Purposeful Ramblings Through My Mind

I'm sharing this lesson in part so that those of you on supply lists can load it onto your MP3 player, throw a set of speakers into your car, and teach kindergarten or grade one music at the drop of a hat. If you find yourself in a classroom with youtube access, you don't even need the MP3 player.

Learning goal: students will associate various elements of music (tempo, timbre, pitch, rhythm in particular) with various animals according to the animal's characteristics. They will move creatively to the music, pretending to be the animals in question.

(You can find expectations to match this in the full day kindergarten curriculum. I suspect you can probably find expectations in any kindergarten program. Bonus points if you can get video of the kids moving, to leave for their regular teacher so she can use the lesson for evaluative purposes.)

Materials: A variety of music tracks or video tracks referencing animals, and the technology to play them.

Set: Who can tell me about hippos? Are they big or little? Are they slow or fast? (This is almost certainly a misconception - hippos are actually very fast. I'd let that go for this lesson.) As you play the song "Mud, Mud, Glorious Mud," ask the students to tell you what it is about the song that reminds them of hippos.

Play the song once. Get a few answers to the question: there are low notes, the music goes fairly slow, lots of talk about mud. Play it again, and this time invite the students to move like hippos. If they're having trouble, get right in there and move ponderously around the carpet.

Then ask the students about bumblebees: how do bumblebees move? Would you imagine high notes or low notes for bumblebees? Quick notes or slow notes? Tell them the next song is about a bumblebee, and when they're done listening, you'd like them to be able to tell you why it sounds like a bumblebee. Then play "Flight of the Bumblebee." (I used a version recorded by Perlman on the violin; this piece has been recorded on everything from a piccolo flute to a tuba, so make sure you choose one to start with that is played on a violin, otherwise you'll confuse the heck out of the kids.) Watch for the kids who move like bumblebees, buzzing and flapping and darting or running in place. It's only a little more than a minute long, so let it go to the end.)

Get answers about why it sounds like a bumblebee. If you're in the regular classroom and have a place for such things, this is a good time to make an anchor chart with pictures and words: a hippo with the words "slow, low", and a bumblebee with the words "high, fast."

Lesson part two, probably during a second class period:

Remind the students of the previous work and the anchor chart. Tell them that this time, they're going to listen to the music without knowing what animal it's about. They get to move to the music and then guess what the animal might be. You can give them some examples: if the music is slow and low, they're going to move one way, and if the music is jumpy, they can jump, and if the music is calm and flowing, they can glide smoothly.

I used several pieces from the Carnival of the Animals, by Camille Saint-Saens. Some of the recordings had a poem about the animal at the beginning, so I set it up ahead of time to skip that part. I played the kangaroo first, and the kids quickly realized it was jumpy music with some calm parts. When asked what they thought it was, I got one kid who said it was a cat, because sometimes cats prowl and sometimes they jump on stuff; that's a level four answer. Another kid guessed a bunny, and another a kangaroo. After that I played the elephant one, which is played on double basses; they got that one quickly, too. Then I played the swan one, and they had more trouble with gliding movements; I got a lot of ballet twirls from the girls for that one, but the answers were about fish and birds, because it sounded like the animal was gliding calmly.

Wrap-up: Students can contribute to the anchor chart about elements of music, and talk about how music can represent movement in different ways. Since the point of the lesson is to explore movement rather than language, it's up to you how much you want them to talk about what they did or learned. It might be valuable to get them to draw their perception of one of the pieces of music, for an art/music connection; perhaps use the Aquarium from Carnival of the Animals for a drawing connection.

It was an awesomely fun lesson to teach and I got a lot of good information out of it.

I was teaching "My Favourite Things" today (I'm starting a series of songs in waltz time, having taught 4/4 time successfully most of the fall) and that led to a Sound of Music-themed earworm.

On the way home, while singing, "How do you solve a problem like Maria?" I realized something: the way the nuns describe Maria in that song is a perfect description of someone with ADHD. The nuns who are driven wild by her behaviour are those who see order as key, and her disturbance of their order a major crime; those who see her as sweet, kind, gentle, and fanciful see character as based first in how she treats others, rather than how she behaves, and they advocate for the good in her.

The result, for me, will be humming that song every time I'm trying to deal with an ADHD child who just will. not. sit. still. They won't get the reference and it will help me deal.

This morning's offering: a brief discussion of This article. Exciting, I know.

The article concludes that drilling for some knowledge, some of the time, in a way that isn't boring, is necessary to academic achievement. I agree. However, I'm going to go one step further and say exactly when material should be drilled, and a little bit of how and how much.

First, when: drill should only take place after the students thoroughly understand how a process works. Drilling is then useful to fix the specifics in their heads. For example, I will never drill multiplication for a kid who can't line up a rectangle made of blocks into an array that shows a given multiplication fact, or draw that same rectangle on graph paper, or group objects in a set number of groups with a set number of pieces to show how multiplication works. While they can probably learn the facts by rote even if they don't understand them, they won't know how or when to use them and they won't be able to manipulate them - for example, they will struggle with reversing the multiplication fact to get a division fact.

This is one of the biggest mistakes teachers make in math: if a student doesn't grasp a concept on the teacher's timetable, the teacher pushes ahead anyway, saying something like, "Just learn it." If they don't have the conceptual framework in place to learn it, then they won't, and years later some other teacher is going to discover they don't know this, and that they don't know any of the things that naturally flow from it, either. When a student truly doesn't get it, the best thing to do is put more and more things in place to help them see the connection you're trying to paint for them, until the light goes on. After they get it, THEN you drill them.

Second, how: for heaven's sake make it FUN. I don't care how you make it fun. Games between two or three students are a great way to do it. Flashy computer games to drill those facts are wonderful if you have access to computers in your classroom. Some kids like flash cards. Most kids like the feeling of being tested briefly on something they know, and getting a reward for it - a quick oral test of one multiplication table gets them a sticker on a chart and they're puffed up with pride. And there's nothing wrong with that. The two keys are that it doesn't feel like heavy work, and once they know a certain set of facts, they stop practising those facts. Giving students work to do that they already know is just as soul-killing as giving them work to do that is way above what they know.

Third, how much: it should be less than 20% of a math program. The bulk of math instruction should be problem-solving, analyzing strategies for problem-solving, and extending the problems. Drill fills in the gaps in this program. It does not replace it, ever, even for low-functioning students, because the studies show that teachers tend to underestimate the abilities of those who came to them with a label of "level 1" student already attached. So we give all students the opportunity to problem-solve, adjusting the numbers or number of steps in the problem rather than eliminating the problem itself, and follow up with drill for those students in about the same quantity as we do for those more able. I can guarantee that sometimes, those level 1 kids will surprise their teachers if they're given a chance to do so.

We've been doing multiplication in math lately.

Now, when I teach multiplication, it's embedded in a visual context that is also easy to represent in a tactile way, so as to reach as many students as possible. So we start with arrays, or area models, more commonly known as rectangles. In the question 3x5, for example, I'd get graph paper and draw a line three blocks across, then five blocks down. The kids can easily visualize that the answer is 15 because they can finish the rectangle and count the blocks inside it. If they can't conceive of it visually, they can use blocks to do the same thing.

Well, we realized while doing the quilt blocks that when you have the same number of blocks across as you do down, the shape you've made is a particular kind of rectangle - a square. We further realized that you could subdivide many squares into smaller squares and still have squares; for example, 6x6 gives you a square of 36, and that grid can be further broken down into nine blocks of four squares each or four blocks of nine squares each.

In the grade five curriculum, the only appearance of exponents is in units of measurement for area and volume; so the kids learn centimetres squared as cm^2 and centimetres cubed as cm^3, but otherwise never see an exponent. But when we were doing square numbers anyway, I figured there was no harm in showing them how people write square numbers.

Fast-forward to about two weeks ago. I was coming up with division questions for my class, focusing on those with no remainders or decimals for simplicity's sake. I gave my kids 196/7. They were to figure it out using manipulatives (blocks) or graph paper arrays (drawing the seven squares across the top and figuring out how many rows they'd have to have in order to have 196 boxes in their rectangle.) Most did this by doing 7x10, and subtracting 70 from 196, then doing that again, then figuring out how many were left over and how many times 7 would give them 56. It's a visual representation of a standard division algorithm.

One kid finished really fast, so as an extension, I asked him: can you use the array you drew to figure out if 196 is a square number?

He cut out his rectangle, cut it in half, put the two halves next to each other, and realized he now had a 14x14 array, so he answered yes, and explained it to the class.

Then I gave them 256/8; how can I tell that 256 is a square number? What is it the square of?

Well, we did a bunch of them, playing with the arrays as the kids developed their understanding of friendly numbers to include square numbers. (Friendly numbers are those that are easy to work with, like multiples of 10 or 25. They're numbers you round to and calculate with when you don't want to do it the long way.) Eventually, we figured out that 7^2x3^2=21^2.

Today, I introduced them to the idea of a mathematical proof. I told them, "We figured out on Friday that seven squared times three squared equalled twenty-one squared. What I want to know is, does that pattern hold true all the time? If I pick two random numbers and multiply their squares, will my answer be equal to the product of those two numbers, squared? I want you to do a bunch of examples and show that this is true in every single one of them." Meanwhile, I asked three of my brightest kids - including the original 14x14 kid - to try the same thing, only with cubes instead of squares.

They did it. A few of them needed to check their multiplication on calculators, but they did it, and they understood it.

Then they asked me why it was important, so I showed them the basics of scientific notation. They grasped that really fast, and the idea that you could express seventy million as 7x10^7 was intriguing to them.

One of them asked if this was still grade five math. I got out my curriculum document and went on a hunt, because, while I was sure we were well beyond grade five math, I wasn't sure how far; it's been five years since I taught grade seven. I discovered that scientific notation isn't an expectation until grade eight, and facility with exponents isn't an expectation until grade seven. So I told them that if I gave them a quiz on this material, and they all did fairly well at it, I could give ALL of them an A - because they're working beyond grade level on this topic. Furthermore, if they didn't quite get it, that was absolutely fine - because I knew that every single person I'd asked to do that problem was fairly adept at grade five multiplication, so I could still give them a B. (There was another group working on multiplication facts, because they've managed to miss them up to this point. If I can get them up to speed on their facts, they may still get a B, but they're almost out of the running for an A because we're running out of time.)

I've come to a few conclusions with this.

First, if you're determined to ask big questions with big connections in math, you're not going to stick to your curriculum.

Second, sticking to the curriculum is boring.

Third, artificial divisions of concepts aren't worth the paper they're printed on. If my kids are able to handle it, and interested in doing it, and coming across the ideas on their own but needing help expanding their understanding of it, it would be bad teaching for me to hold them back because it's not in the curriculum. The curriculum is a guideline, not a Bible. Nobody is going to care if I cover too much math, anymore than they've ever cared if I didn't cover quite all of it. (I'm in no danger of that. I still have fractions and probability to do, but they won't take me four months and I'll be through the curriculum in plenty of time to do some grade six or seven stuff.)

Next up: x^2 times 2^2=36. Solve for x. (The grade five curriculum doesn't call for variables in expressions requiring multiplication either. I don't care.)

When I started teaching, I taught the way I had been taught.

In fact, regardless of the quality of teaching in a faculty of education, most teachers start out this way, because that first year of teaching is a trial by fire and when under stress, people fall back on what they know. So, having come through a school system where grades were used to determine who was smart and who wasn't, where the only way to prove your knowledge was to write about it, where marks were taken off for poor spelling or less-than-perfect handwriting or not underlining the title in red with a ruler, I taught that way, too.

The paradigm was deeper than those things, which were just a surface expression of it. The paradigm said that school needed to teach the basics to everyone as a baseline for the middle class, but it also said that some people would never achieve in school because they were not smart enough. It said that basics had to be taught before enrichment, and that enrichment - as the name implies, which is why I no longer use that term very much - was for the top students, the ones who proved they could do it. If you hadn't mastered the basics, you were doomed to read books and regurgitate their information, practise handwriting and multiplication, and get more and more bored of the whole thing until you eventually dropped out of school, thankful that it was finally over with. The only ways to master the basics were through rote learning, called "drill and kill" by its detractors. You had to learn either orally or through reading or writing - but orally didn't mean talking, it meant listening or repeating by rote. The whole class was given the same things to learn and it was the students' job to keep up with the teacher.

It worked for a lot of people. It worked for me. In fact, I was probably one of the kids for whom it worked best, because I had a good memory for lecture-style learning and I loved to read and write as my primary modes of learning. There were a lot of other people for whom it worked sufficiently, like my brother, who managed to come out of school with a massive chip on his shoulder from the methods used on him that didn't work, but also with a good education to take with him to university. (I'll talk more about this second group further down.) It worked all right for a lot of people in the middle - people who assumed that B's and C's were fine, they were just that kind of student, and they could get what they needed out of education so long as they were allowed to drop math (or some other subject they were having trouble with - math was the most common example but not the only one.) Many of this group did just fine in university later on - they'd learned to play the game of school and they'd learned enough about how to learn to make up for the deficits in their actual knowledge and problem-solving skills. In fact, having to work hard and work through boredom usually worked to their advantage later on. (Again, more about them later.)

Then there were the ones for whom it didn't work. Some of these people would have been invisible to me as a student. It wasn't until the early 1980's that Ontario law even required that this group all be in school, so some of them wouldn't have been part of my primary experience until much later - middle school, IIRC. Some of them didn't appear different, and I know I didn't really notice most of them when I was a student, but they were there. There were the kids labeled as stupid because their learning disabilities were such that pencil-and-paper, drill learning was the worst possible way to get them to learn. There were the ones who spent most of the time in the principal's office for not doing the homework, or for acting out in class, or whatever it was. For whatever reason, an estimated 20% of kids did not receive enough education to be considered literate when they left school. A further 10-15% were able to function at literacy tasks, so long as they weren't asked for anything too strenuous. The system's answer? Of course some students aren't going to do well. It's normal, and we can't give them intelligence if they have none. They'll find perfectly good jobs as workers and they'll be fine. (More on this later, too.)

The paradigm shift started just before I entered teacher's college. Most of it hadn't made its way into the faculty of education at that time; it was still new enough that the research was being done but the application wasn't. It was being applied in Australia, consistently, via a program called First Steps - an early developmental model of learning that has had profound effects on the research over the last two decades, though Australian teachers don't use its specific formulae anymore. The National Council for Teachers of Mathematics had come out with a document some years earlier that espoused the new paradigm - and it was met with such stiff resistance by everyone involved that it's still being slammed nearly twenty years later.

In Canada, textbooks were being written that would work well with the new methods, but since the new methods were not yet being taught to teachers with any consistency, many teachers didn't know how to use them to best effect (if such a thing were even possible - it's actually really hard to use a textbook as a primary resource in the new paradigm, but those textbooks were a good starting point to do so.) It took several years for the paradigm shift to make it as far as the revised Ontario Curriculum, which started to be released shortly after the current Liberal government took power in 2003. (This paradigm was in the documents that the Harris government released starting in 1997, but it was not fully-developed in them and that government didn't support the teacher learning that would have been necessary to ensure full implementation.) I've looked at similar curriculum documents for three provinces and half a dozen states, and all of those that have been revised within the last ten to fifteen years contain varying levels of the new paradigm. (Interestingly, the further south one went, the less of that paradigm was evident in the curriculum for the public school system.)

The new paradigm states that all students are capable of learning, given high expectations and high support. It grew out of psychological research begun by Piaget and continued in theories of multiple intelligences. (One of the great tragedies of the former paradigm was that it looked to Skinner's behavioural model instead of Piaget's developmental model for its research base.) The key points in this model are:
1) Students learn in many different ways. It's the teacher's job to find out how his students are learning and design lessons that will activate as many different ways of learning as possible, so that the maximum number of students will be able to access the learning.

2) Students bring a great deal of knowledge and experience of the world into the classroom with them. The corollary here is that all new learning is built on previous learning (this is called "constructivism" and it's the central tenet of this educational paradigm.) The teacher's job is to find out what the student already knows and help them build on it, again by accessing as many modes of learning as possible.

3) All students can and need to learn to think critically and present their thinking to others in a variety of ways. Critical thinking and problem-solving are not add-ons for after kids have mastered the basics; they're vehicles through which the basics can be taught. Bloom's taxonomy has pride of place in this model, because the upper levels - especially synthesis - are where student should spend the overwhelming majority of their time.

4) When students aren't succeeding, the correct response is to increase the support through group work and/or individual help, paired with resources that meet the student at their level. It is not to dumb things down, go back to low-level questions, or limit the student to drill and practice. Those things perpetuate the problem instead of solving it.

Now, I'll be the first to admit this is a dramatic shift in focus. It changes the entire purpose of public education, which has traditionally been to prepare the middle class to be worker drones in an industrial society. The new purpose of education is to maintain the middle class while also preparing students for an information-based economy. In other words, it's not good enough now that nearly 50% of our students will graduate unable to function in an information-based economy. Our society will collapse in on itself in a few generations (if it's not already) if we continue to teach in ways that encouraged that rate of failure. At its best, this paradigm should allow students to acquire basic skills while pursuing the topics that are of interest to them, developing a deep level of reflective learning to support future learning - starting from the very beginning. (Yes, I do ask my three-year-old reflective questions about what she's reading, starting with, "Why do you think x happened?" And yes, she answers them in a three-year-old way. I'm not expecting rocket science, but I am definitely expecting that she will learn her ideas have value and should be expressed and used as the basis for new knowledge.)

It's not surprising, though, that the people for whom the old system worked just fine are up in arms about it. They don't see the need for change, because it worked just fine for them and would probably work just fine for their children. (If their kids turn out to be LD, they often change their tune on this.) Their worldview when it comes to human intelligence and psychology leads them to believe that basics first, enrichment second, works better than a problem-solving approach. They're building on their previous knowledge, which isn't broad enough to support the need for change. They didn't see the kids for whom the system failed, or they did but felt it was acceptable, or inevitable, for them to fail. Or they blamed the failing students for their failure rather than the system. Or some combination of the above - I've known them to switch back and forth on these points, apparently not realizing that some of them are contradictory.

The second and third groups - the people for whom the old system worked all right - may see the need for change. They may also have a lot of trouble with certain elements of the new model, especially the bit where advanced students get more independent work while less-advanced students get more individual attention. They're right to have concerns about this, because it's one of the stickiest areas of the new model. When they express those concerns, they often fall back on what they know again, which is streaming into advanced classes for those who are capable of handling it. Most of this should be unnecessary if the new model is being implemented fully - only the top and bottom 3-5% should need more than the classroom teacher can provide. But this group is also the group with stable jobs, the group most likely to vote, and the group most likely to write to their political representatives or their newspaper. They're the group most likely to show up for parent-teacher interviews or to take concerns to the principal. And they often don't realize that they're operating under a different paradigm from their children's teachers. So when they ask for something that would have been forthcoming under the old system and find that it's no longer available, they get upset. It's a lack of communication on the part of the school and school board, but it's a serious one because it leads to parents thinking the new paradigm isn't working.

The last big problem: paradigm shifts do not happen quickly. They take a bare minimum of fifteen years, according to some business theory experts. So far, in education, we're at 12 years and counting in Ontario, and we're not there yet. My student teacher last year had a placement in a grade two classroom before she came to me. That teacher had her class sitting in desks arranged in rows, doing worksheets and then being tested on their contents. Sound familiar? That's the old paradigm in action. My student teacher was totally flabbergasted to realize that the number of fill-in-the-blanks worksheets I gave in a year could be counted on one hand; that kids sat in groups not because it made better use of space, but because I paused lessons every three or four minutes to get the kids to discuss an idea or problem amongst themselves; that anytime she suggested a drill-style activity, I was going to veto it and suggest ways to add higher-order thinking into it.

The reason I am so well-versed in this paradigm is quite simple: my school used to be one of the ones failing under the old paradigm. The Ministry and Board of Education decided to pour money and training into our school and others like it, to make them models of the new way of teaching. They did this right across the province, with the result that perhaps 20-30%% of Ontario teachers have now been immersed in the new model for several years running, and have seen its results. Teachers' colleges are actually teaching it now, though they still have trouble finding mentor teachers who know these methods well enough to mentor all the new teachers. (My school was approached by three different teachers' colleges for this fall, and I've got two who want me to take a student teacher this year. I know of another school where teachers have taken on three or more student teachers EACH per year, so great is the need for teachers who understand these methods and apply them well.)

I used to teach the old way. I did not simply accept everything I was told by a faculty of education. I am not a parrot. I worked through the old paradigm, and it did not work for what I needed it to work for - educating my students. Gradually, I switched to the new paradigm, adding pieces, discussing, reading, arguing about pieces I felt were wrong, and eventually coming to the place I'm at now. I can look back at the route that brought me here and know beyond a shadow of a doubt that I'm serving my students far better than I ever did before; that I'm serving my students better than any of my own teachers ever did; and that the shift of paradigms must continue, because it works. I look at the road ahead of me and know, again beyond a shadow of a doubt, that my career will be spent teaching within this paradigm and teaching other teachers to implement it; that my master's degree will investigate this trend and suggest ways of speeding up the implementation process; and that at the end of it, I may not win accolades in the profession at large, but I will have contributed to society at large on a much broader scale than a classroom teacher gets to do.

That's not arrogance, as some have alleged. It's professional expertise, and it was hard-won.

The question always comes up in debates about teaching mathematics: how come the Japanese do so well at math, when they stress rote memorization, if memorization is the wrong way to go about teaching math? (It's usually said in a tone of smug satisfaction, as though there's no possible comeback other than to admit that a constructivist approach to mathematics must clearly be wrong because the Japanese aren't doing it.)

The answer: a little from column A, and a little from column B.

The Japanese do indeed expect kids to memorize a lot of facts, but that doesn't mean they teach by modeling procedures and then assigning students practice questions to follow those procedures. No, they present a problem which the students do not yet have the skills to solve, then the kids work in small groups to figure out the problem. At the end of the lesson, the teacher summarizes what was learned, and students are assigned a small number of questions to practise on at home.

In other words, they use EXACTLY the method we're being told to use - with the exception that they do not allow calculator use.

The single biggest difference between North America and Japan is the number of school days - Japanese students have far more. The next biggest difference is in average expectations. When American moms are asked what mark is acceptable in math, they generally say a B or a C. Japanese moms expect an A.

So - high expectations, a constructivist and problem-solving approach to mathematics, high support in the form of parental help and extra tutoring - that's nine-tenths of the items we're expected to include in our mathematics programs.

Oh, and I should point out that the rumours about this kind of math instruction ignoring basic computation skills are false. We do drill math facts; we just make sure to drill them AFTER students have achieved comprehension, rather than before or instead of.

For future reference: http://www.gphillymath.org/ExempPaper/TeacherPresent/Mastrull/SMastrull.pdf

I haven't been posting all my discussion posts here, but here's another one for those who care.

( Doing Math )

My assignment: First Journal entry
Think About It
Take a few minutes and jot down your thoughts in your Journal. Do you think we teach children mathematics or do you think they learn mathematics? What images do you think of when you think of teachers teaching? What images do you see when children are learning? Is it possible that learning can happen on its own or from a peer, with the minimum involvement of a teacher?

Teachers play an important role in the process we call learning. They become the "guide on the side" not the "sage on the stage." Showing and telling are not teaching. Rather, we hope that children can be actively engaged in learning, questioning, analyzing, predicting and constructing knowledge from meaningful contexts and real-world experiences.

My answer:
I don't think this should be an either/or question; that is, I don't believe there is a great divide between teaching and learning. They are two parts of the same whole. Facilitating learning is called teaching, and it can take many forms. When I think of teaching, I generally picture direct instruction, the teacher modelling and then the students practising with guidance. Then I think about the activities I have students do that look nothing like that - jigsaw activities, where my whole role is to write questions on chart paper and let groups come to their own answers through exploration; whole-class discussions, where my role is to moderate who speaks and when, and possibly ask leading questions if the discussion stalls; critical literacy and exploration activities, where I provide materials and guide students to see them in light of certain key questions. I'm not doing much modelling or direct instruction in any of those situations - and together, they make up well over half of the classroom activities I plan.

I don't think the traditional teaching methods - imparting knowledge to students who lack it, in a top-down model - is really as traditional as recent research would have us believe. It seems to me that the method we're being told to consider, including the one hinted at in this journal entry with that very leading question, is just a variation on the age-old Socratic method, where we teach by asking questions that lead students towards more, deeper questions, and the knowledge they require to ask the next level of questions. Indeed, in the Socratic method, the students ask at least as many questions of each other as the teacher asks of them, resulting ideally in a depth of discussion that is totally lacking in so-called traditional educative methods. (I call it a variation because the Socratic method is mostly a thought exercise, with the students studiously avoiding getting their hands dirty, whereas the modern version of it requires kids to get up to their elbows in manipulatives of all kinds.) Most learning has been done like this since the beginning of time; it was the people we think of as traditional teachers who changed it, bringing Skinner's behavioural model of teaching into the forefront of pedagogy.

So, to bring this journal back around to its point, students learn mathematics in a variety of ways, including direct teaching, exploratory learning, peer interaction, and observation of the world around them. My job as a mathematics teacher is to highlight the connections between mathematical concepts, to ask the questions that will lead to students deepening their understanding of those concepts and their connections, and to facilitate their exploration of their world through the language of mathematics.

I've discovered that Twitter is actually a good tool for summarizing notes, because it forces me to be very succinct; however it's not a good tool for storing those notes, even with tags attached, so I'm transferring them here starting now. ( Cut for those who don't care to read teaching summaries; also, cross-posted to Ontario_teacher. )