Purposeful Ramblings Through My Mind

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.

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 ankh_f_n_khonsu for the link.

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.

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?

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. )

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... )

But funny. Very, very funny.

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. )

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.

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.