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The Toronto Star has begun a series on this topic. The first article in the series is entitled Why Teachers Matter, and makes the case that training teachers well, paying them well, respecting them, and then letting them teach, is the best way to maximize the results of the teaching relationship. I tend to agree.

The second article strikes me as far more controversial, and I admit to arguing the same case while falling victim to it as a parent. It's called Too Many Choices and deals with school choice, and the end result of too much choice: to bring down the generic public education by removing the people most likely to insist that it be improved. I can attest to this on a personal level. French is important to me, and I teach it better than many and far better than it has traditionally been taught in Ontario, but as a parent, I didn't choose core French. Why? Because I want to be sure my kids learn French, and the advocacy required to make sure the core French system works to teach kids French just isn't there. Canadian Parents for French focuses its efforts almost exclusively on French Immersion and Extended French programs, while the core French programs that reach every kid languish in disrepair. If there were no French Immersion, would Core French be under pressure to actually teach kids to speak a second language? Probably - certainly more than it is now.

He also points out that private schools tend to perform exactly as well as public schools, and charter schools often fare worse, despite the hype associated with them in the States.

I'm looking forward to the next article; these ones were interesting and I agree far more than I disagree. Long-time readers of my blog will see many familiar themes.

x-posted to two teaching communities; sorry to those of you who see it twice.
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"It's not the school's job to cater to [insert student with a specific difficulty which they may or may not have brought upon themselves here.] It's the student's job to figure out what they have to do and do it."

The specific scenario, in this case, was a teen mom who gave birth two weeks before the end of the school year. She got up from her hospital bed, left her baby in her mother's care, and went to write a couple of tests so she could graduate.

I have a really, really big problem with this.

First, I can't imagine who the school thought they were serving by requiring this. Most women are not at their best intellectually or emotionally a few days after giving birth, so it's not hard to imagine that the young woman in question might have seen her marks suffer when she wrote those tests. That makes the assessment invalid, because it doesn't match her usual abilities. If the test is not a valid measure of her abilities, then it's not serving her needs for her to write it.

She wanted to graduate and go to college in the fall, so the argument could be made that the college needed her marks to know exactly what she could do and to decide on admissions. I'm not buying it, again for the reason of the test's lack of validity: the college was getting a skewed view of her abilities unless she managed to pull some excellent grades on that test. So an invalid test doesn't serve the purposes of the institute of higher education, either.

So whose needs were being served? The school's, of course. The flexibility required to let her graduate without the week or so's missed work required extra work on the part of the school, and a lack of (what the school would call) fairness to other students. They might have to recalculate a GPA to exclude those tests, so she wouldn't be penalized for missing them, or they might have to give her an alternate, less-stressful assessment, or they might have had to plan in advance for her to finish her schoolwork (or at least finish enough of it that she could be said to have been evaluated on the full content of the course) a bit early due to the likelihood that she'd deliver around the time of her final exams.

I don't believe that's what real fairness looks like. Real fairness evaluates students in a variety of ways, giving them lots of opportunities to show what they know and can do. Real fairness can and should look different for different students. A rigorous adherence to a marking system based on tests and GPAs is inherently unfair, not just to our new mom in the example but to every kid who has test-taking anxiety, or a learning style that makes test-taking a problem, to name a couple of possibilities.

If some bureaucracy is inevitable in a public school setting (a debate for another day) then the least schools can do is ensure that what bureaucracy they have is essential to be fair to the students.
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I followed some links on a friend's page and found a PDF of the new Core Common Standards, which are now adopted by 34 states and the District of Columbia, making them the basis for curriculum across most of the U.S. For the sake of simplicity, I'm starting with the math standards. Language standards tend to be harder to compare because they often differ by only a word or two from one year to the next.

I was hoping to find that the standards were based on the NCTM standards for primary/junior math. I confess myself somewhat disappointed in this regard. While the language all points to a constructivist method of instruction, it seems to move faster than the NCTM standards in a couple of areas, specifically multiplication and division, and much slower in others. There's no introduction of probability at all until grade seven, no statistics at all until grade six, no measurement of angles in any capacity until grade seven. These are all key mathematical understandings that can be taught effectively much earlier, and provide a context for other mathematical understandings.

Meanwhile, the standards for multiplication and division lead me to suspect that the authors of the curriculum subscribe to a back-to-basics model. The level of multiplication and division expected of grade fours is very high, and without the added context provided by a firm basis in measurement and probability, it looks like the constructivist language is nothing but lip-service. It's too fast, and the size of the numbers the students are expected to master goes past the level of abstraction that most kids in grade four are developmentally ready for.

As I go deeper into the primary grades, I'm looking for references to constructing understanding using manipulatives. I'm not finding them. There are occasional uses of the word "represent," which could mean anything from writing number sentences to elaborate models. It's so vague it might as well not be there at all. But the absence of specific expectations related to manipulatives really worries me. It strikes me that it would be very, very easy to teach to these standards using nothing but pencil-and-paper activities. Manipulatives should be an accepted part of the curriculum at all levels - yes, even high school, though obviously they'd be very different manipulatives there - and the lack of references to them is another indication of a back-to-basics philosophy. It's destined to fail because it pushes students to a level of abstraction they aren't ready for, without giving them the opportunities they need to move from concrete concepts to representations of those concepts to symbolic and then to abstract reasoning.

In short, after this very brief look at the core common standards, I'm beyond unimpressed - I'm actively concerned for the colleagues whose attempts to teach constructively are about to be undermined and for the children. They're going to get the kind of math instruction that led to a society where it's perfectly acceptable to ask someone else to calculate your portion of the tab in a restaurant because you're not very good with numbers. They're going to get that instruction on the basis of a political climate that sees knowledge in an outdated way that fits a certain political agenda, and the U.S. public education system will continue to be undermined by it as they see that, exactly as has happened before, it doesn't work.

Has anyone taken a closer look at the other subject areas?
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Gender gap persists at highest levels of math and science testing

The authors of this study point out that the achievement gap between boys and girls, when testing gifted seventh-graders, has narrowed dramatically in the last thirty years. When it was studied in the eighties, the number of boys scoring above 800 on the math SAT outnumbered the girls 30 to 1, and that gap has narrowed to about 3 to 1. That happened in the first fifteen years - that is, the 3:1 gap has been consistent since 1995.

So the authors are postulating that the persistence of this 3:1 gap indicates a difference in innate ability between boys and girls in math and scientific reasoning (where the same gap is evident.)

I'm not buying it. Here's why.

First, for every elementary school teacher who is well-versed in constructivist teaching methods as they relate to math, there are a bunch more who aren't. The NCTM (National Council for Teachers of Mathematics) put out the original version of their constructivist curriculum in 1989; I suspect if one were to break down the changes further within that thirty-year time span, it would be the years between 1990 and 1995 that would show the biggest change. But the uptake is, at best, piecemeal. Teachers still teach from textbooks, which short-circuit the problem-solving process by their very nature. Manipulatives still start to disappear from ready availability in classrooms as early as grade four. The higher one goes in math, the more likely it is that manipulatives will disappear almost entirely from the classroom, to be replaced with purely abstract problems and procedural methodology - not because those are the end goal of mathematics instruction but because that's how the teachers were taught, and when they get out of the comfort zone of their pedagogical instruction which is generally aimed at the middle of the expected outcomes for their grade level, they tend to fall back on what they know.

In short, how much of this is the fact that girls learn mathematics differently, and their learning styles for mathematics aren't being supported in their gifted classrooms? My gut instinct says that's a huge part of the reason for the gender gap, but of course I don't have the stats to back it up.

Second, their base data is of twelve-year-old gifted kids. Leaving aside the selection criteria for giftedness (which honestly I question, knowing as I do dozens of people who are very clearly gifted academically but were not identified as such in school) there's the question of socialization. Girls are still socialized away from mathematics, more subtly perhaps than they used to be and less often by teachers, but it still happens. Twelve-year-olds are at the point in their lives when they're really struggling to figure out their place in the world. How many of those gifted kids have already decided that math isn't their thing, due to a couple of years of the poor teaching I mentioned above? How many of them will be talked out of that thinking once it starts to establish itself? Or will it simply be seen as her choosing what she's best at, and hey, there are great careers in language-based subjects, too, so what does it matter if she gives up on the highest levels of math?

In short, socialization has been downplayed as a reason in this study, probably erroneously. The cultural myopia of the data selection is in my favour, here: there is no gender gap in several Asian countries when it comes to mathematics, which makes me question why there should be a 3:1 gender gap here. But the study is done entirely on American students using American tests.

Third is the issue of NCLB. It started in 2001. It short-circuits attempted improvements in instruction because so much of the testing is knowledge-based rather than based in a problem-solving model. Because the testing has such very high stakes attached to it, teachers teach to the test, meaning that improvement in instruction has been stymied in favour of getting the test scores up. You'd think that wouldn't affect gifted education, but school culture affects everything, including the kids who otherwise might not have to worry about it. If the teachers' PD is all about getting test scores up, the teacher of the gifted students effectively is getting no PD at all. His kids are going to do just fine. But he's not then getting trained in the enrichment methods which would really serve everyone much better and are absolutely essential for the highest-functioning kids.

In short, if you want to see problem-solving in students, you have to ask for problem-solving on the tests. The US as a whole is not doing that, so the level of problem-solving isn't improving.

Should I email the authors of the study and point out the problems in their methodology? :)
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The assignment has become a standard one for grade one students in Ontario: gather a collection of 100 items to bring in and explain to the class on the hundredth day of school, usually in mid-February. It filled my six-year-old with glee. She knew exactly what she wanted to bring in. I was less thrilled, because the collection she chose did not belong to her, but to me: my dice.

Nevertheless, we spent an hour one Sunday afternoon gamely dividing dice into baggies, categorizing them by number of sides, then by colours. Along the way, we explored concepts relating to the base-ten system and the absolute basics of multiplication.

My daughter has plenty of experience with dice. They’ve been a part of her life since birth. Plush dice, foam dice, and the vast array of dice used for roleplaying games by both her parents and all their friends, were her first introduction to numbers that weren’t on her fingers. For her, dice represent fun times with friends, groups of people laughing and telling stories around the dining room table, the adults who take an interest in her life even though they aren’t related to her – and math. So it was natural that when she needed a real-life collection to bring in for Hundreds Day, her first thought was dice. Dice are math as it is in her life.

There is a great deal of emphasis in mathematics education on making math real, on finding the ways to make the numbers concrete, tactile, visual. This emphasis is a vast improvement over numbers that never left the page, because it does facilitate a deeper understanding of mathematics, and that is the ultimate goal of mathematics education. (1) But my daughter’s experience with dice is evidence that it doesn’t go far enough. For her, dice are not something a teacher brings out to show how numbers work; dice are real life that we describe using numbers. The educational establishment has been getting it backwards. The goal is not to make math real. Math is already real. The goal is to teach how reality can be described using math.

Paul Lockhart, in his article, “A Mathematician’s Lament,” discusses how mathematics is the art of pure idea. (2) When we teach it procedurally, we strip from it the inherent creativity and beauty of it; but when we use it to describe our ideas, and engage students in describing progressively more complex ideas with mathematics, we find that everything is math. There’s no need to make it real because it already is. As teachers, our vision for our students should be to bring their mathematical understandings into the classroom. Where is the math in their lives? What forms of art exist in their cultures, and in the culture to which we’re introducing them, that can further their understanding of number and pattern and relationships? It is when we follow students’ mathematical understandings and extend them that we get the deep understanding of mathematical ideas that creates lifelong learners and problem-solvers.
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Edgerton Ryerson, the father of public education in Canada (and arguably in Western culture) called it the Hidden Curriculum. Back then, it consisted of neat, clean uniforms, washed hands and face, shoes, standing when spoken to, lining up, and other niceties of polite behaviour. It didn't change too much for a very long time.

No one really questioned the rules at the time, with the possible exception of the children themselves. Middle-class people believed in cleanliness and obedience for children and expected their schools to mirror those values. Lower-class people got no say in the matter. Upper-class people were sending their children to private schools that had similar rules for public behaviour, with a few additions (such as uniforms.) Since the middle class was driving the public school system and middle-class values were self-regulating their homogeneity, it was understandable and expected that middle-class values would dominate the school system. This was right and good.

We still use school to instill middle-class values in kids. We still teach them the rules of behaviour. We still expect them to speak politely and work hard and obey their teachers. But some of the other expectations we have for public education are changing, and they're creating a conflict.

Pedagogical research is clear: the deeper and higher the question, the better the learning that accompanies it. That means that a book about the Underground Railroad won't just be discussed in terms of theme and characters and history; it will also spark questions about points of view, both missing and present, the meaning of freedom, and rights. A book about the water cycle will be discussed in terms of what children can do to conserve water in their own country and to help other people around the world who don't have enough fresh water. A poster about diamond mining may lead to discussions of diamond cartels, sweat shop labour, and capitalism. (All of these examples come out of my own classroom, and I'm not alone; I'm using the materials bought for me by the school board at the behest of the Ministry of Education.)

Middle-class liberals are familiar with these topics. They come up in their perusals of the internet, they get discussed amongst friends, they spark donations and outreach in their communities, and they appear in newspaper articles aimed at this group. But other middle-class groups may not have the same values. Will a discussion about diamond mining be neutral territory and fair game in a public school in the Yukon? Will a discussion of water cycles and pollution be fair game in Sudbury or other heavily-polluted mining towns? It's extremely difficult to moderate that type of discussion in a classroom without the teacher's own views coming through; how much politics in the classroom is too much? When does enculturation - the introduction of children to their own culture - become acculturation - the introduction of children to a dominant culture not their own?

The clearest indication of this disconnect at the moment in North America is the Conservative movement to homeschool. The reasons for homeschooling are complex, but they boil down to a belief that liberals (especially secular liberals) are using the school to brainwash kids into accepting liberal values without question. The belief is that teachers' job is to teach reading, writing, and arithmetic, and maybe some history and geography. In other words, a teacher's job is to deliver facts and basic skills. According to this view, it's the parents' job to give these facts and skills a cultural context. Children should be studying culturally neutral material and learning the skills everyone agrees they need. The critical thinking, the higher-order questions, the debates - all these things are too culturally charged to be left in the hands of a group well-known for their liberal leanings.

I'm certainly not going to pooh-pooh their concerns. They're right in several ways. We have changed what we expect kids to learn in school. We have changed the behaviours we expect of them. We have changed the questions we ask and the materials we ask them of. And "liberal" is indeed a good way to describe much of it. While the intention is to get children thinking critically about what they read and view, the fact is, teachers' biases are going to come out in those discussions, and the students - beginning critical thinkers as they are - will not always realize it.

How much right do teachers have to go against parental requests in the name of better teaching?
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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.

Doing Math

Jun. 28th, 2009 09:02 pm
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I haven't been posting all my discussion posts here, but here's another one for those who care.

Doing Math )
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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.
velvetpage: (Default)
I occasionally (or not so occasionally) come across people who believe that the way they were taught math before all this problem-solving mumbo-jumbo was superior, not because it worked better at the time but because, after learning the steps without understanding them, they developed the ability to problem-solve and reason mathematically later on. Their reasoning: kids don't need to learn to reason mathematically from the outset; they'll learn to think procedurally and then as their brains develop, they'll fit their procedural knowledge into a growing problem-solving framework. The end result will be adults who reason well AND are good at arithmetic (these are usually the people who believe that New Math creates kids who can't add or multiply, to which I say: then the person who was teaching it didn't really know what they were doing.)

I used to think this way, because in math, this is what I did. School taught me the traditional algorithms, my dad taught me to add and multiply quickly in my head, and I made connections between concepts without being encouraged to. (When I first learned fractions, they were already old friends - I'd been seeing them on sheet music for two or three years at that point.)

It occurred to me last night, in conversation with [livejournal.com profile] wggthegnoll, that there is a better metaphor in my life for old math versus new math: music.

Now, I have some natural talent for music. There was never a time when I couldn't carry a tune, and some of my earliest memories are of singing interesting and difficult music at the Salvation Army, sometimes before I could read. I started taking piano lessons at the age of eight, and continued taking them until I was sixteen, at which point I started teaching piano instead. My training on the piano was classic. I was given a piece of music in a book geared to the level I was supposed to be at. I read through that music with my teacher there to point out anything I didn't get or some new technique for fingering. Then I went home and practised that piece of music until I could play it well. When I was done most of the songs in that book, I moved on to the next one. My education in music theory was mostly taken care of at music camp, and I ended up skipping Grade One Rudiments entirely (though I did do the Grade Two exam - it was necessary to get a high school credit for my piano lessons, along with my grade eight piano exam.) I learned my scales, I did fingering exercises, I knew how to play tonics and dominant sevenths and minors harmonic and melodic.

However, I was not the kind of kid who thought outside the box that was presented for me. I didn't start listening to the radio until years after my friends did, and I was fifteen and already had my grade eight exam under my belt before it occurred to me that I could play music that didn't come out of a Royal Conservatory Repertoire book. I rarely tried to figure out for myself how something was played, and other than a few hymn tunes, I had little experience with playing anything that wasn't classical. In short, I never learned to play by ear, and more importantly, nobody ever drew for me the connections I would have needed to develop that talent (which, btw, I have when I'm singing, though it's undeveloped.) I never learned about modes; no one pointed out to me what standard chord progressions were and how I could play with them to make them sound different using those arpeggios and scales I'd memorized so assiduously. Though my dad pointed out that I didn't have to play a hymn tune exactly as it was written, I could break up the notes for better rhythm, no one showed me what the chord notations were or how to read them or how they connected to each other. The advanced harmonics, which I was perfectly capable of playing, were not taught to me as something I could reason my way through.

I was taught the arithmetic of music. I was taught, "If you follow these steps, you can play any piece of music you pick up. You'll learn how they work later." I was not taught to think critically about the form of the music I was playing. All of that was saved for after I had the rudiments; it would have come about during Grade Three Harmony, which I started but never finished. It was assumed I'd pick it up on my own - but I never did.

Because of all that, I'm a very limited pianist. I know the most basic chord structures, the ones that show up in most pop music and a wide variety of hymn tunes, and I can generally figure out how to make them sound cohesive if I work at it. I can read music, though I can't sight-read very well - that is, if you put a piece of difficult music in front of me and ask me to play it, it's going to take me a week of painstaking work to get to the end of it and be able to play it back to you. (That one is a confidence issue: I don't like to keep going if I make a mistake at something, but when you're sight-reading, that's exactly what you need to do so that the person you're accompanying doesn't have to stop in confusion when you break rhythm.) I still know very little about modalities and harmonies. I still can't play by ear or even chord by ear most of the time, though I can come up with vocal harmonies with no trouble at all. I can hear in my head what it is supposed to sound like, and I'm frustrated trying to bridge the gap between how it should sound and how I'm able to make it sound.

I'm going to bring this up with my uncle when he starts teaching Elizabeth in the fall. I've taught the way I was taught, and passed on these same errors out of ignorance, but I'd prefer that my daugthers' talent for music ends up better developed than my own, so I'm going to talk to him. I want to figure out how to apply New Math strategies to elementary music, and I won't take the platitudes - they'll figure it out on their own later, don't worry about it, one step at a time, basics before critical thinking - at face value. While it may work for some kids, it didn't work for me. I don't settle for, "They'll pick it up later" when I'm teaching math. Why should I settle for that when teaching music?
velvetpage: (oxford comma)
This is from Piet's journal, still screened over there because he's getting a better night's sleep than I am, on the education thread from a few days ago. I don't think [livejournal.com profile] professormass will mind me reposting his comment, and I'm pretty sure [livejournal.com profile] oakthorne won't mind being referenced in it, either. I'm leaving it unlocked because those gentlemen aren't on my friends list and have a right to see this. And I'm posting it here because, until Piet unscreens the comment, I can't answer it over there. :)

First, [livejournal.com profile] professormass's comment:

Something occurs to me (and I apologize for butting into the conversation -- as you know, velvetpage, I'm keenly interested in education):

The sweeping generalizations and the arguments against making those generalizations are missing a key point — the education system must address generalization, because it's trying to work for the mythical "average student," casting a net that catches as many kids as it reasonably can. The exceptions will always and must always be the issue. No bureaucratic system can account for the wide variety of learning styles present in the complexity of human nature.

People oakthorne and myself are exceptions. So, yes, much of pyat and velvetpage's arguments hold water, with the percentage of the population who aren't exceptions.

I think that the biggest point of difference I'd have with them is what percentage of the population represents exceptions to things like "
A middle-class person who doesn't get that education might be able to keep their middle-class status with a job that doesn't require it", where "requiring it" is a highly subjective thing, in most cases. My field, for example, routinely requires anywhere for 6-12 years of degrees, diplomas and certifications; I have none, and still operate at an executive level.

A friend of mine, a schoolteacher, told me that he thought the percentage of exception was something like 1%. I think it's more like 25%.

Modern school systems have almost always served the needs of the majority. When pyat says "it's getting better," I read, "it's serving a broader swath of the majority."

There will always be exceptions to the rule. After having done much research, I'd tend to say that public education has succeeded in catching a slightly broader swath than when I was trapped in the system. I don't think it will ever catch all the exceptions.

So, really, the question is: what to do with the exceptions? What safety net can be cast for people like oakthorne and myself? Can one be cast?

Now my reply:

Arguably, Piet and I are exceptions, too. As I believe Piet stated somewhere else, he was identified “gifted” but nearly flunked out several times, getting by with barely-passing grades. I was at the opposite end – I excelled with so little effort that I spent much of my class time in elementary schools with a novel open under my desk, because I was bored silly. And yet we managed to make system work for us, in our own ways.

That said, you’re right – the education system works best for the people who test out as average and slightly above-average in intelligence. It generally works all right for those slightly below-average, because they’re able to access extra help that is sent their way, and it often works just fine for those at the top of the intelligence scale because they learn to play the system. But for all the special placements, resource help, gifted classes, and what have you, that the school boards put in place to cast that wider net, there will always be those who don’t quite fit it. Most of those will benefit by taking everything they can out of the education system and then going their own way. But the fact that it doesn’t work for them doesn’t diminish the value of education overall; it only speaks to the need to address individual needs as broadly as possible, or as you say, to cast a broader net.

In terms of the number of kids with a diagnosed exceptionality (at the top or bottom – this number includes gifted) you and your friend are both wrong: it’s between ten and fifteen percent, statistically. But the school board makes concerted attempts to catch most of those within their net.

I believe my school, and for that matter a fairly large chunk of the schools in Ontario (not all, yet, but we’re moving that way) are doing a better job of this than ever before. I now routinely teach to four or five different levels in my classroom at a time. I have smart kids who are feeling challenged and rewarded, and I have low-average and below-average kids who are learning as fast as their brains will let them, and the kids in the middle aren’t being forgotten, either. I have a learning-disabled gifted kid (neither of those are official diagnoses, the first because his parents don’t want him labeled and the second because the LD got in the way of the intelligence testing when we did it) who is enjoying school for the first time in his life. I’m teaching him to game the system – how to get what he needs from it as he goes on to grade six, what it’s important to do, what can be ignored – because there’s no reason this kid can’t succeed at the highest levels and get the kind of career you only get through education. (He wants to be a lawyer.)

Part of the reason he’s going to make it is that nobody’s telling him that school isn’t important, or that many people can succeed without it, or that the system is out for its own benefit. Those things are true some of the time, but they’re not helpful overall. They’re excuses for people who did not succeed within the education system. Some of the time, those who didn’t succeed within the system manage to succeed outside of it, as you and oakthorne have done. More often, that is not the case.

And here we get to the crux of the matter. I’m quite willing to admit that school doesn’t work for everyone, and that some people succeed just fine without it. What I’m NOT willing to admit, and indeed will argue against with all the force at my command:

1) that this is the rule for most, even for most of those we would classify as exceptions;
2) that the existence of holes in the net in any way diminishes the value of the educational system;
3) that teaching the conclusion we’ve been arguing against (that advancing your education through traditional channels is a worthless endeavour) is going to help the people who take the lesson to heart;
4) that in fact, people who hear that lesson and learn it well stand an equal chance of succeeding at their various endeavours in life, as measured by the level of control they have or can access over their own workplace and community, as those who remain within the educational system.

Please note that I think it’s possible to enter the education system at differing points and still succeed within it. A child who is homeschooled until high school often ends up doing better when they finally do access formal education, in large part (I believe) because their parents took a very active interest in their education and made sure that they understood the value of an education – traditional or otherwise. But sooner or later, most people who succeed at the highest level they’re capable of, do it by making use of some facet of the education system.

So, to answer your question: I think the net that is catching more and more kids still has something of value to offer to the exceptions, especially the exceptions at the top of the spectrum. The clearest evidence of this is the fact that three exceptions have now come forward in this journal or another, to argue their case. They've done it with varying degrees of rhetoric, but they've all done it with a good grasp of written English. Put simply, they're attacking the educational system with tools that they accessed through an educational system. It didn't fail them as far as they say it did. (Yes, I include you, [livejournal.com profile] professormass, in that assessment.) The higher up one goes in education, the smaller the net the system is attempting to cast, for exactly the reasons you've stated - not everyone needs it. For an adult, the choice to work within the system or circumvent it (or simply ignore its impact) is a choice. There should be (and are) mechanisms in place to help those who wish to access that system but are having difficulty doing so. But the onus is on the student to take what they need from the system - not on the system to offer whatever the student needs. The focus shifts further and further away from the responsibility of the system and more and more towards the responsibility of the student. The student who either fails at that responsibility, or decides not to take it on, needs to take a hard look at where the problem was. Many of them cut their own hole in the net.
velvetpage: (teacher)
Also, thanks to [livejournal.com profile] anidada who suggested I get this book. :)

"Notice that this argument for the abolition of traditional grades isn't based on the observation that some kids won't get A's and, as a result, will have their feelings hurt. Rather, it is based on the observation that almost all kids will come to accept that the point of going to school is to get A's and, as a result, their learning will be hurt."

So, what happens to a "good kid" like me, who internalizes that second message?

First, one of the things that made me a good kid was that I almost never experienced a lack of understanding; I could do pretty much anything I needed to do with minimal effort. So I was mostly spared the anxiety of a fear of failure. Even so, it reared its ugly head a few times in my school career. There was my father's jocular habit of looking at a grade of 95% and remarking, "Not bad - but it leaves room for improvement." There was the internal, quickly-suppressed panic when I didn't understand something instantly. I suppressed it because it was imperative that NOBODY KNOW I didn't understand. I knew I was capable of pulling the wool over everyone's eyes and making them think I understood just fine, until I figured it out. There were my siblings' efforts to be different from me, often by underachieving so as not to compete.

Even so, when it came to school work, I put in minimal effort to get the grades I wanted. Grades were everything in school, from - as best I can remember - about grade three. That was partly because my grades at the end of grade two were among the lowest I ever experienced in my school career. My parents' and teachers' disappointment in the grades - not in the learning they represented, because I actually knew everything they were teaching, notably in phonics in which I got a C - was like a slap in the face. I knew even then that I'd been shafted. The teacher hadn't marked how well I understood the phonics - I was reading "chapter books" fluently by then. She'd marked the answers in the workbook, which, because I was bored and slightly depressed from a recent move to a different province, were incomplete. Not wrong - INCOMPLETE. What did I learn? I learned that what you knew in school didn't matter, unless you answered the questions correctly, no matter how boring you found them.

So I coloured my title pages, underlined my titles in red pen with a ruler, whipped through spelling exercises without ever paying much attention to them - I was a natural speller with a good grasp of phonics, so this didn't matter - and spent my daydreaming time with a pencil in hand, writing stories. My teachers loved that, and the rote learning was so boring that I had plenty of time for cooking up stories, often while my pencil was busy with the "real work." The stories always got A's.

Time passed, and I began to read more outside of school than in it. This is where the really interesting part comes in, in terms of pedagogy. I was a reader by nature and nurture. While I didn't read earlier than kindergarten, I did read better, faster, than most of my peers. When I learned, it was with a sudden light bulb rather than a slow progression. The important part is that I didn't see reading as a school activity. I saw it as something people do for fun, because that's how my parents saw it and how they encouraged me to see it. The things I saw as school activities were still done well, mostly because I was too much of a pleaser ever to risk the displeasure of the adults around me by doing a half-assed job. But most of my learning didn't come from those activities. I can't remember more than a handful of activities I was actually assigned in school, nor more than a few things I was told to read. The books I remember, the books I learned from, the books that informed my worldview and gave me the historical background colour on which to pin real historical understanding - ALL of that came from reading that I did for the love of reading.

Which makes me wonder: what happens to the kids who don't grow up with the knowledge that reading is something people do for fun, and who don't have the benefit of reading early and well? I can answer this, this time from looking at my students: they see reading as an in-school activity, and they see it as something on which they will be graded. The "good kids," that is, the pleasers, will do fairly well at it in the context of school work - but they'll stick to reading material with no meat to it, whenever I let them get away with it. They'll refuse to challenge themselves, and they won't think about what they're reading unless I can get them to forget about the grade. The not-so-good kids, that is, the ones who don't read early or well, will start to give up by the end of grade one, sometimes sooner. They'll have their failure to read on the school's timetable reinforced as a failure of their ability, and each time they fail at a reading task (in their own estimation) a failure becomes more likely the next time.

I continued in much the same pattern through high school and even into university. Coursework was a slog that I had to get through - often even if the topic was fascinating. I managed not to read half the books I was supposed to read, because I knew I could pass the tests by regurgitating what the professor had told me, especially since my own ideas were unwelcome and got lower marks on those tests. If the prof only wanted his own ideas given back to him, why would I read the book and risk getting some of my own? That way lay frustration - so I avoided the frustration by eliminating the learning in favour of getting the grade.

The courses where my own ideas were welcome got far more of my effort and taught me far more lasting lessons. Still, much of what I've learned about subjects that interested me came because I read about them on my own. The more I think about it, the more I realize that my success in school is more an indictment of the system than an advertisement for it. I succeeded at school while learning as little as possible in it.

If the goal of school is to educate, then we need to consider: are our methods of assessing students undermining that goal? Are kids learning what we want them to learn, or are they learning to please teachers and parents while avoiding practically all valuable thinking? Are we setting them up for failure by grading their successes?

And if so, how do we fix it, in such a way that ALL students come out educated?

June 2017

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