Reflection


 * Reflection**

Not going to sweat the formatting right now, until I get a better handle on this
 * * posting reflections in your Wiki after each class
 * posting reflections on practicum Wiki
 * 1-2 pages short essay on any topic you wish to reflect on relating to the teaching of math
 * summary of a reading you might have done in teaching/learning/math ||


 * August 31, 2010**

First class. Didn't get into much specifics regarding teaching math-only, but rather discussed some general strategies such as:
 * Bell Work
 * Gaining attention
 * Lesson plans and outlines
 * Transitions

One point that I found well worth noting, was the notion that using a raised voice as an attention getter carries a significant risk of loss of authority and subsequent embarassment if it fails to work. I'll have to try to be careful to avoid this technique since I think that it would be the most natural for me personally.

The assessment component of creating a Wiki is a little unsettling for me because:
 * I haven't really interacted such things (ie personal wikis) before
 * I'm not really a "write-things-in-my-journal" sort of person, so it seems a little unnatural

The good news is that I am now using OneNote for all of my courses, so this will help with the above.


 * September 1, 2010**

We reviewed some interesting clips of Alfie Kohn lectures. The whole notion of intrinsic vs extrinsic motivation was quite interesting, and just what I was hoping to get out of teacher's college. That being said, although Kohn's presentation is provocative, we have yet to see any solutions or balances offered by him. At some point, it is not enough to be a "pot-stirrer", but rather some concrete suggestions must be offered. Otherwise we are just imagining an idealistic non-reality. Furthermore, although reward and punishment may be de-motivating on a psychological level, it is not realistic to conceive of eliminating all extrinsic motivations - for example "grades" and "promotion" must exist, and I don't think that these are necessarily bad things.

It was also noted that, for many, the material that is learned in school is often forgotten later, at least partly because it lacked intrinsic value. However, I can think of at least one personal example that highlights the notion that youngsters are not necessarily fully capable of autonomously filtering what is "interesting" vs what is "boring", and consequently, what may hold value or impact them later in life. In my case, I specifically remember being disinterested in Literature classes. I also have a vague memory of a teacher describing the literary terms of "character development" and "suspension of disbelief". Ho-hum... So What?. Well, it was not until much later in life that I became interested in reading novels. I developed this interest while encouraging my young son to read - by reading with him, and talking about the novels that we read. And, what I found most striking, was that my enjoyment of any particular novel was very strongly tied to what I could recognize as strong "character development" and subsequent "suspension of disbelief". So, in fact, despite my extrinsically-motivated formative education, I actually developed an intrinsic connection and value on some of the material that I was exposed to.

More good news: I posted a comment on the discussion board of the course Wiki. Chalk one up for Community!
 * Sep 13 2010**

In class today we did a role-play dramatization of a stand-off between a student and a teacher. Although I suspect that the teacher's part was somewhat scripted, it did hit home for me because the stand-off situation is one thing that has worried me about teaching. What if a student doesn't respect my authority? I offered in class that it would be best to avoid the escalation in the first place, but I can easily see how it can get out of hand before you even know it. Although it was suggested that a closer rapport could have helped in the model situation (the student had had a traumatic experience earlier in the morning where her mother kicked her out of the house and called her a slut), I have my doubts that one can expect to consistently ascertain such personal insights with a practical (and probably appropriate) level of rapport.

I also had a side conversation with Robin regarding "punishment". I argued that there is a difference between consequences and punishment. Robin offered that the difference is largely semantic. After some reflection, I can think of examples where I believe the difference is more distinct. One such example: My AT tries to offer his grade 7 class the opportunity to be treated like adults. For instance, he permits the use of water-bottles in class. However, he established that if inappropriate behaviour such as splashing water on others occurs, then he will simply revoke the privilege. It is clear to me that this consequence is by no means intended to be a punishment. Rather it is simply a necessary measure to remain true to one of the classroom rules which was "maintain a safe learning environment" (the AT had previously discussed that a safe environment is possible when each student keeps his body, objects and comments to himself)

Another score for Community: I posted a thought on the Math Practicum Wiki.


 * Sep 15 2010**

So, for bell work today we were asked to reflect on a situation where "in our hearts we know that the math concept that we are teaching is not relevant or important to our students". I actually don't really buy into the notion that I can make such a determination. This is because any given student may pursue an academic stream that requires the foundation. I'm not a math major, so I can't arbitrate on "what has appliction" and "what does not". Even for weak students who are unlikely to progress beyond simple addition, it would not be right for me to treat them as such. I brought up my thought, and Robin did make a good point that we would like to be armed with a reasonable, and concrete answer to the question "Why does this matter?". I think that the moral of the story is to try to research for some application, no matter how simplistic that can be used.

We worked on Fermi problems again, and I think that I came up with a good idea: "If you ask your Mom to drive you to the Mall, how much CO2 will you add to the atmosphere?" I like this because it has so much relevance to the present hot-topic of global warming, and has a somewhat surprising answer. The answer is surprising because we tend to think of car exhaust being "lightweight" (after all, it's just a little bit of gaseous emission).

I also got the chance to work with someone that I haven't yet met before, and I think that this is good practice for my personal and one-on-one skills. I know that I sometimes tend to overpower conversations when I have an idea that I want to express - and I think that I did this again. Oops.

I devised what I think is a good Minds-ON exercise for the topic of dividing by fractions, and I made a video of it since it works best as a teacher-guided exercise. See the Content page for a link. I'll also post it on the course wiki.


 * Sep 20 2010**

Today we worked with manipulatives. I don't have a problem with manipulatives as a //learning// tool, but I think that there might be a tendancy for these to become a crutch for some students. Think of the //original manipulative//: counting on one's fingers. I can't buy an argument that holds that counting on one's fingers is okay for grade 7's.

One of the difficulties that I may have is in understanding the effect that some manipulatives may have on some students. That is to say, if I can just "see" something quite easily, then I may have trouble recognizing how a student interprets the same thing. Some of the manipulative exercises that I think might be useful are:
 * Using linking cubes to validate edge length and surface area (this gives me an idea for a high-tech linking cube: cubes that can light-up of edges and sides, to assist in counting)
 * Using pattern blocks for recognizing fractions and also for dividing fractions
 * Using algebra squares for practicing negative number arithmetic


 * Sep 22 2010**

Today we worked with the following applications: Not much to say except that I think that each of these can be useful in the classroom. I'll add them to the arsenal.
 * Clickers (CPS)
 * Geometer's Sketchpad
 * Tinkerplots


 * Oct 5 2010**

Okay, so we watched another Alfie Kohn video - this one arguing that homework has no effectiveness. It is further argued that those students who "get it" won't benefit from "doing it over and over", while those who "don't get it" won't benefit from practising doing it wrong. There is some merit to both facets, however I feel that there are plenty of cases where the real intent of homework is to provide practice - and this is especially true in math. Here are some simple examples starting from the earliest stages to the latest:


 * Multiplication tables - there's nothing much here to "get". Those that don't learn continue to struggle with math and life. Practice at home is one of the best opportunities to master these.


 * Multiplication and Long Division. Again, everyone can benefit from practice. I'll agree that dozens of problems aren't necessary. I'll also agree that the student who doesn't "get it" won't be able to do it. However, no-one is really going to "get it" by doing one or two problems in class. We all need some practice. And those who don't "get it" - well they probably didn't learn their multiplication tables - yes, now they need a lot more help! But let's not pretend that the average student isn't going to benefit from some practice.


 * Function Integration. Again just practice, even at the senior level. The test is going to have some of these problems, and you won't have much success if you haven't practiced. There is no point in saying "I won't have this kind of test", because I can assure you that these problems will be on your 1st year calculus exam!


 * Application Problems. Even into my 4th year of undergrad, all (successful) students took the initiative to practice solving (unassigned) textbook problems. Practice may not make perfect, but it certainly breeds familiarity and confidence.

My son plays competetive hockey, and I can assure you that most players don't "get it" after one or two practice attempts. That's why we run certain drills over and over in practice. And for anyone that says "physical routines are different than mental routines" - let me throw this back in your face: "You're probably also the same person who supports learning styles theory - so you can't have it both ways!"


 * Nov 1 2010**

I watched the FAT city video that I missed during class 10. The video is about a simulation of what it is like to have a learning disability.

I must say, the video was very enlightening and compelling. I am especially inspired to check myself if I ever feel like telling any student to "look harder". A little outside of the focus of Math Methods, but very worthwhile.


 * Nov 8 2010**

We watched a video that illustrated several teaching strategies - particularly in the context of math classes. A few of the things that appealed to me were:


 * Using a motion detector DAQ to re-create a position-time graph (kinesthetic activity)
 * Using peg-board to plot points and see slope (finally a non-silly use of Geoboards!)
 * Attachment of topic to career choices
 * Layered topics rather than isolated units -eg use fractions throughout all topics
 * Brain teasers
 * Pair outloud concept reviews
 * Use movement (eg coming up to get a handout or manipulative) to break up tedium
 * Students re-teach each other in groups or a coach-player as part of the lesson structure
 * Individual whiteboard challenge problem hold-up and share (correct) answers with rest of class (if can't solve problem, then just write a happy message)
 * "Learning Log" Re-Write procedure and then follow it (or get partner to try to follow it) to check for missing or incorrect steps

Of these, I am particularly enthusiastic about the Learning Log. Here's my reason: In practicum, I was teaching the grade 7 unit on adding and subtracting integers. As we approached the mid-unit test, it was apparent that many students still didn't really "get it". So I planned and delivered a lesson where I explained a simple and fool-proof procedure. All the students seemed to be on board with this, and yet, when marking the test, it was clear that many reverted back to their prior misconceptions and did not take advantage of the simple procedure. This actually made me feel quite depressed and defeated, as I wondered if it was even possible to help some students learn achieve.

Now, having seen the Learning Log strategy, I really believe that it could have made all the difference. The students needed more guided practice on communicating and executing the procedure! Seeing the procedure work, and hearing it from me several times was not enough! They needed an exercise to internallize it. I will definitely put this tool at the top of my box.


 * Nov 15 2010**

Today we presented our lesson plans for peer feedback. In developing the lesson plan with Vicki, I continued to build on the (scary) feeling that much of the math curriculum has little application other than to lay foundations for learning and careers in technical fields such as Science or Engineering. I was especially disappointed by what I found in the current grade 9 textbook. I don't recall any of my highschool textbooks attempting to provide "real-life application" problems in the exercises. In the contemporary textbook, however, it seems that such an attempt has been made, and I'm not impressed. The examples in the book go beyond "contrived" and enter into the realm of "silliness". For example, here is one such application question: //"The lengths of the sides of historic Fort York can be represented by the expressions x, x+171, and x+156..."// Sure, the lengths //can be represented// by such expressions, but there is no reason to //ever do so//! It seems to me that if this is the only type of application type question that can be presented, then it would be better to not do it, but rather just present basic canned, non-application questions. Beyond the silliness, I think that this type of application question can actually do some harm to the student's ability to develop a solid mathematical and scientific mindset. The reason is that much of Science and Engineering relies on developing, and understanding the limitations of, mathematical models of physical situtations. I find that quite a few scientists and engineers have a poor grasp of the fact that they are frequently using //models// in their computations, not God's-own formulae. If we start presenting nonsensical models to young students, then I fear that this will only further cloud their understanding of the //modelling process// as they progress in their education.


 * Nov 24 2010**

We watched another Richard Lavoie video. In this one, he presented a folklore tale about fixing a cracked polished stone. The analogy is that we will be disappointed if we try to fix or hide the flaws in students (particularly special needs students). Rather we should accept the flaw and try to work with it and "make it beautiful". A very worthwhile video, and one that I'll remember. Really, though, both Richard Lavoie videos should be made part of the syllabus of the Core Methods course.

We also discussed the general state of our math education system, with the precept that if so few students are successful, then we must be teaching it ineffectively. A notion was also raised that "we are on the cusp" - the idea being that we have the opportunity to either change our approach or else condemn the system to failure. I'm not sure that I really agree with either of these notions. It is easy to agree that the way that math is traditionally taught is ineffective for a large portion of students. I also agree that there are strategies (engagement, context, etc) that could be implemented to increase the effect of our teaching on a wider population of students. The more I think about it, however, the more I suspect that the purpose of math education is NOT to educate the masses. Rather, there is a small population of students who benefit from traditional instruction, and these are the same students who will actually do anything with their math knowledge. The rest of the students aren't going to do anything with their (lack of) math knowledge anyway, so maybe it doesn't matter what their benefit is (or is not). As to being on a "cusp", I think maybe it is now the same as it ever was. Again, perhaps the real purpose of math education is to educate those who naturally 'get it'; to allow them to choose, if they so desire, to pursue further education that requires a foundation of math. By this token, I wonder if there is a danger in modifying our approach to better suit the masses: could this possibly adversely affect the quality of education of those select few who would have otherwise been successful with a traditional approach? I haven't made up my mind on this, but I have an uneasy feeling that this may indeed be the case. As a teacher, I want to reach everyone, and to have everyone achieve success - but this may not be possible - in which case we need to decide who should be the beneficiary of our efforts.