Funtions from Kindergarten through sixth grade

Willoughby, S, S. (February 1997. Funtions from kindergarten through Sixth Grade. Teaching in Middle School, .pp. 197-201.

My article talked about the way students can interpret funtions as inputs into a machine and getting something out of it and how to solve problems such what do I have to put into a machine that when I add 7 to it and then multiply that quantity by 6, that I end up with a given specified number. It went through thte various grades and how each grade thought about it a little differently and how the thinking evolved as the grade level increased.

I think this is important to relate things like funtions to applications that might be easier for the kisd to comprehend and to understand. A funtion is like a machine that when you give it an input it will give yuou a specified output. This is useful for children in teaching them what a funtion is.

How Does Your Doughnut Measure up?

I read the Article entitled How Does Your Doughnut Measure up

Bibliography

Paula Maida Michael Maida, Natrional Council of Teaching Mathematics: Mathematics Teaching in middle school. Vol 11. NO. 5 How does your Doughnut Measure up? December 2005/January 2006.

The main ideas of this article is to communicate that math has real life practical applications to every day things. They showed this by going into a class room and having the kids calcule the area, volume and surface area of a doughnut. They asked the students to first find the diameter of the inner and outer circles of the doughnut measuring the height and so forth. Then they asked about accuracy whether there calculations were an over or under estimate of the actual doghnut dimensions. this gave the kids a practicul real life aplication of math with every day things such as doughnuts.

I think it is very important for math to have real life practical applications for sudents and especailly young students so that is is not just a boring subject that they are being forced to learn. It is important to make math interesting for the learner. The children in this class room were able to learn a lot about math while they were having fun with douhnuts. This way math becomes more real to them and not this abstract idea and concept they have to put up with. Math you can eat! Thats my kind of math! (reading this article actually made me hungry for a douhnut!)

Students Learning without Algorithms

In the article some advantages to teaching math without telling the students the proceedures or even the right answers is that they will thik things out for themselves. This is an advantage because the students can become good problem solvers and learn how to fish (do math) and not be handed a fish (given the answers.) As in the case of the article where the students were asked to divide 1/(1/3).

A disadvantage to not telling the students the right answers and the procedures is that if they are getting the right answers for the wrong reasons this can de detramental to their learning because they will not know the proper way and later in their math carreers because their foundation is sandy, they will fail bif time. We can see this by the 2 different trains of thoughts the children had when asked to divide 1/(2/3).

Von Glasersfeld and Constructivism

I wish to discuss what Von Glasersfeld means when he talks about constructivism. He uses this term contruct knowlege as opposed to aquirering or gaining knowlege is to show that knowlege is something that we bulid on that we "construct." It conjures up the image that knowlege is something we must buld from the ground up and not something we can add to our arsinal of learning or take in through some other means. To construct knowlege is to go through life and based on your experiences know things the way you do. Experience plays a vital role in constructing knowlege. If your knowlge is that women are pretty, it because you have experienced what ugly to you means and you know what pretty to you means based on your experience with what it means to be ugly or in other words not beagtiful. The conditions in which we construct knowlege is when we come across something that challenges what we know by showing us a case were our knowlege does not hold, and we make changes to our knowlege and come to know something new based on the thing that caused us to question what we thought we knew. This is also the same reason Von Glaserfeld refers to knowlege as theory because a theory is something we believe and think we know to be true but there is a chance it is false just like with our knowlege. The reason knowlege is viable rather than correct is that being viable allows knowlege to adapt and evolve with our experienceses and and it is not fixed and unchanging with what is what is implied when one says knowlege is correct.

Assumng I believe in Constructivism (shich I do) I wish to also say how it would effect my role as a teacher. Hypathetically If I were a teacher and asked my students to turn there home work in at the end of class because I had a knowlege that some students came late to class and they might like me better as a teacher if I did not penalize them for being late. In so doing I have the experience that the students will then procrastinate their homework assignment and do it while I am lecturing, that would challenge my knowlege and I would adjust what I know, (Construct my knowlege) so I would ask for their home work at the beginning of class so a.) People might be on time. and B.) They will pay attention to me during class. This is a constructivism view point because I am able to change what I know, what I know bases on my experiences and what I know is viable meaning it is a theory that based on my experience with the class, I know is a wrong theory and I must make changes to make it a more correct one.

Erlwanger's Article; Benny

I think the main idea of Erlwanger's article is that the system of teaching students is broken. The evidence I have for this opinion if that the author spends a lot of time discussing how Benny understood math and how it was the wrong way to do problems but to him it made sense. Benny is a really smart kid, bases on his knowlege he was doing things correctly. However the system said he was understanding the math correctly in fact he was one of the best students in the class. This shows that the system is broken because it passes kids through who don't know what they are doing.

I think the main point in paragragh one is that people who the schools and teachers say are passing math classes and understanding math problems really arn't. This is valid today because a lot of children in today's schools just remember enough to regergitate information on the test but the forget it and do not understand what they are doing and why they are doing it. this system is broken and must be improved by better teaching methods and better more conceptual understanding based tests.

Scemp's Article: Relational and Instrumental Understanding

This blog entry is to summarize Skemp's ideas about relational and instrumental understanding. What each one is, how their related (if at all) and the pros and cons of each one. According to the article instrumental is to have an equation or concept and how to plug things into it or use it but to not know where the equation or idea came from. Relational understanding is to know not only how to use the equation but to know where the equation or idea came from and how it related to other ideas in Mathematics.

There are overlaps between these two types of understanding. Meaning that these two understandings are not mutually exclusive. Relational understanding of mathematics is a deeper form or Instrumental understanding. With instrumental understanding you can know how to use the equation. Relational understanding takes instrumental understanding a step further. It lets you know where the concept and equation comes from and why plugging in numbers into the equation works.

There are pros and cons to each type of understanding. Some pros for instrumental understanding is that it is quick to learn at first. If the problem does not deviate from the set pattern, answers can be obtained fairly simply. Some of the cons for instrumental understanding is that not knowing why an equation works or the general ideas behind it and where it comes from can create problems when presented with problems out side of the set equation. Another con with instrumental understanding is that you will have to remember every equation because you might not be able to see where each equation comes from and the a general idea can be formed to give you various individual equations.

Some pros for relational understanding are that it allows you to see where the equation comes from and why it works thus allowing a student to do problems that are outside any given equation and students can derive the equations from understanding and not have to memorize each equation for every situation like they would have to do if their understanding was only instrumental. The main con for relational understanding is that it takes more time to learn than instrumental and that some times relational understanding might not be as obvious to understand as the instrumental idea of "here is the equation, it works, use it."

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.