In Stephen and Marion’s article “Problem Posing in
Mathematics Education”, five topics are mentioned: (a) some math questions are
posed abstractly so that they do not make sense to students; (b) besides solving
a question, there may be some other educational works that students might
engage in, i.e., create questions using newly learnt math concepts; (c) a
question with detailed descriptions may be reasoned similar to other questions,
which means these questions reflect different sides of a coin. Students may
benefit from exploring these questions; (d) based on part (c), students may
learn more if they can construct new problems using the given conditions. This
may broaden their understandings of the knowledge; (e) knowledge is constructed
on social context, and so should be the math problems.
What we are doing as math teachers is to compress math
thoughts of hundreds of years on students’ brains so that they can pass exams. The
object here is so clear that solving the question is much more important than
experience the process of learning. For those who can understand math concepts
easily, they more or less follow this thinking process, and we believe that
they can relate math concepts to their living experience. However, for those
who cannot do math problems well, can teachers really “teach” the living
experience they need? Even if we can teach the experience, are students willing
to learn? Or they prefer to learn the “fast, accurate, and efficient” ways that
can help them pass exams?
I spent hours writing the above paragraph, and I realize
the situation here may be not because of the way we teach, but the object we
teach and students learn. If the issue here is that math teaching and learning
are exam-driven, students will no doubt focus more on methods of solving
problems instead of methodologies used in mathematics (correct me if I use inappropriate
words). The math education in Finland may be an example of showing how math can
be taught in an exam-free educational system.
Hi Shan! :)
回复删除Yes I totally agree with your last paragraph. The way the education system is set up in North America does not support the idea of learning for the sake of learning. It is a system set up to compare knowledge of one child to another or one school to another or one state to another and so on. I have just recently learned that Finland has an exam-free education system, and I would like to know more about how their education system operates.
Shan, you make some interesting comments about the "experience of the process of learning" in your second paragraph. I think part of the problem is that students learn at different rates. I know that the research done at the Khan academy shows that students who struggle at first can sometimes flourish later on. This only works if they are given enough time to develop slowly at the beginning. Otherwise they fall behind, which in a subject like math can be a real difficulty, since one topic often builds on the previous topic. Here's a must see talk by the founder of the Khan academy (at end Bill Gates says "You may have just seen the future of education"):
回复删除http://youtu.be/gM95HHI4gLk
With regards to the Finnish system … they do have many nice features, such as excellent pay for high school teachers. :) They still lag behind the asian countries in terms of PISA math scores. They only 12th overall and 5th among western countries on PISA math results (2012):
http://en.wikipedia.org/wiki/PISA_2012
The first 7 spots are all asian countries.
Change your webpage to English!
回复删除In the first paragraph you mention the idea of students creating their own word problems. I think this is an unexplored area of mathematics education that has a lot of potential. It is a chance for students to use their creativity, as well as confirm that they understand how problems are set up.
回复删除I'm currently involved in a project at UBC that involves students creating their own physics questions:
http://blogs.ubc.ca/phys101/
It is amazing the creative problems students come up with. It's also a window into their thinking.