velvetpage: (punctuation saves lives)
I showed up about half an hour before the end of the school day. Within seconds, I was lamenting that I'd forgotten to bring a deck of cards. Ever resourceful, I pulled out the coloured tiles instead - little squares of plastic, uniform in size, in five different colours.

"Gather round while I show you something cool!" I said. I laid out one green tile and surrounded it with red ones, one for each edge, making an x shape. Then I took a few more green ones and surrounded the red ones, again so that no red edges were on the outside.

"How many red ones are there?" Four, came the answer. "How many green ones?" Nine. "Oh, those are interesting numbers. Four is two times two, and nine is three times three. Let's see what happens when we add another." We kept going, finding to their surprise that the pattern continued - 16 red ones after the next round, 25 green ones after the following, 36 greens. Then we took the two colours apart and I asked them to see if they could form them into a square of each individual colour.

In fifteen minutes of messing around with two colours of tiles, I taught a bunch of grade threes about square numbers up to 36, and they never even noticed they were learning it. I also managed to keep them from noticing that one of their classmates puked about ten feet away from where we were playing. And all of them were in the middle of massive sugar-highs at the time.
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This article suggests that attempts over the last twenty years to improve mathematics education in North America have been wrong-headed non-starters, and that people don't need as much math as the education establishment says they do. I'm going to address the second point first.

I can certainly see the case for not needing the full gamut of mathematics available to the end of high school; that's why they're called electives. But the author (who, sadly enough, is a professor emeritus of mathematics) has made a fundamental error in logic: he has taken the mathematically-illiterate society we have, where being unable to calculate a tip is seen as normal and acceptable, and claimed that people don't need math to get along in that world. He's right. The reason, however, is not that math is useless; it's that we're comfortable as a society with the failings we've built into the system to accommodate people who can't do math.

The better question is: what harm does rampant mathematical illiteracy do to our society, that could be remedied with better mathematical literacy? This is a much harder question, because it requires both a diagnosis of a problem that currently exists and a proposed solution.

From my somewhat less-exalted position, I see several ills in society that could be improved with better mathematical literacy.

The first is our societal inability to understand and use statistics. This begins with journalists and trickles down to everyone who reads what they write. When experts in various branches of science or finance or politics begin to talk about the effects of their discoveries, most people can't parse the numbers. Even those who can figure out what those numbers mean in context can't always break them down to decide for themselves if the numbers seem reasonable, which means they're unable to use their own knowledge to verify or deny the claims with any accuracy. Does this mean they don't try? No, it doesn't - it means when they do try to parse those claims, they get it wrong, and most people DON'T NOTICE. The result is an inability to separate a well-spun lie with statistics from a truth that could improve our lives.

One lie perpetuated this way is the myth about stranger danger, leading to us bubble-wrapping our kids far later than any society in the past has ever done. Politicians use these lies all the time to drum up support for this or that cause - I suspect the recent health care debate in the U.S. is part and parcel of this problem, and I'm quite certain the Tea Party movement is based in large part on the participants' inability to work the math.

The second, closely related problem is the general scientific illiteracy in society. Again, it leaves us unable to interpret scientific data with accuracy, which then leads to devaluing it. Thus NASA is defunded while the war in Iraq continues to grind on.

The third is our societal inability to deal with finances. When money is almost completely computerized, the inability to manipulate numbers and understand what's happening to our money is a significant problem for many people. This is evidenced by the willingness to overspend using credit.

Our society would be better off if every young person got to the end of grade ten with a thorough understanding of all the math concepts presented up to that point. Which brings me to my second point: that the good professor is overstating the change in mathematics education and therefore has understated its impact on understanding.

Sea changes in education take decades. We've been engaged in this particular change since the mid-eighties, so a little more than two decades. I'm in a position to know much more than he is: many teachers haven't caught on yet. They think they're doing it right when they use manipulatives, but they haven't got a firm grasp on the entire span of changes to teaching that needs to happen for the most effective math learning to take place. It's still very common for teachers to stop using manipulatives almost entirely in the junior grades. It's still very common for teachers to be unable to diagnose the fault in understanding that a child is experiencing well enough to scaffold their learning and fix the problem. When teachers are stressed, they fall back to teaching the way they were taught.

At the moment, most standardized tests in America are asking for factual knowledge of discrete facts. Teachers need to teach to this because their jobs are on the line if their kids aren't doing well. So they teach using methods that will increase discrete knowledge and skills. The research into mathematics education (and all other branches of education, for that matter) says that this is backwards; we need to be teaching to the big ideas and then engaging our students to break down those ideas to get to the little bits. This is not yet a consistent thing in North American classrooms, and part of the reason is that the test scores are being misused to blame teachers instead of supporting them. (And we're back to the first point: this is a classic example of the misuse of mathematics to make a political point, and a huge section of society has swallowed it whole, unable to break it down because they don't understand the statistics.)

Mathematics education has changed, but not enough. Not every teacher is using it. Not every teacher has the supports in place - notably the classroom supplies - to use it. Many, perhaps even most teachers still rely on textbooks that short-circuit real problem-solving in favour of giving an example and asking students to do it that way. We can't see yet what a society full of mathematically-literate people looks like; at least, not in North America.

My vision is to see how a mathematically literate generation of students would change the face of society. How would our civilization be different if more than half of our kids were able to read a series of political stats and call out the person who deliberately misrepresented them as a liar? How would our society improve if every young person determined that living off credit was a deal with the devil and they weren't going to do it? How would our society be better if a huge scientific discovery in the area of astrophysics generated a demand for more funding to pursue it further? How would our society be better if math were again seen as art, integral to and interconnected with other arts?

As in any discipline, if you ask the wrong questions, you get the wrong answers. The author of the article asked the wrong questions, because he lacks a vision of the way things could be.
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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.
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American students tend to misunderstand the meaning of the equal sign far more than international counterparts.

The article is decent, and the study says exactly what I'd expect it to say. The idea of balancing numbers on both sides of the equal sign is crucial to algebra, but most students taught procedurally understand the equal sign to mean, "This is where I put my answer."

However, the article attributes the problem to poor textbooks. While this is probably a factor, I'd call it correlation rather than causation, because over-reliance on textbooks for mathematics instruction is symptomatic of the poor teaching that leads to the misunderstanding of the equal sign. Studies have shown that teachers who teach from the textbook most of the time generally rely on the textbook to lay out their plans for them. They'll spend exactly as much time on a topic as the textbook will - even if their students don't yet understand. Any mathematical concept that the textbook is unclear on, the students will be unclear on too, because the teacher is unlikely to address it outside the framework of the textbook.

Solution: get kids as young as grade one working on addition and subtraction sentences that involve balancing equations: 4+2=9- ___, for example. For primary classrooms, have a graphic of a teeter-totter with the equal sign on the fulcrum, and make it clear that the idea is to balance the teeter-totter in the middle. Do this all the way through the primary grades with increasingly complex problems and manipulatives.

By grade five, kids are ready to be introduced to the idea of a variable to take the place of the blank; they're also ready to solve problems by making tables of values that rely on one thing being equal to another: "A spider travels 19 cm every second. How long will it take her to travel the perimeter of a room that is 3m x4m?" One logical starting point is 1 second = 19cm, and the table of values can be built up from there, provided the students know they have to keep counting the number of nineteens in order to balance the equation.

Thanks to [livejournal.com profile] ankh_f_n_khonsu for the link.
velvetpage: (pi)
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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Calling all my knitty and mathematically-inclined friends: this website is absolutely fabulous. I want to make all of them but I'd never have enough time.

Pascal's Triangle is in there. They didn't do anywhere near as ambitious a project as I was planning.
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Did anyone here replace their original edition of Settlers when the new one came out, and keep the old one around?

If you would be willing to part with an old copy, or if you upgrade at any point in the next, say, six months, I would love to have your old copy. I'd like to run a games club at school in the fall, and Settlers is great for basic probability and strategy. It can also be played start to finish in about 45 minutes once you know the game, and our lunches are 40 minutes long, so kids could conceivably play an entire game during a lunch hour if they were quick about it.

If I don't get any games this way, I'll be holding a game raffle in the fall to raise money to buy some games for the club. The list of games I'd like to teach the kids includes, but is not limited to:

Mancala
Checkers
Chess
Settlers of Catan
Carcassone
Monopoly
Payday
Cribbage and other card games

Some of these - chess, checkers, mancala, cribbage - can be bought quite cheaply, for a few bucks apiece. I could budget thirty bucks for ten to twelve kids to play at once and supply those games. The others are more expensive, with Settlers and Carcassone topping the list. I doubt I'll get much of a budget for this - I'd be surprised to get fifty bucks - so any help anyone would like to offer would be valuable.

I'll repost this in August when I'm sure I've a) got the job, and b) looked at my own supplies to see what I can come up with. Right now it's the genesis of an idea. I'm thinking six weeks per grade, starting with the older grades around the end of September. We'll see how it works.
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I'm sure there are more than a few people on my friends list who think Pascal's Triangle spells doom without adding anything to the mix, but bear with me.

My student teacher and I took a grade four activity about probabilities in coin tosses (what is the likelihood of throwing two heads on two coins? How about three heads on three coins? Two heads and one tail on three coins?) The probabilities with coin tosses just happen to fit in beautifully with Pascal's Triangle, one of the more famous (and complex) number patterns out there. The National Library of Virtual Manipulatives" has an excellent starting point a ways down that page.

Well, it occurred to me that this might make an interesting quilt pattern. Could I choose colours that would reflect certain multiples and go from there? I realized pretty quickly tht making it in the shape of a triangle was at best problematic, but I could make it more like fraction strips (which I'm sure you can find if you poke about on that website a bit) so it would be a rectangle cut into fractions to represent the numbers in the pattern.

Well, so far, so good. The problem came when I realized I didn't just want to divide the strip in half, then in quarters, then in eighths, with the numbers of the sequence in each piece. What I wanted to was to represent the numbers themselves as part of the fractions. So, the zero line is one whole strip; the first line has two ones, which can be represented as halves of the strip; but the second line is 1, 3, 3, 1. I don't want to represent that as quarters; I want to represent it as eighths, where the two threes in the middle are three times larger than the ones at the side and set apart from each other using colour or texture. The next line after that is 1, 4, 6, 4, 1; that's a total of sixteen. The ones can remain the same colour as the other ones to give the impression of a triangle as I descend the quilt, and I kind of want to bring multiples into it, so the six would have to be a similar colour scheme to the two threes in the line above, and the two fours would have to be totally different.

It then occurred to me that the possibility of pulling colour theory into this exists in primary, secondary, and tertiary colours (though it gets a bit hazy after that and may need to be accomplished with texture and multiples of those colours.) So, if one is black, I could start with red for multiples of three and blue for multiples of two. So the zero strip would be black; the first would also be black but halved using texture somehow; the second would be black, redredred, a different redredred, black; the third would be black, royal blue in a four square, purple (for 2x3) in six parts, blue again in a four square, and black.

The next row has another prime number, five, which then needs to be - yellow, I guess, to use up the primary colours? - then there are tens in there so yellow + blue = green. The next row has six again, so purple, followed by fifteen, which is 5x3 to yellow + red = orange, then twenty which brings in gray for the first time, multiplying 2x3x5, or all three primary colours. Or I could skip the primary colours for five and go straight to a tertiary colour to make it more interesting.

My head's starting to hurt but I just CAN'T STOP.

Can anyone else follow this at all?
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I think maybe I'll make this a weekly feature, just for the heck of it. It reminds me that I'm doing a good job.

So, this week in math, we started by reviewing what little we'd learned about equivalent fractions before the March break. Since exploratory learning is starting to become de rigeur for my students (as it should be) I took them to the computer lab and let them play around with the fractions manipulatives on the national library of virtual manipulatives website. The differentiation was easy. Don't get this at all? Try the basic fractions manipulative. You've already figured out the pattern rule? Great, do the grade six one.

On Monday, I gave them an assignment to do. They would get four or five pieces of regular paper in different colours, their choice. On each one, they'd write a different common fraction. The example, which I made up with my six-year-old on Sunday, was one half. Then they'd come up with three or four equivalents for it, and represent those equivalents in pictures. They had to explain the fractions and how those fractions were connected to the original fraction. There is a rubric - let me know if you want it.

The kids who were having trouble even accessing one half came for some guided math with me. We did fraction strips, and played around a bit with blocks, until they got to the point where they understood that if you divided all the pieces the same way, you had an equivalent fraction.

Then, because an abstract understanding is important to work towards, we took the equivalent fractions they'd come up with and analyzed them. How do you get from 1/2 to 4/8? If they expressed it as adding, I told them they were right, but they'd see the pattern faster if they thought of it as multiplication. So what would they multiply by? Some figured out the pattern independently; others needed to be shown several times, and finally got it with the help of more manipulatives. Then I encouraged them to use the pattern rule they'd found to come up with what they thought was an equivalent fraction. They had to come up with the proof that they were right by drawing a representation they could connect back to the original fraction.

As of this writing, most of my kids have finished two or three sheets of equivalent fractions. The rubric gives an A to kids who can connect their equivalents to decimals or percents, so I've explained the idea of repeating fractions to the kids trying that for 1/3. I suspect most will finish on time - the assignment is due on Monday.

Next Week: comparing and ordering fractions is so much more interesting when the denominators are different.
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The assignment was to try something completely new for five hours over the last few weeks, and write about where the math was to be found in this new thing, and how learning it gave you insight into your students' issues with trying new things in math.

Fortunately, I've started learning to play hand drum in that time period. )
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I don't think I posted this here before, so I'll do it now. It was one of the initial assignments for my specialist course.



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.

Resources

1. Carpenter, T.P, Hiebert, J., Fennema, E., Fuson, K.C, Wearne, D., & Murray, H. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth. Heinemann.

2. Lockhart, Paul. (2008) “A Mathematician’s Lament.” Mathematical Association of America Online. http://www.maa.org/devlin/LockhartsLament.pdf
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Attention, parents of school-age children! Are you frustrated by the fact that your child's math homework doesn't bear even a passing resemblance to the method you were so painstakingly taught? Is your child frustrated at getting one method at school and another at home?

Never fear, help is here!

Read more... )
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It appears to be primarily aimed at the homeschool crowd, but it does a fabulous job of portraying mathematical models in a way kids can grasp. I'll be using it my classroom soon.

http://www.naturalmath.com/
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But funny. Very, very funny.

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No, this isn't what I do in math class. It's the polar opposite of what I do in math class. But Tom Lehrer never fails to make me giggle, so you're getting a video! Aren't I nice?

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Attention, parents of school-age children! Are you frustrated by the fact that your child's math homework doesn't bear even a passing resemblance to the method you were so painstakingly taught? Is your child frustrated at getting one method at school and another at home?

Never fear, help is here!

New Math, the way your kid is learning it. )
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The grade five curriculum requires kids to develop a basic understanding of a variable for the first time: as an unknown number and as a series of values dependent on each other.

My kids already have the first definition. So yesterday we started on the second.

I put up on the board ___ + ___ = 17 and asked them to find all the ordered pairs that would fit in the blanks. I showed them how to write an ordered pair (0, 17) and then let them have at it in their groups. Meanwhile, on a piece of chart paper, I drew the upper right portion of a co-ordinate grid (because my kids haven't learned integers yet and I didn't have a lot of room, anyway.) When my kids were acting like they were done, I explained that a co-ordinate grid was a way of using math to draw things; it was just a format that everyone would use in much the same way, like a language, so that anyone who saw it would understand how it worked. I rewrote the equation on the board as x + y = 17, and asked one kid to come up and plot a point on the grid. She took (0, 17) and pointed to seventeen on each axis, saying that they were different points that were written the same way. She drew both points. I told the class that this was a problem; how could the same two numbers indicate two different points? How would that help anyone to figure anything out? I let them stew about it for a minute, and discuss how the points were different, until someone finally said, "But in the equation, x came first." I got that person up to the chart to explain what she meant, and we figured out that (0, 17) was a different point from (17, 0).

After that, we plotted each of their ordered pairs on the grid in turn, and I showed them how to label the points correctly. When we had about half the points up there, I got one of the girls who was complaining she didn't get it to come up. I handed her a metre stick and told her to play connect-the-dots.

I had expected the lesson to end there, but it didn't, because when my students notice something, I run with it. Half a dozen of my kids looked at the resulting triangle and said, "That's a right-angled triangle!" One more said, "It's a half-square triangle like in our quilt blocks!" We discussed the other two angles - both 45 degrees - and I asked the kids, "What do you think you'd have to do to get a triangle that wasn't a half-square triangle on this grid? What would the equation look like for that?"

Well, they tried x + y = 12, though my gifted kid told them right off the bat it was going to just make a smaller version of the same triangle. It did. When it seemed they were stumped, I took the metre stick and drew a line from (7,0) to (0,14). Then I asked them to figure out the relationship between seven and fourteen. They eventually got it, with a bit of help to get the form right: 2x=y, where x=7.

Every kid in the class understood the basics of the co-ordinate grid. Every kid as far as I could determine understood the ordered pairs and how they were graphed. A few kids couldn't wait to pull out the graph paper and try some equations on their own. They were tickled by the idea that I did this math in grade nine and it's currently in the curriculum in grade eight.

Today's lesson: Explain why x + y = a whole number is always going to produce a half-square triangle, no matter what the whole number is.
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Want a way to bring statistics home, to make them comprehensible in their magnitude?

Here it is.

I'm using a few of these for my global citizenship unit later in the year.
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My kiddo is smart. She just gave me a perfect explanation of how there were ten sets of ten in 100. We segued into her three times table, showing 3 on each die and counting the number of dice. Then by grouping the dice, we figured out that 12x2 is the same as 8x3. For the record, she's in grade one - basic multiplication is a grade three topic.

Then we put all the dice in my dice box into baggies in sets of 10, for her hundreds day project. She's supposed to take in a collection of 100 items. Later this week we'll write a short speech for her in French to explain her collection. It will include things like, "These twenty dice all have ten sides, while these fifty dice all have six sides. This bag of ten dice has six eight-sided dice, three four-sided dice, and one thirty-four sided die."

While we can't beat [livejournal.com profile] doc_mystery's and his daughter's one hundred zombies for its sheer awesome, I think 100 dice is a pretty decent hundreds-day project.
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The grade five curriculum says that students are only supposed to measure angles up to 90 degrees this year. In grade six it's 180 degrees, and by grade seven it's the whole circle.

Of all the ridiculous and counter-productive divisions of concepts they could have made, this one has to be near the top of the list.

My kids were having difficulty understanding exactly what it was they were measuring with a protractor, and understanding the units was even worse. So I drew a co-ordinate grid on an overhead (no numbers, just the lines.) They told me it was a t-chart, it was an X, it was four triangles - that's when I stopped them. Where are the triangles? So one kid came up and drew the lines that connected the four points of the grid into a square, bisected diagonally twice. We measured the angles in the middle and the angles on the outside and discovered they were ninety degrees and added up to 360 degrees.

Then we talked a bit about the Babylonians, who came up with a number system with a base of sixty. They figured out that they needed a way to divide up a circle into even fractions. Since they did things in base 60, when they divided a circle their first instinct was to divide it into six parts, each of which was 60 degrees, and the 360 degree circle was born. We talked about why you might need to measure angles, and how that was used. I gave them the vocabulary words they needed: bisect, radius, diameter. At this point I'm well into the grade seven geometry curriculum, and they're eating it up.

Then I pointed out that you can draw a triangle in a circle by going through the centre point of the circle, using the radius as sides. We measured the inside angle of that triangle and found it to be 58 degrees. With one more line, I showed them the complementary triangle: the one made with the 122 degree angle.

They spent the rest of the period exploring triangles as fractions of circles and investigating the number of degrees involved. I have three different kids who realized that the sum of the outside angles would be greater than 360 because they were measuring a radius but also a line that went through two points on the circumference. I had a fourth kid who realized that if she drew the X so that the angles weren't all 90 degrees, she could figure out the measure of all the other angles if she knew the measure of any one, by doubling it for the opposite angle, subtracting that number from 360, and then halving the result. (Convoluted but it works just as well as figuring it out using 180 for the straight angle. Efficiency will come later; right now I want comprehension and wonder.)

I'm going to get those four kids to present their findings to the class tomorrow so the rest of the class can be exposed to those ideas and play around with them themselves. So basically, I can give a B to any kid who can measure an angle less than 90 degrees accurately, and an A to those who can explain how angles are related to circles and why triangles add up to 180 degrees.

I love it when lessons become kid-directed explorations of important mathematical ideas.

June 2017

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