Reflection: 'Free Write' & Math Poem Analysis

Exerpts from free-write:

"Zero is an ultimatim..."
"...it is the absense and it is comforting."
"It is the producer of uncertainty, goal of asymtotic behaviour"

Analysis:

I found my free-write to be too confusing to pull much from. I certainly didn't use any direct quotations from the free-write for my poem. But to give feedback on the effectiveness of this activity, I have created a pros & cons list.

Pros:

• Fun exercise that made me think a lot about pure numbers and their relations
• Good practice of applying knowledge of mathematics to an artistic medium
• The free-write did provide some useful ideas for future work

Cons:

• I needed some more inspiration before I did my free-write
• I would have prefered more variety in topic options
• I would have loved writing the poem if I had the freedom to rhyme

Math Poem

"Over Zero"

What do you give something that has
nothing? Only everything for which it
has the capacity, even if it's zero.

Is there a limit, a point where
we squeeze the answer up into it?
Rollerpin it flat and toss it on top.

Nothing has a specific size and
can be played with in this way.
That's how we found the answer.

Reflection: Micro-Teaching Presentation 14/10/09

For our presentation, we chose the topic of logarithms. We then had 15 min. to present the subject in an engaging fashion with our students. Given these conditions, it proved rather difficult to approach, as logarithms are such a large topic and we had to pick and choose what to present.

Our presentation started with a brief history of the inventor of logarithms and his motivation for creating them. I felt this was a very valuable aspects of teaching the subject, but in retrospect, it used up some precious time for teaching. Given a week to present logarithms to a class, I would devote a reasonable time to the history and more abstract qualities. But in 15 min. we really ought to have stuck to the basics. And so in our rush to fit in the rest of the material we had planned, we received some negative feedback regarding the organization, flow and fun of the presentation.

Next we reviewed many of the laws of logarithms with the students and derived the Multiplication and Division Laws. In retrospect, since we had agreed that the students would already know the basics of logarithms, we shouldn't have spent so much time reiterating them.

Lastly, we wanted to go through a typical logarithm problem and by having the students reply with the answers, slowly graph it. This would have worked well, because we were going to split a large logarithm problem into some shorter steps and have the class solve each one. Then we would simultaneously be plotting these answers to show the general trend. That way the students would understand the relationship between the mathematical and visual interpretations of the logarithm. Again, we received negative feedback because we ran out of time.

In future, we will work to be more careful with what amount of information can be transferred from teacher to student in the given time and prioritize accordingly.

BOOPPPS Lesson Plan - Exponentials & Logarithms

1. Bridge: We will begin our micro-teaching presentation with a 'hook', something to grab the students attention. In this case, we have a brief history of the invention of logarithms to show the class.
2. Teaching Objectives: We plan to cover the laws of exponentials and logarithms and then work through a problem along with the class. This will comprise of a number of smaller problems taken from the larger problem and posing them individually to the class for response. Finally, we take all those responses, put them together and graph the result.
3. Learning Objectives: We plan to have the students apply the various logarithm rules to solve a question and display their understanding.
4. Pre-test: We will go over the rules of exponentials and logarithms. We will then have the class answer a few simple questions. We will then show a graph comparing logarithms in various bases.
5. Participatory Activity: We will work through the long logarithm problem with the class and produce a result that everyone is happy with.
6. Post-test: Time permitting, we have a second (shorter) problem for the class to work on and solve.
7. Summary: We will briefly recap the equations and concepts that we have covered.

Citizenship & Democracy in Mathematics

Response to: "Citizenship Education in the Context of School Mathematics" ~ Elaine Simmt.

Elaine Simmt argues in her article that by studying mathematics, one will develop critical thinking and problem-solving skills, that are crucial for a democratic, informed member of society. I certainly see the potential validity of this statement. Having met, and studied under, many mathematicians, I know that people who study math passionately are conscious and critical thinking. There are many personality traits (reserved, patient, analytical, cautious) associated with mathematicians that would suggest that those who study math are affected positively by their extensive study.

But perhaps Simmt has missed a step between critical thinker and global citizen. As much as there are 'problem-solvers' in the world, there can be 'problem-posers', or in a social context, 'problem-causers'. Do we know any bad 'math people'? What about internet hackers, inside traders, card sharks, money lenders and ruthless businessmen? Have they not done the math? Having the skills to manipulate our highly quantized world doesn't pose any limit or moral barriers to those seek wealth or power.

I do agree with Simmt though, that math is very much a part of the human experience. And those who use math, do 'transform' our reality and not just 'transmit' it. Math is a double-edged sword. For every opportunity for constructive processes, there exists the possibility for destructive processes; for every problem solved, a problem posed. The beautiful symmetry of math means that everything has an inverse. Math gives us the skills to examine the world at its most fundamental levels, but the choice of good or bad still exists. All I can hope for, is that I can show my students the beauty, symmetry and good in mathematics and prepare them for the true problems that life will present them.

Review of "What-If-Not" strategy from "The Art of Problem Posing"

This review is in reference to pages 33-65 of the "The Art of Problem Posing" by Stephen I. Brown and comments on the strategy of "What-If-Not".

The term "What-If-Not", refers to a creative thinking process of examining mathematical/problem elements with a fresh perspective. With each concept, we identify key ideas & problems and then we try to expand on them in new directions, incorporating our own perspectives and thinking processes.

I really like this strategy for problem posing/solving. It's all about asking the right questions. Taking a theorem and changing a variable or condition to create a new situation and then examining it to see where it has taken us. This can be used extensively in the classroom, as we have begun to discover in our other classes. The students learn through inquiry and compound ideas and ultimately will have a more concrete understanding of the concept using this strategy.

We first explored this in SCED 316a with the aptly named 'UnDemo' in which the students were in control of the experimental process and merely guided by the teacher. In this way, the students were allowed to ask "what-if" and then examine "what-if-not". If students can understand what something 'is not', then they have a better understanding of what 'it is'. We next explored this in our Principles of Teaching lecture in which we talked about what it means to define a chair. And then from that, looking at a plethora of objects that were 'chair-like', but not chairs (i.e. thrones, stools, lounges, couches, etc.). In this way, the pythagorean theorem is a chair, but the other equations that were plucked from it, where 'pythagorean-like' and equally valid for examination. And by understanding what the pythagorean theorem wasn't, a student would have a better understanding of where it fit in their theoretical framework.

The only limitation to this method of inquiry, however, is that fact that this type of exploration can often be off 'topic' or time-consuming and may even lead to confusion. We cannot use this strategy like a blunt tool, or it will damage the process. Instead, we must use it as a fine instrument that requires skill and precision in utilizing for best results.

10 Questions/Comments on "The Art of Problem Posing"

This response consists of 10 questions/comments regarding pages 1-32 of "The Art of Problem Posing" by Stephen I. Brown.

1. I valued most highly, the idea that problems are very much context-based. The difference between a lousy problem and a superb problem (on the same topic) lies almost solely in context. And I think a student can tell/appreciate when a truely good/unique/intriguing question has been asked (as opposed to a conventional one).
2. I often see my own contextual view of problem posing and solving as being quite narrow. What are some strategies or literature that might best help broaden my contextual view?
3. I like the notion of posing an idea and not a problem. The open-endedness of it is very freeing and allows the mind to explore new possibilities and solutions. I was amazed at how many answers there were to a given idea.
4. I also thought it was very interesting, the idea that the person being asked to respond to an idea like x^2 + y^2 = z^2, was, in a sense, posing and answering their own questions and that the problem-solver imposes their own views on an idea and becomes a unique problem-poser.
5. What do you do when the a student wants the proof to an equation or principle and the explanation of the proof would be too far above that student's current knowledge level? Often times, I think math teachers are compelled to have their students take their word for it, because the beautiful simplicity of an idea is too complex to explain to the student.
6. How strongly should I be emphasizing pattern recognition, rare sequences and phenomena in amongst the standard curriculum?
7. What license do I have as a teacher to assign value/marks to a student's answer that is a unique approach/solution to the problem for which they put the wrong answer? i.e. If I want the answer 5,12,13 and I get the answer -5,-12,-13?
8. I found it very funny in the hand-shake example of the jurors, because my mind immediately jumped to the problem of how many total handshakes. I guess that is my narrow contextual perspective.
9. How far can I probe into a simple concept before I run the risk of undoing the knowledge I taught and confusing my students?
10. What IS the use of examining phenomena?

Response to Timed-write 2

Well, I wrote this timed-write really fast, so I'm not sure how accurately it portrays my feelings. But I do know that it sheds some light on what I think about my future students and what my future performance might look like. A lot of what I wrote is stuff that I've already been thinking about and trying to problem solve. I sit on the bus dreaming up ways to approach topics with kids and how to best teach difficult concepts. Some of my ideas are way out there and that's why it was easy to write a positive and negative 3rd person perspective.

With the positive perspective, hypothetically written by a former student, I wrote it in much the same style that I might have written an actual letter to my former teacher. I admired many of his attitudes on the subject of learning math and I hope to embody some of those same teaching styles. Whether a student ever realizes the quality of my instruction is immaterial, so long as they receive the benefit from a well-suited instruction and curriculum.

With the negative perspective, again written by an imaginary former student, I try to shed light on some of my fears as a new teacher. I can imagine some of my creative teaching methods as being counter-productive, confusing or only benefiting a small portion of the class. So, I'm still struggling to find the right idea, or a least a good working-idea of what an effective (interactive) teaching approach is. Plus, this negative perspective mirrors some of the comments in the positive perspective and attempts to reveal that there can often be goods and bads to any approach. There will sometimes be disparity amongst students as to who really benefits from my teaching.

So, as you can see, my focus continues to be to find an effective and interactive teaching style. I like the picture I portrayed in the positive perspective and I'm going to try to find one-on-one strategies to avoid the picture I portrayed in the negative perspective.

Timed Write - 10 Years From Now

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.