From the Principles and Standards (The Learning Principle)
"A major goal of school mathematics programs is to create autonomous learners, and learning with understanding supports this goal. Students learn more and learn better when they can take control of their learning by defining their goals and monitoring their progress. When challenged with appropriately chosen tasks, students become confident in their ability to tackle difficult problems, eager to figure things out on their own, flexible in exploring mathematical ideas and trying alternative solution paths, and willing to persevere. Effective learners recognize the importance of reflecting on their thinking and learning from their mistakes. Students should view the difficulty of complex mathematical investigations as a worthwhile challenge rather than as an excuse to give up. Even when a mathematical task is difficult, it can be engaging and rewarding. When students work hard to solve a difficult problem or to understand a complex idea, they experience a very special feeling of accomplishment, which in turn leads to a willingness to continue and extend their engagement with mathematics."
Please reflect on the above and your experience - perhaps within mathematics, more likely in your other life, of something you have learned or some problem you have solved. Place that learning in the context of the statements above.
For example, I personally might reflect on how I learned how ro sail.
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I have come upon many situations in which learning something new is very hard and frustrating. I can still remember my 10th grade year of high school taking Ms. Yarbourgh's geometry class. Up until that point I had not gotten lower than a B in mathematics. My first report card I got a C and was very upset. I was upset with the fact that I had trouble leanring proofs and the concept itself did not make sense. I was determined to learn how to write a proof even it was the last thing I would do. Everyday when I came home from school I would write a proof on my minature chalk board and my mom would check it. I would ask the teacher for more proofs, take them home, and work them out on my little chalkboard. Finally it was time to take my midterm exam. I had never been so proud of myself than the day I got my midterm exam back with a big A on it. After receiving such a good grade I was confident that I could write a proof for any problem.
This is a perfect example of self learning and being able to learn from your mistakes. Many times the teacher can teach, the parents can teach, but until a child teaches themselves will the light truly come on and ignite a autonomous learner. My experience taught me that if something is not going the way I want it to then I have to make the change. I have to be willing to become frustrated in order to truly learn. The easiest things learn are usually the easiest things forgotten.
This is very true, especially for my GATE students. Very rarely will I use direct instruction with my gifted students. Instead, I will assign activities in which they are required to explore the information, manipulate the data, and arrive at a solution on their own. The MathThematics textbook used in the GATE math curriculum allows and requires students to think above and beyond a typical math problem or situation. Each lesson is full of questions that spiral from the previous lessons. Many standards or skills are taught at a single time rather than focusing on only one concept. Allowing the students to work in groups or pairs and arrive at the solutions on their own does instill a sense of accomplishment and presents the challenge that they crave and need.
After tutoring a high school student over the summer the thought of using this method would concern me. The student I tutored for ACT math did nothing to help himself prepare for the test except get a tutor. He knew his future education was in his own hands but still he expected me to do all the leg work to help him prepare. This is just an example of how students today rely too much on teachers to help them get through their education. If all the control was put into the students hands I am uncertain of how well they would fair.
Ah - proofs!
Question - do proofs make sense now?
Good response!
So - to Kelly - and all - how can we get them to take responsibility for their own learning? No answer - it might be worth a brainstorm. But we do have to try!
Yesterday I was helping my 9 year old daughter with her math homework and she told me that the teacher did not tell them how to do the problems. She was working on place value, so she had addition problems with large numbers that had a lot of zeros at the end. I proceeded to tell her that she learned how to add in the 1st grade so her teacher did not have to teach that to her again because it was her responsibility to know how to add. Students have to realize that teachers are responsible for presenting information to them and guiding them on how to do the work, but they cannot force them to learn it. The student has to take that initiative for themselves and put in the time and effort that is necessary to master the skills.
The phrase that sticks out to me in this passage is "When challenged with appropriately chosen tasks, students become confident in their ability..." When reading this part of the passage I can reflect on my time of taking Calculus II and III. I thought what a great phrase that sums up my time in those classes. I can distinctly remember my Cal II teacher finishing up her syllabus and saying "oh and by the way, no calculators." What!! How could she? Cal II is definitely a course where one needs to work hard and bring things learned from the past like trig, alg, cal I, etc. I thought it was a great course because it brought up and used so much of what we previously learned along with new stuff. I remember spending hours doing and redoing homework assignments. I found other books and did more problems. Of course the math was difficult but it became easier once you understand what is going on and it just makes sense. There's nothing like doing one of those problems where you need 2-3 pages to solve it, one of those problems where it takes you on all these different paths, and you look at the back of the book and you get it. Not only the right answer but you understand it and it makes sense. The same way with Cal III, which was just a nice culmination to it all. And with no calculators :)
I enrolled in university with the intent to major in graphic design. As I entered the program I learned a lot about both myself and the “grown up” world. As I progressed in the program I learned that often I had to forgo my own interests in place of the interest and vision of the hypothetical client. Soon my interest faded, and I became a creature of rote technical procedure. I felt that my personal goals were not being addressed. After switching majors a few times and finally settling on statistics and psychology, I continued art, but on my terms. I discovered a new understanding of myself, my work, and the development of my technique. Instead of following another’s rules and aesthetic tastes I gained the courage to pursue my own voice and to develop self discipline in achieving goals that I set for myself. In reference to mathematics education, I believe individual motivation is very important. When a student has an area of special interest, and when they are allowed to explore the possibilities in which math can be applied to that area (in a structured environment), learning with understanding is sure to follow.
I think the hardest thing I had to learn was how to be an effective manager. When one thinks of management they think of running the daily operations of a business. However, it turns out the most important aspect of management is managing people. How to deal effectively with different types of people and different personalities. Also, learning how to put people in position to be successful. I believe the most important aspect of management is putting people in a position to be successful. Learning to manage people is a skill that has to be learned through experience.
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