Author: Lensyl Urbano

From novices to experts

Socrates teaching (from Wikimedia Commons).

The primary role of an instructor is to transform a novice into an expert within a given subject area. – Cooper (1990)

The above quote comes from a paper on instructional design by Graham Cooper. I don’t quite agree with it entirely since it does not seem to allow for a well rounded view of a student as an individual, or the Socratic ideal, but it does seem applicable to the more strictly academic areas in the middle school curriculum.

In order to figure out what distinguishes experts from novices, cognitive scientists have spent a lot of time observing the two groups. Their key finding has been that as you become an expert on a topic, you construct mental pictures (or schemes) of the shapes of problems, so when you encounter a new problem you can just fit the new problem to the mental pictures you have and see which best fits. It’s a bit like learning rules of thumb that apply to different situations. When a problem comes up, the expert can quickly whip out the right rule of thumb from their mental back pocket while the novice, though equally smart, needs to figure out all the steps with some degree of trial and error.

This is a nice perspective when it comes to teaching something like solving equations, but I think one important distinction of the Montessori philosophy is the belief that adolescents should also be learning flexibility, and be capable of dealing with novel problems. Because adolescence is when students are just becoming able to think abstractly (at least according to Paiget), and abstract thinking needs to be practiced, it is necessary that students encounter novel, challenging problems on a regular basis.

Pangea breakup in reverse (adapted from image in Wikimedia Commons).

A lot of creative and problem solving thinking comes from hashing out new problems. In a globalized world, where technology is capable of dealing with routine tasks, be they constructing a car or solving a series of equations, creative problem solving is becoming a more and more valuable skill. Especially now that “The World is Flat“.

Math map

When introducing the x,y coordinate system using a world map it is most effective if you use the right type of map projection. Most maps use a Mercator projection which stretches thing out as you get closer to the poles. You can see this in how the latitude lines get further apart as you get closer to higher latitudes and Antarctica and Greenland get all stretched out.

Mercator map from Wikimedia Commons.

The Mercator map is great for navigation, because if you measure angles on the map (bearings) between two places, you get the right bearing to sail your ship.

The better map to use for demonstrating the coordinate system would have an equirectangular projection, where all of the lines of latitude and longitude are equally spaced. These are harder to find, so I’m posting here a background map with the latitude and longitude marked on it (in 10 degree increments).

Equirectangular world map. Click on the image for a larger version. This is image is free to use if you cite this website (see cc-by-nc).

Notice the difference in sizes between the maps and the size of Greenland.

I’m also posting here, because it might be useful, a simple linear graph in Excel (or for OpenOffice) that simply plots lines if you enter the equation in slope-intercept form. There are lots of websites that let you do some really cool things (I’ll post about them later), but sometimes the simplest is what you want.

The importance of thinking about thinking

Know thyself - Socrates (and others) from Wikimedia Commons.

Personal World is the time for introspection. Every day students get a chance to think about themselves and, hopefully, about how they think. Thinking about your own thinking is called meta-cognition, and there is growing scientific evidence that it is an important life skill that leads to better decision making.

According to Jean Paiget’s theory of cognitive development students only begin to develop the capacity for abstract and meta-cognitive thinking in the middle school years. However, further research has shown that only about 30-35% of high schoolers actually develop the skill (it’s called “formal thinking” or “formal operations“). The brain develops the capacity for formal thinking in early adolescence, but people do not naturally move into that stage; school and the right environment are important.

For formal operations, it appears that maturation establishes the basis, but a special environment is required for most adolescents and adults to attain this stage. – Huitt, W., & Hummel, J. (2003)

The requirement that students spend some time in introspection is one key way that Montessori adolescent programs try to promote formal thinking.

Finally, Jonah Lehrer, a journalist who writes a lot about how people think and makes decisions, points out the importance of metacognition in the qualities that make for a good president.

some studies suggest that when confronted with a complex decision – and the decisions of the president are as complex as it gets – people often do best when they rely on their gut feelings …
However, it has also become clear that listening to your instincts is just a part of making good decisions. The crucial skill, scientists are now saying, is the ability to think about your own thinking, or metacognition, as it is known. Unless people vigilantly reflect on how they are making an important decision, they won’t be able to properly use their instincts, or know when their gut should be ignored. Indeed, according to this emerging new vision of decision-making, the best predictor of good judgment isn’t intuition or experience or intelligence. Rather, it’s the willingness to engage in introspection” – (Lehrer, 2008; emphasis mine)

Other sites linking to this post

    Waves and earthquakes

    There are a lot of Earth Science applications that deal with waves. Seismic waves from earthquakes are a major one that is particularly pertinent after the recent Haitian earthquake. There are quite a number of lesson plans dealing with seismic waves at Larry Braile’s website. Most of the lessons are as practical demonstrations pdf’s and some use downloadable software (Windows only unfortunately), but there are some online applications as well.

    In terms of online resources, the IRIS network, produces nice maps of recent earthquake locations. It also has a good page with “Teachable Moments” regarding recent earthquakes. These include the above video of why the Haitian earthquake did not produce a tsunami.

    Although it’s not directly related to waves, I particularly like the thermal convection experiment on Braile’s website. It provides, with a baking dish, a sterno can, some water and some thyme, a great example of the convection in the Earth’s mantle that drive plate tectonics.

    Mind-Mapping software (Free)

    VUE mindmap

    I recently received a tip about the VUE mind-mapping software. It’s free from Tufts University and it’s open source, though you do have to register to download the software. A quick test that produced the image above shows that it’s pretty easy to use and can easily incorporate images.

    Free software. It’s better than coupons.

    Publishing: The Blue Pencil

    In searching for venues where my more literary-minded students might get a start in publishing their work, I came across The Blue Pencil. It’s student edited and produced, so it fits the Montessori Philosophy very well. My thoughts based on their website based on my own criteria for publishing literary work:

    • The Blue Pencil “is edited and produced by the students in the Writing & Publishing Program at Walnut Hill School for the Arts“. It is aimed at young writers (12–18) around the world. You submit online and your work is evaluated on an ongoing basis during the school year and they say the try to respond within a month. Unfortunately you typically do not get feedback if your work is not accepted. On the plus side, they only have a one month embargo of your work after they’ve published it before you reacquire the publishing rights.

    Students publishing their written work

    I am constantly amazed by the quality of the writing my students turn in. The honesty of the reflection, the clarity of the writing and the elegant turns of phrase. Certainly better than anything I’ve ever churned out. I sometimes find it difficult to advise them on their pieces because their work is often so good that they’ve gone past issues of mechanics and my comments are purely my subjective opinions (and I’m clearly no expert). I have some serious writers in my class, so I was not too surprised when one of them asked me the other day if there were any venues to publish their work.

    Byron (from Wikimedia Commons).

    A little online searching turned up a few places that might by worthy venues for the serious writer, however, I have a few biases that narrow down the possible candidates even more. Of course the publication needs to be serious about publishing good work. In the online world anyone can self-publish, but because of the glut of information out there it is pretty hard to find the good stuff. A good, selective editorial staff is your friend there. It certainly makes it harder to publish, but it also helps you create better pieces.

    The second key criterion is that the publisher should not retain all (or even most) of the rights to your work. Signing away the rights to something you spent a lot of time and energy creating never sat well with me when I was choosing journals for publication. I’m pleased to see that an increasing number of online scientific journals are rejecting the practice, but it remains to be seen if they will be successful in the long term.

    With that in mind, I’ll post a what I find as I find them under the tag “publishing“. At the moment, I’m relying on two key resources, the Alliance for Young Artists and Writers Blog and Publish Me a website from the University of Tennessee, Knoxville, for teens who want to publish.

    Video Game Addiction

    Boys tent to have more problems with games (From Hauge, Marney R., Gentile, Douglas A., (2003, April)).

    While the American Psychiatric Association does not yet include it as a diagnosis (as of 2009) video game and internet addictions are problems I’ve seen first hand, and, given my own plugged-in-idess, are topics I personally think about when I reflect on my own computer use. The web is a powerful tool so it’s not so unusual that we’d spend a lot of time using it. There is a point however when it becomes compulsive and takes so much time that it becomes a detriment to our other work.

    There are any number of website and online resources about the topic and even a few commercial sites that offer treatment. For anecdotal descriptions there is the Berkley Parents Network website. which has a few examples from parents dealing with the problem. The National Institute on Media and the Family has a good page describing video game addiction. For adolescents they describe the symptoms as;

    • Most of non-school hours are spent on the computer or playing video games.Falling asleep in school.
    • Not keeping up with assignments.
    • Worsening grades.
    • Not telling the truth about computer or video game use.
    • Choosing to use the computer or play video games, rather than see friends.
    • Dropping out of other social groups (clubs or sports).
    • Irritable when not playing a video game or on the computer.

    Interestingly, this tends to be more of a problem for boys. And one solution recommended is wilderness therapy. I think that may be a bit extreme. Another suggestion was:

    … the experts … said the best way to cure kids’ video game addiction is to set strict limits. They suggested not allowing kids to have computers, PlayStations, TVs, etc. in their bedrooms if monitoring their behavior is a problem.

    The Montessori classroom is a closely knit community and, especially in a small classroom, falling asleep in class and being irritable (especially more so than normal teens) can be very disruptive to the entire class. It is, therefore, essential that the problem be addressed as soon and as quickly as possible.