There should be a distinction between what we want students to learn for the short term and skills we want them to be able to achieve for a lifetime.
If I instead I ask you to draw me a map from how to get from a new area to 124 Main Street, without ever letting you see the area or a map, that would be more difficult - you would have to walk down all the streets in the area, measure the distances and document the landmarks in order to create that map. In this scenario, there's a good chance you will not only remember our appointment, but you will also probably remember the area in so much detail that you could describe it to any visitor who needs directions.
The first task takes a moment to remember. The second task would take you time to learn and develop the skills to complete the task but it will be time spent developing the way your mind thinks, reasons, makes comparisons, and then communicates what it has learned.
In school if a student is asked to remember facts to take a test, the chances are that when the test is over the student will have trouble retaining the information he/she just "learned"because the purpose of the work has already been fulfilled. In contrast, if a teacher spends time to take that student through a learning process, the test will be just a mark in the road where the student was able to show the teacher what was learned so far. The first is temporary learning, the other, hopefully not.
The architect Frank Lloyd Wright was a master of proportion. When he designed houses he knew the basics of his craft but his intuition helped him know how to DESIGN STRUCTURES OF BEAUTY. It was this sense of artistry that set him apart from other architects.
A great teacher knows the same. A great teacher knows the difference between teaching the basics, as if directly from a textbook, and bringing the student's mind alive by engaging it in learning.
Today I had the pleasure of observing a teacher who is dedicated to doing the latter - to bridging the gap between what students need as "skills" and what they need to become deep and meaningful thinkers. In essence, he's creating an unstructured, inspired learning environment in what is traditionally a very structured math classroom.
Remember back in school when you had to work on those pesky equations and convert them into lines on graph paper? Something like 3x + 2 = y? If I'm bringing back bad memories, I'm sorry, but maybe that's why this is important to read. When I was in math class I had to graph what seemed like hundreds of equations and I never knew why I was doing them ... or their purpose...until many years later. (Bryan Jossart, where were you when I needed you?)
In Lasse Eronen's analytic geometry class he was using a different method with his 15-year old students - in essence, he was making his students "walk the neighborhood, measure the distances and document the landmarks" - and then find out how they were all related so they could understand the concept.
Students were able choose what they wanted to work on - he called it a "buffet" of choices - and figure out how they related to one another - without any prior instruction other than knowing 3x + 2 = y! Some examples of what they could choose from were how an equation relates to a line on a graph (by drawing or by using a graphing calculator), relating a word problem to an equation, relating an equation to another type of equation, or relating a graph to another graph.
Lasse understands the work of learning to think is hard, and he understands that 75 minutes is a long time to work on these problems without a break. He lets the students leave the classroom when they want to take a break, and many go to play billiards for five or so minutes in the common area and return to do more work when they are ready. Lasse noticed a great benefit in this freedom, especially for the students who are less motivated. He said (paraphrased), "For students who sit in the class and only do two problems in a whole class period - two problems that shouldn't take very much time - I have learned that if I let them take breaks they will structure their time better. They may complete five problems, then take a five minute break at the billiard table, and come back and do five more problems. Overall, they're more productive."
I asked Lasse if this laborious thinking process was helping his students achieve what he wanted them to learn and he assured me that most of the students were already showing a deep understanding of the concepts.
When I asked him if he had to get permission to teach in this manner - he grinned.
"This freedom is the beauty of Finland. When the students are in my classroom, they are my responsibility."
- Lasse Eronen -
Background for math enthusiasts: Two years ago he taught these same students about equations like 3x + 2 = y. He may even have taught them how to take a word problem and form an equation from that word problem, but that was a long time ago. The students were never taught anything like y = 3x + 2 or x/y = 4.)
Sample problems from this lesson: