Finnish Schools and Montessori Education

The BBC has a fascinating article on the Finnish educational system; specifically, why it consistently ranks among the best in the world despite the lack of standardized testing. A couple things stand out to me as a Montessori educator.

The first is the use of peer-teaching. There’s a broad mix of abilities in each class, and more talented students in a particular subject area help teach the ones having more difficulty. It’s something I’ve found to be powerful tool. The advanced students improve their own learning by having to teach — it’s axiomatic that you never learn anything really well until you have to teach it to someone else. The struggling students benefit, in turn, from the opportunity to get explanations from peers using a much more familiar figurative language than a teacher, which can make a great difference. I give what I think are great math lessons and individual instruction, but when students have trouble they go first to one of their peers who has a reputation for excelling at math. In addition to the aforementioned advantages, this also frees me up to work on other things.

A second thing that stands out from the BBC article is how the immense flexibility the teachers have in designing their teaching around the basic curriculum coincides with a very progressive curriculum. This seems an intimate consequence of the lack of assessment tests; teachers don’t have to focus on teaching to the test and don’t face the same moral dilemmas. Also, this allows teachers to apply their individual strengths much more in the classroom, making them more interested and excited about what they’re teaching.

E.D. Kain has an excellent post on the video The Finland Phenomenon that deals with the issue specifically. It’s full of frustration at the false choices offered by the test-driven U.S. system.

(links via The Dish)

Celebrity Charities?

“Very few sports stars, other than Lance Armstrong, actually donate to their own charities,” says a tax adviser. “Most of them say, ‘My fans will donate.’ Their attitude is ‘I’m contributing my celebrity to this cause.’”

— Vanessa Grigoriadis (2011): Our Lady of Malawi in the New York Magazine.

Andrew Sullivan extracts the crucial point in Vanessa Grigoriadis’ article on the failures of many celebrity charities.

As we work on social action this cycle, it’s important to consider why we’re contributing, and what it takes to make a meaningful contribution.

Solving Quadratics

Solving quadratic equations requires finding the factors, which is not nearly as easy as multiplying out the factors to get the unfactored equation.

Instead, you have to do a bit of trial and error, to figure out which pairs of numbers multiply to give the constant in the equation and then add together to give the coefficient of the x term.

Factoring quadratic equations.

It gets easier with practice. Or you could use the quadratic formula, where if the equation were:

a x^2 + b x + c = 0

The solutions would be found with:

x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}

So in the equation used in the diagram:
x^2 + 7 x + 10 = 0

you get:

  • a = 1
  • b = 7
  • c = 10

Putting these values into the quadratic equation gives:

x=\frac{-7 \pm \sqrt {7^2-4(1)(10)}}{2(1)}

which simplifies to:

x=\frac{-7 \pm \sqrt {49-40}}{2}

x=\frac{-7 \pm \sqrt {9}}{2}

x=\frac{-7 \pm 3}{2}

With the whole plus-or-minus thing (\pm), this last equation gives two solutions:

(1): x=\frac{-7 + 3}{2} = -5

and,

(2): x=\frac{-7 - 3}{2} = -2

Now, you may have noticed that the solutions are negative, but when the equation is factored in the illustration, the result is:

(x + 5) (x + 2) = 0

The difference is that, although we’ve factored the equation, we have not solved it. When I say solve the equation, I mean find the values of x that would result in the left hand side of the equation being equal to the right, which is zero. Since multiplying anything by zero will give you zero, and the two factors multiply each other, the left-hand-side of the equation will equal zero when either one of the two factors equals zero.

So:


(x + 5) = 0
x = – 5

and:


(x + 2) = 0
x = -2

Finally, we can plot the line:

y = x^2 + 7 x + 10

using the Graphing Calculator Pro app, or this somewhat crude Parabola-Line Excel Graphing Worksheet, to show that the line crosses the x axis at -5 and -2.:

Note the curve crosses the x axis at -5 and -2.

$25 computer

Here’s a real computer, the Raspberry Pi, for only $25. It has only two ports, one for a monitor and another for a keyboard. I’d suggest it needs one more USB port so you could hook it up to external devices (like robots), if you can’t split the single USB.

Its intention is to bring computer hardware and programming into schools. I’d love to get hold of one.

(Articles from BBC and geek.com).

Illustrating the Multiplication of Quadratic Factors

Each term needs to multiply the other two terms in the opposite parenthesis, so start with the reds, then do the two mixed colors, then the blues, and finally, combine the mixed colors.

I’ve been playing around with ways of showing how to multiply out quadratic factors like the one above. I’m still not perfectly happy with these animations but they’re the best I’ve come up with so far. A smoother, Flash or svg animation might work better though.

In this second version the terms being multiplied are highlighted. I like how the highlighting gives some more stability to the animation, but I’m always leery of putting too much color or bells and whistles because they tend to complicate the picture. In this case at least, I think all the colors have meaning and are useful.

Multiplication of quadratic factors.

Kindness and Community

Just as each species in a biological community contributes something that helps sustain the community, people need to contribute to each other in their communities to to keep them stable, productive, and happy.

We’ve been talking about social action this cycle. Students have been finding and reading articles, and thinking about what they could do — themselves right now — to promote social justice. The articles have come from a number of different places: local stories from the Memphis newspaper, the Commercial Appeal; national articles from the New York Times; and even international things from the from BBC. Now, for Personal World, they’re thinking at the really small scale, about what they do for their classroom community.

The objective is twofold. First, I want them to contribute more to each other, and think about what they’re contributing, to maintain a healthy community. A little self reflection should help them realize if what they think they’re doing for others is actually helping. But, secondly, I also want them to recognize the efforts of their peers for what they are: attempts, even if futile or misguided, to be helpful.

It’s sometimes easier to think about doing charitable things for people far away, because it’s impersonal. There’s little risk of being embarrassed. But even the smallest groups need some altruism to grease the wheels of community.