engaging question for the class to do
For 15 minutes this was way too ambitious of a time-line. The history went a bit longer than expected taking up ~50% of our time, not giving much time to work on the lecture portion and student centered activity.
The big issue from what I gleamed from the comments was that there was not enough activities for the students. So how would I change it for the future?
I would cut the history to 2 minutes max and quickly write up the basic rules of logarithms, such as for log(0), log(1), when base is the same as what is being logged. For this type of presentation, we can assume that we are just doing a review of the rules and the students have already learned it before.
I would cut out the derivations for the multiplication and division laws. In fact, I don't think I'd include those laws at all for this presentation. For 15minutes we need to keep it extremely short. What I would do is write a function on the board, starting with y = x and ask the class, 'without using a calculator how would I graph y = log(x). The purpose of this activity would be for the students to recognize that when x <= 0, there would exist no y. When x = 1, y =0. When x = 10, y = 1 etc. I would then get them to do the same graph but with a different base instead of 10 and see how the different graphs compare.
I could then give them an equation such as y = x^2 or x^2 - 3 and we could repeat the above process. They can check their results using a graphing calculator after wards. The purpose of this activity would be for them to recognize that a log of a negative number or 0 is always undefined and a log of 1 is always 0. Also, to graph it without a calculator, the students would have to recognize how to choose easy number to work with when dealing with logarithms. For example, if you were working with a base 2 logarithm, you would want to solve for when the inside is 2^n.
Example, for y = log(x^2 - 2) [base 2]
If y = 1, then x^2 - 2 = 2. Solving for x you get 2. Next you do for y = 2, then x^2 - 2 = 2^2 = 4. Solving x you get root(6). It's even easier to graph the logarithm if you were t first graph y1 = x^2 - 2 first, and then look for key locations on y1 such as when y1 = 0,1,2,4,8,16, 1/2, 1/4, etc.
That's all I have to say.
1) Ralph do a short history as a hook
2) Write up some of the rules on logarithms
3) Derive multiplication and division laws for logarithms
4) Have the students do two quick questions to review the logarithm rule blog(x) = log(x^b)
5) Have an engaging question for the class to do
For 15 minutes this was way too ambitious of a time-line. The history went a bit longer than expected taking up ~50% of our time, not giving much time to work on the lecture portion and student centered activity.
The big issue from what I gleamed from the comments was that there was not enough activities for the students. So how would I change it for the future?
I would cut the history to 2 minutes max and quickly write up the basic rules of logarithms, such as for log(0), log(1), when base is the same as what is being logged. For this type of presentation, we can assume that we are just doing a review of the rules and the students have already learned it before.
I would cut out the derivations for the multiplication and division laws. In fact, I don't think I'd include those laws at all for this presentation. For 15minutes we need to keep it extremely short. What I would do is write a function on the board, starting with y = x and ask the class, 'without using a calculator how would I graph y = log(x). The purpose of this activity would be for the students to recognize that when x <= 0, there would exist no y. When x = 1, y =0. When x = 10, y = 1 etc. I would then get them to do the same graph but with a different base instead of 10 and see how the different graphs compare.
I could then give them an equation such as y = x^2 or x^2 - 3 and we could repeat the above process. They can check their results using a graphing calculator after wards. The purpose of this activity would be for them to recognize that a log of a negative number or 0 is always undefined and a log of 1 is always 0. Also, to graph it without a calculator, the students would have to recognize how to choose easy number to work with when dealing with logarithms. For example, if you were working with a base 2 logarithm, you would want to solve for when the inside is 2^n.
Example, for y = log(x^2 - 2) [base 2]
If y = 1, then x^2 - 2 = 2. Solving for x you get 2. Next you do for y = 2, then x^2 - 2 = 2^2 = 4. Solving x you get root(6). It's even easier to graph the logarithm if you were t first graph y1 = x^2 - 2 first, and then look for key locations on y1 such as when y1 = 0,1,2,4,8,16, 1/2, 1/4, etc.
That's all I have to say.
Tuesday, October 13, 2009
Booppps Lesson Plan - Exponentials and Logarithms
Bridge (The Hook) - The history of logarithms, who invented it and why was it invented.
Teaching Objectives - To cover/review some of the laws of exponentials and logarithms before going over a more challenging problem together as a class. If there is additional time we have an extra problem that we can challenge the class with.
Learning Objectives - Have the students apply various log rules to solve a question, displaying their understanding.
Pretest - Go over the rules of exponentials and logarithms and then have the students answer a couple simple one or two step questions.
Participatory Activity - Work through together as a class a challenging logarithmic problem. Graph, without a calculator, y = log(x+2) - log(x-1) where the log is base 3.
Post Test - The participatory activity can be considered as the post test since there is only 15 minutes for the lecture. If there happens to be time we have an additional question to give.
Summary - Tell the class there will be an exam next week!
Teaching Objectives - To cover/review some of the laws of exponentials and logarithms before going over a more challenging problem together as a class. If there is additional time we have an extra problem that we can challenge the class with.
Learning Objectives - Have the students apply various log rules to solve a question, displaying their understanding.
Pretest - Go over the rules of exponentials and logarithms and then have the students answer a couple simple one or two step questions.
Participatory Activity - Work through together as a class a challenging logarithmic problem. Graph, without a calculator, y = log(x+2) - log(x-1) where the log is base 3.
Post Test - The participatory activity can be considered as the post test since there is only 15 minutes for the lecture. If there happens to be time we have an additional question to give.
Summary - Tell the class there will be an exam next week!
Sunday, October 11, 2009
Flatlander
Just finished the book 'Flatland'. I'll talk about it more later, for now I'd like to just put up this link which someone sent to me a long while back: http://www.tenthdimension.com/medialinks.php.
I remember reading a Sci-Novel a while back which involved a 3-D world and a 2-D parallel world. This was a long time ago and could possibly just be part of my imagination... I'm going to see if I can find.
I remember reading a Sci-Novel a while back which involved a 3-D world and a 2-D parallel world. This was a long time ago and could possibly just be part of my imagination... I'm going to see if I can find.
Mathematics and Citizenship Education
The article "Citizenship Education in the Context of School Mathematics," by Elaine Simmt, states that mathematics has a role in citizenship education and is crucial in the development of informed, active and critical citizens in our society. "Mathematics will help us understand ourselves... [and] understand the world in which we participate." Mathematics is embedded into our world from business, economics, time, space, communication, transport and war and our youth must be educated to understand and critique the 'formatting power' of mathematics.
So, how to we teach math to accomplish this. Right now, math is usually taught as:
1) a set of facts, skills and processes - creating consumers who know how to leave a tip.
2) as facts and fact - pushing around numbers and writing them in different ways suiting the teacher.
3) as right or wrong - reinforcing the notion that mathematics must not be questioned and creating no confidence in individuals.
Elaine gives possible methods such as:
1) focusing on problems and questions arising from student interaction with math using a strategy called 'posing variable-entry prompts'.
2) having the students explain their reasoning attempting to prove their assertions. The necessity of student contribution.
The example in this article about making rectangles using 36 square tiles reminds me of the recent video we saw in class about solving two challenging exponential/logarithmic problems using 2 or more methods.
These types of questions would likely take up a lot of time and therefore I would probably limit each unit to one, possibly two activities of this nature. It would depend on the unit and how much worth I see the students actually getting out of the activities.
These activities encourage group work and negotiation, forcing the students to pose questions and hypothesis on how to approach open ended questions. I like the idea of groups of students writing down their solutions on poster paper and then having to present their results. I completely agree that it is important for students to present their results explaining their reasoning's and approaches to the problems. Also, being in groups will provide them more confidence when discussing the results.
So, how to we teach math to accomplish this. Right now, math is usually taught as:
1) a set of facts, skills and processes - creating consumers who know how to leave a tip.
2) as facts and fact - pushing around numbers and writing them in different ways suiting the teacher.
3) as right or wrong - reinforcing the notion that mathematics must not be questioned and creating no confidence in individuals.
Elaine gives possible methods such as:
1) focusing on problems and questions arising from student interaction with math using a strategy called 'posing variable-entry prompts'.
2) having the students explain their reasoning attempting to prove their assertions. The necessity of student contribution.
The example in this article about making rectangles using 36 square tiles reminds me of the recent video we saw in class about solving two challenging exponential/logarithmic problems using 2 or more methods.
These types of questions would likely take up a lot of time and therefore I would probably limit each unit to one, possibly two activities of this nature. It would depend on the unit and how much worth I see the students actually getting out of the activities.
These activities encourage group work and negotiation, forcing the students to pose questions and hypothesis on how to approach open ended questions. I like the idea of groups of students writing down their solutions on poster paper and then having to present their results. I completely agree that it is important for students to present their results explaining their reasoning's and approaches to the problems. Also, being in groups will provide them more confidence when discussing the results.
Wednesday, October 7, 2009
What-if-not
What-if-not? Yes, what is that?
It is a strategy to assist you in problem posing. It is a systematic approach that has five levels.
How can I use this technique in preparation for my groups micro-teaching lesson next Wednesday on Logarithms and Exponentials? At the moment, I'm not really sure how it will be of use. However, I am sure we'll be able to use this problem posing strategy to think of some questions that could be useful for the presentation.
As a level 0, we could possibly begin with one of the rules for logarithms, such as the Power Law, or Multiplication/Devision Laws. From there, who knows where it will go?
The strengths of the What-if-not method is that it is a pretty direct way to formulate a lot of questions from a single starting point. It is a tree that keeps expanding and is interconnected with itself to create even more possibilities. Another positive aspect is that fact that it is a systematic approach to problem posing. It walks you through the steps making sure you really analyze the starting point, the topic you chose, by listing all the attributes you can see. For every question you get a what-if-not question and you are forced to analyze to prove/disprove (find out when true/false) each statement before continuing. You will really able to flesh out the topic you are looking at using this method.
The weaknesses of this method is that it can take a long time, and I mean a LONG time. And after putting a lot of hard work into the process, you may find yourself with nothing to show for it. For example, you may create a lot of what-if-not statements similar to Attribute 8 on page 50. Those arn't going anywhere and will prove to be a waste of time (If I'm wrong about that last sentence, please let me know). Nothing useful may come out of the process meaning I could have watched an episode of House instead of using the strategy.
I'll try using this method during the preparation for the micro teaching lesson. I'll see how it goes.
It is a strategy to assist you in problem posing. It is a systematic approach that has five levels.
- Level 0 Choosing a Starting Point: ie. Pythagorean Theorem
- Level 1 Listing Attributes
- Level 2 What-If Not-ing: Take an attribute and ask yourself, 'what if that attribute was not true and was instead something else?' List as many alternatives as you can think of. ie. What if the Pythagorean Theorem was instead written as a^2 + b^2 <>
- Level 3 Question Asking or Problem Posing: What questions come to mind when you write down this alternative form? When is it true? Is it significant?
- Level 4 Analyzing the Problem
How can I use this technique in preparation for my groups micro-teaching lesson next Wednesday on Logarithms and Exponentials? At the moment, I'm not really sure how it will be of use. However, I am sure we'll be able to use this problem posing strategy to think of some questions that could be useful for the presentation.
As a level 0, we could possibly begin with one of the rules for logarithms, such as the Power Law, or Multiplication/Devision Laws. From there, who knows where it will go?
The strengths of the What-if-not method is that it is a pretty direct way to formulate a lot of questions from a single starting point. It is a tree that keeps expanding and is interconnected with itself to create even more possibilities. Another positive aspect is that fact that it is a systematic approach to problem posing. It walks you through the steps making sure you really analyze the starting point, the topic you chose, by listing all the attributes you can see. For every question you get a what-if-not question and you are forced to analyze to prove/disprove (find out when true/false) each statement before continuing. You will really able to flesh out the topic you are looking at using this method.
The weaknesses of this method is that it can take a long time, and I mean a LONG time. And after putting a lot of hard work into the process, you may find yourself with nothing to show for it. For example, you may create a lot of what-if-not statements similar to Attribute 8 on page 50. Those arn't going anywhere and will prove to be a waste of time (If I'm wrong about that last sentence, please let me know). Nothing useful may come out of the process meaning I could have watched an episode of House instead of using the strategy.
I'll try using this method during the preparation for the micro teaching lesson. I'll see how it goes.
Sunday, October 4, 2009
10 Questions or Comments
1) From the text, 'The Art of Problem Posing', I like the line on page 3, '"The answers Gertrude, what are the answers?" - whereupon Gertrude allegedly responded, "The questions, what are the questions?"'
It reminds of a scene in the book 'The Hitchhikers Guide to the Galaxy'. Deep Thought, a super computer created by a pan-dimensional, hyper-intelligent race of beings, was faced with the task of figuring out the answer to Life, The Universe and Everything. 7 and a half million years later he eventually comes up with an answer, 42. Deep Thought then replies, "I think the problem, to be quite honest with you is that you've never actually known what the question was."
There are all these answers and facts out there but there isn't only one correct path to these answers. So what are the questions?
2) x^2 + y^2 = z^2. An interesting way to begin because it is so open ended. You could choose anything, there are an infinite number of possibilities. The book asks you "What are some answers?" and the only way to come up with any is to pose some questions your self. I simply chose some simple numbers so I could move on with the reading. After reading some of the answers written though it got me thinking about the vast number of responses, and the fact that the responses didn't have to do with math at all.
3) When I see x^2 + y^2 = z^2 I think of a circle first then Pythagoras and right angle triangles. It's a tool to get an answer. The triangles that come to mind are the standard 'special' triangles with ratios 1:1:root2 and 1:2:root3.
4) On page 14 they show a triangle with squares on each edge. I have seen this before, but I have never known the reasons, it seems pointless. How does it help? After writing this question I think I may have figured it out. Is it simply a reminder that to relate each edge length you must square each length first?
5) Do you ever give us any of the answers to the questions you pose? Some of the points brought up are interesting but I guess it isn't the purpose of the book to give answers. That's what the internet is for eh?
6) The hand shaking question on page 19. When I hear this problem I instantly think 'the number of people, choose 2'. nC2, where n is the number of people shaking hands. So, with 9 people, we get 36 handshakes.
Yes, I am one of those guys who came up with the question, "How many handshakes were there altogether."
7) It has been so long since the first time we actually learned the properties of isosceles triangles and Pythagoras, etc. It makes me wonder, how did I first learn these concepts? What went through my mind when it was introduced to me? Did I ever think of interesting questions such as "can I make a bicycle hub out of congruent isosceles triangles?" When I read this chapter today, I certainly didn't come up with any questions like that. I agree with what is written on page 22 whole heartedly, those who have been exposed to geometry "find great difficulty coming up with much more than observations relating the equality of the lengths to the equality of angles."
8) If I was given a Geoboard of 25 nails I would get an extremely stretchy rubber band and try to make the most complicated looking picture possible using all of the pins without the elastic ever crossing over itself.
9) Coming up with unique questions for these pythagoras, geoboard, isosceles and handshake problems is tough. I'm sure after teaching a few courses in math and seeing what students come up with I'll be more capable of thinking of questions, both in and out of context.
10) The handy list of questions on page 30. Add ons:
What is the origin of the formula? Is it continuous? Discrete? Is there an average? Are there visual representations? For whom is this useful to? Practical or theoretical? Can we prove it? Do we need to prove it?
It reminds of a scene in the book 'The Hitchhikers Guide to the Galaxy'. Deep Thought, a super computer created by a pan-dimensional, hyper-intelligent race of beings, was faced with the task of figuring out the answer to Life, The Universe and Everything. 7 and a half million years later he eventually comes up with an answer, 42. Deep Thought then replies, "I think the problem, to be quite honest with you is that you've never actually known what the question was."
There are all these answers and facts out there but there isn't only one correct path to these answers. So what are the questions?
2) x^2 + y^2 = z^2. An interesting way to begin because it is so open ended. You could choose anything, there are an infinite number of possibilities. The book asks you "What are some answers?" and the only way to come up with any is to pose some questions your self. I simply chose some simple numbers so I could move on with the reading. After reading some of the answers written though it got me thinking about the vast number of responses, and the fact that the responses didn't have to do with math at all.
3) When I see x^2 + y^2 = z^2 I think of a circle first then Pythagoras and right angle triangles. It's a tool to get an answer. The triangles that come to mind are the standard 'special' triangles with ratios 1:1:root2 and 1:2:root3.
4) On page 14 they show a triangle with squares on each edge. I have seen this before, but I have never known the reasons, it seems pointless. How does it help? After writing this question I think I may have figured it out. Is it simply a reminder that to relate each edge length you must square each length first?
5) Do you ever give us any of the answers to the questions you pose? Some of the points brought up are interesting but I guess it isn't the purpose of the book to give answers. That's what the internet is for eh?
6) The hand shaking question on page 19. When I hear this problem I instantly think 'the number of people, choose 2'. nC2, where n is the number of people shaking hands. So, with 9 people, we get 36 handshakes.
Yes, I am one of those guys who came up with the question, "How many handshakes were there altogether."
7) It has been so long since the first time we actually learned the properties of isosceles triangles and Pythagoras, etc. It makes me wonder, how did I first learn these concepts? What went through my mind when it was introduced to me? Did I ever think of interesting questions such as "can I make a bicycle hub out of congruent isosceles triangles?" When I read this chapter today, I certainly didn't come up with any questions like that. I agree with what is written on page 22 whole heartedly, those who have been exposed to geometry "find great difficulty coming up with much more than observations relating the equality of the lengths to the equality of angles."
8) If I was given a Geoboard of 25 nails I would get an extremely stretchy rubber band and try to make the most complicated looking picture possible using all of the pins without the elastic ever crossing over itself.
9) Coming up with unique questions for these pythagoras, geoboard, isosceles and handshake problems is tough. I'm sure after teaching a few courses in math and seeing what students come up with I'll be more capable of thinking of questions, both in and out of context.
10) The handy list of questions on page 30. Add ons:
What is the origin of the formula? Is it continuous? Discrete? Is there an average? Are there visual representations? For whom is this useful to? Practical or theoretical? Can we prove it? Do we need to prove it?
Friday, October 2, 2009
The Distant Future...
Letter from a student who liked you.
Hello Mr. Thiessen,
My name is Slarty Bartfast and I was in your class 10 years ago in grade 10. You changed my life. I wasn't always a fan of math. In fact I hated it! After your class I was inspired. You showed me how math could actually be and taught. Your student centered lessons really got me involved and had me understanding the material in no time. It made me continue onwards into grade 11 and 12 and beyond.
You most likely recognize my name because I've been on the news constantly this past week. The man who successfully made the calculations for the first intergalactic warp drive system. It was actually my PHD paper.
Thanks Mr. Thiessen. Because of you, mankind can now travel to the stars.
- Slarty
Letter from a student who hated you.
Mr. Thiessen,
My name is Troy McClure, you may remember me from high school grade 10. Math, I hated it and due to that I didn't give you a chance when I walked into your class. Math was tough, equations and equations of nonsense. Plug this number into x and sub that number into y. It made no sense and that's how I wanted it to stay.
During your lessons you'd often talk about understanding this, and understanding that. I didn't care about any of that. I knew that once math was no longer a prerequisite I would no longer enroll into it.
I didn't want to discuss ideas in groups or do any projects. The bare minimum to pass, that's all I wanted. This was the last mandatory year for math and once I finished high school I was set to pursue my future career in TV broadcasting.
Which is what I did. No thanks to you,
Troy McClure
My hopes and worries...
I hope to be a teacher who passes on an understanding of math to my students. I didn't fear math, I enjoyed it, and I hope I can pass that feeling on to my students by being a effective and engaging teacher. However, I know there will be students who will never like math and I hope I can inspire them enough to get them through the courses.
Hello Mr. Thiessen,
My name is Slarty Bartfast and I was in your class 10 years ago in grade 10. You changed my life. I wasn't always a fan of math. In fact I hated it! After your class I was inspired. You showed me how math could actually be and taught. Your student centered lessons really got me involved and had me understanding the material in no time. It made me continue onwards into grade 11 and 12 and beyond.
You most likely recognize my name because I've been on the news constantly this past week. The man who successfully made the calculations for the first intergalactic warp drive system. It was actually my PHD paper.
Thanks Mr. Thiessen. Because of you, mankind can now travel to the stars.
- Slarty
Letter from a student who hated you.
Mr. Thiessen,
My name is Troy McClure, you may remember me from high school grade 10. Math, I hated it and due to that I didn't give you a chance when I walked into your class. Math was tough, equations and equations of nonsense. Plug this number into x and sub that number into y. It made no sense and that's how I wanted it to stay.
During your lessons you'd often talk about understanding this, and understanding that. I didn't care about any of that. I knew that once math was no longer a prerequisite I would no longer enroll into it.
I didn't want to discuss ideas in groups or do any projects. The bare minimum to pass, that's all I wanted. This was the last mandatory year for math and once I finished high school I was set to pursue my future career in TV broadcasting.
Which is what I did. No thanks to you,
Troy McClure
My hopes and worries...
I hope to be a teacher who passes on an understanding of math to my students. I didn't fear math, I enjoyed it, and I hope I can pass that feeling on to my students by being a effective and engaging teacher. However, I know there will be students who will never like math and I hope I can inspire them enough to get them through the courses.
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