November 2010 – Page 4 – Montessori Muddle

Explaining algebra: concrete to abstract

Concrete to abstract, or abstract to concrete? Bottom up or top down? Introducing new concepts can be done either way, but which way is best? Students tend to learn better when they’re building on an existing scaffolding. However, some students are more adept at seeing the bigger picture first then analyzing the details, while others favor seeing the smaller details and constructing the bigger picture of the pieces of the puzzle.

In yesterday’s lesson introducing algebra, I started concretely with weights on the scale for variables and the fulcrum as the equal sign. But I worked with them abstractly. Instead of telling students I had 200 g, 100 g and 50 g weights, I said we had three types of weights and called them a, b and c.

The weights were set on the scales as above so we wrote out the equation:
$! a + c = b + b + c$
and simplified to give:
$! a + c = 2b + c$

Then we talked about balance. Whatever you do to one side you have to do to the other, so I took the c weights off and showed how it solved for a:

$! a + c -c = 2b + c - c$
$! a = 2b$

Now you can solve for b by dividing everything by 2 to get:
$! \frac{a}{2} = \frac{2b}{2}$
$! \frac{a}{2} = b$

However, since you can’t exactly divide the 200 g weight into two, you can’t exactly demonstrate this, but at least you can show the math. You can also now substitute in the values for the weights. If b = 100 g.

$! a = 2b$
$! a = 2 (100)$
$! a = 200$

At this point, students could look at the weights and see the numbers.

Concrete to abstract and back again, I like how the lesson turned out, although, today I had to go over how to show their work again.

Time

“… he was purchasing time, than which nothing is more precious to a man bent on great achievements” – from Plutarch‘s Life of Sertorious

Without grades and extrinsic rewards, students build much more durable commodities: strength of character and self-motivation. But as I try to manage a classroom there are so many things that could so easily be considered rewards. The most important of these is time.

I feel the significance of time most of all when we have to reschedule P.E. for the end of the day instead of just after lunch. We only do it then when there’s a lot of work that I want to make sure the students get done. It sits there, dangling at the end of the day, if only they’re focused enough, if only they work smoothly and efficiently enough.

It’s clear from the literature, and from my own observations when I do this, that extrinsic rewards reduce creativity and devalue both the work and the reward itself. So I suppose I may just have to say, on those days with too much to be done, that we’ll have to skip P.E. and eliminate the expectation altogether. It’s something I’d prefer not to do, but it’s unrealistic to expect students to give their full though and concentration to a subject while glancing at the clock every five minutes.

Sundial_PSF — Image by Pearson Scott Foresman from Wikimedia Commons.

First frost

greenhouse-reset — Transporting plants into the greenhouse.

The temperature dropped below zero (Celcius) for the first time last Friday night. We’d put back up the greenhouse’s plastic cover, which had blown off a couple weeks ago in a wind storm, but that was not enough to save a couple tomatoes and a squash plant.

It was a good illustration of the effects of freezing on plants not adapted to the colder weather. The leaves all turned black and flopped over, probably because the expanding ice ruptured the cell walls.

It also indicates that I need to get at temperature data-logger so I can monitor the temperature inside and outside the greenhouse. In the spring I hope to start a bunch of plants inside but put them into the greenhouse at the first opportunity, but I’ll need to make sure that greenhouse can support them. The data logger will also allow for some interesting experiments.

Saudade

Saudade:
Pronunciation: (from Forvo)
Definition: Portuguese – One of the most beautiful of all words, translatable or not, this word “refers to the feeling of longing for something or someone that you love and which is lost.” Fado music, a type of mournful singing, relates to saudade. (from Jason Wire at MatadorNetwork)

The beauty of the words in Jason Wire’s list, “20 Awesomely Untranslatable Words from Around the World” is that they express somewhat complex emotional concepts.

Last week I had to explain the English word, nostalgia. Its meaning was a little difficult to convey because, when you think about it, to feel nostalgic you need to have had a certain amount of self-reflection. Self-reflection is typically not a strong point of early adolescence, which is why we have Personal World every day.

Then I came across Jason Wire’s list, and there are some wonderful words on there, but the one that resonates right now is the Portuguese “saudade”. I like how it is subtly different from nostalgia, but I also like that there is an entire genre of music, fado, that embodies the word.

NPR has a great review of fado artist, Ana Moura:

Also, in looking up the pronunciation of the word I came across the Forvo website. It has recordings of people saying words from around the world, so you can hear the sounds of words from native speakers. I chose the one in this post, a female Portuguese voice (), because it seems to capture the poignant emotion of the word quite well.

Imagemaps with the GIMP

I’m just testing out a simple image map created with the GIMP. The GIMP is a free image manipulation software, a bit like Photoshop, not quite as sophisticated, but free. I used GimpTalk‘s very helpful guide. I though it would be easiest if I used something from a previous post as a test.

You should be able to click on the cell walls, chloroplasts, vacuole and nucleus. The links take you to the associated Wikipedia pages, but that’s just because this is a quick and dirty example. Image maps have been around for a long time, but I believe this is the first time I’ve ever created one. Now I just need to animate it a bit.

Unfortunately, this image is not easily scalable, though it should not be too hard to find (or write) a script to do just that.