Mathematics is the way in which numbers fit together. It is a way of counting things and solving problems. Mathematics allows you to figure out any given problem relating to numbers and how they relate to each other.
I think that I learn math the best by first having a concept explained to me. Then seeing an example problem that relates to the concept being learned. Then being given a similar problem and being asked to solve it to see if I have learned the concept being taught. If I don't quite have the answer to the problem right, I then ask questions of someone who knows (an instructor for instance) who can point out to me the part of the concept (equation, whatever) I forgot to apply while doing a sample problem of the initial concept explained. Then trying another similar problem and repeating the process.
I think this is true because hearing a concept is not enough, you have to practice what you think you are learning so you can know if you missed anything and learn again what you may have missed the first time. You must do problems to test you otherwise you can not be sure you are learning the information.
Current practices in school mathematics classes that are being used to promote student learning are having students do problems from a book or an other source to see if they have understood what the instructor has been saying. Also mathematics is promoted in classes by having students do problems related to concepts learned which really make them think and challenges them so they are no just mindlessly plugging in numbers into given equations. By doing problems and not being spoon fed answers students can learn mathematics because it challenges them to think for themselves and not be given an answer from the instructor. Although the instructor is there in case the students get really stuck.
Some of the current practices used in mathematics classrooms that are detrimental to students learning is giving them equations for solving things and having them mindlessly plug in numbers into that equation. This is detrimental to students learning mathematics because this does not allow them to see where the equations come from just how to use them or plug numbers into a calculator. Students should have a basic understanding of where equations come from AND how to use them, not just how to use them. If you know how to use the formula a^2+b^2=c^c and know it is referring to the sides of a right triangle that is good, but it is detrimental to just plug in numbers into this equation not knowing where it comes from or what the symbols represent. So just teaching equations and not where they come from is detrimental because any one can plug in numbers into a calculator, but to know where the concept behind the calculator, but not knowing general ideas behind it is detrimental to a students learning.
I agree that thoughtful (as opposed to plug and chug) practice can be very beneficial to student learning. I like your example of the Pythagorean Theorem, and that students should know its about a right triangle. Are there other things that students should know about the Pythagorean Theorem, or other things students should think about when they use the Pythagorean Theorem?
ReplyDeleteWhen you use the term "concept," I wondered what you meant by that. It seemed to me by the way you used this word that you really meant "procedure." What do you think? Are there other things that students should be learning in their mathematics classes besides these "concepts"?
Yo I love examples too! In fact, that is how I do most of my math learning. However, I think that sometimes it is not the best way for me to learn.
ReplyDeleteAlthough examples are very helpful, I know that, in my case, I tend to use them as a crutch. I learn much better by really understanding what I am doing and trying to figure it out myself rather than plugging in different numbers. They might be the best way to teach some material but some students might use them as the easy way out.
So I agree with what you said about plugging and chugging being detrimental to students learning (because that is kind of how I use example problems:/)