Goodman, T., (2010). Shooting free throws, probability, and the golden ratio. Mathematical Teacher, 103 (7), 482-487.

The main point of this article is to show how a contextual problem can be used to provide students with many opportunities to apply the mathematics that they've learned. The author goes through and shows how shooting free throws and probability can go hand in hand. The students reasoned about the likelihood of a student making a certain amount of points if they usually could make a certain number out of so many. This exercise allowed the students an opportunity to create graphs and models that showed the data that they collected. It also helped increase the students ability to solve problems.

I agree with the author in many regards but somethings were just confusing to me. I agree that giving the students a contextual problem such as this can be really beneficial and allow students to see how math can fit into life situations. I also think that it was a good exercise for students to collect data and represent it in forms of graphs. It also has potential to help students increase their understanding of the importance of math and help them to develop skills in problem solving. I was confused by the presentation of the experiment that the author made. I found myself struggling to understand the article when the principles seemed much more simple then the way they were presented.

Johnston, C. (2008). What do bouncing tennis balls have to do with algebra. On-Math, Online Journal of School Mathematics, 6(1).

This Article describes how bouncing tennis balls directly correlate to algebra. Each student bounced a tennis ball for a designated period of time and recorded how many bounces they got. Using this information, they later were able to use all of the varied results in the class to find the mode. After unifying the numbers in the results, they discussed what was the constant in the experiment which they found was time. They discussed whether or not time should be on the x-axis or the y-axis. Once deciding that it should be the x-axis, they were able to plot their points. Through this experiment, Christopher Johnston was able to explain how just by bouncing tennis balls, students could then take a real world experiment and directly connect it to math. He shows how through this one experiment one can find slope, mean, median, mode, range, meaning to graphs, x-axis, y-axis, and so much more.

After thinking about this activity, I have come to the conclusion that this would be a very beneficial thing for students to do. At first I thought it was too much like a science experiment but when Christopher Johnston showed all of the mathematical uses to the experiment, I decided otherwise. This activity shows students how something so simple can be turned into a mathematical thing. One of the big issues today is that kids don't see why they need math and this is a fun little activity to show them that math is all around them, even in really random ways like bouncing balls. It also got the kids engaged and working together. Lastly, it encompasses so many different mathematical concepts and ones that tie into each other really well.

Blog #5

There are many advantages to teaching mathematics without using algorithms. One is that the students have to use their own logic to come up with ways how to do things and how to make sense of it. They have to go through an understanding and discovering process rather than a using process, using meaning using different rules and equations. The advantage of the teacher not ever telling if the answer is right or wrong is that the kids will continue to second guess their logic until they have come to a conclusion that it could not be any other way. This kind of teaching is also beneficial because it gets kids involved and interacting rather than just sitting and listening to a lecture.

The disadvantage to this is that kids may come up with the wrong answer, even after much work and debate on it. If the teacher never tells them it's wrong, they'll continue to do things wrong. I think that the better way to approach this kind of teaching is to sit back and watch the kids construct ideas and learn interactively but when they are sure that they have the right answer, but it is really wrong, the teacher uses questions to have them think through the spots that they went wrong. That way the students are still are able to use their own logic because no particular answer or process was given to them, but they'll be able to construct the correct processes and knowledge.

Glasersfeld meant by constructivism that we construct knowledge, it is not given to us, but rather it builds upon itself through experience and through different things that we've observed. We only learn that we are wrong when we experience a contradiction to what we had previously concluded. According to Glasersfeld, there is no truth because any thing considered "truth" is just something that has not been contradicted.

Knowing and understanding the principles behind constructivism helps me be able to better see how I need to approach things when I'm a teacher. I think it's really important to ask my students a lot of questions because I cannot assume that they have constructed the correct knowledge and the only way to find out if their thinking is correct is to ask them questions. I think it also helps me to realize that one thing needs to build upon the other through a variety of experiences because according to a constructivist perspective all knowledge is gained through experience.

Relational and Instrumental Understanding

Thinking and learning processes occur in two main different ways known as relational and instrumental understanding. Where one is easier to learn and to teach than the other, the other proves to be the most important thing that needs to be the focus of every teacher, especially in mathematics.

Instrumental thinking, in short, is knowing how to get from point A to point B. It gives a step by step process that can be followed. This is the easier one to teach because it doesn't explain why, just how. Students also typically prefer this method because it's easier for them to remember and it's not as difficult to understand but this is what limits them because they don't learn how to see how things are connected and how they can be transposed into different situations.

Relational thinking is knowing how to do something and why it is done. It's knowing how to take different routes to find answers and connecting not only A and B but also the rest of the alphabet. This is harder to learn simply because it's not just an easy answer. It requires reasoning and it also requires more time.

Instrumental understanding is not related to relational understanding, but relational is related to instrumental because instrumental doesn't include any relational understanding. It's just knowing a process. Relational understanding, on the other hand, includes instrumental understanding plus more.

Mathematics is how numbers apply to every day life whether that's in the grocery store or in a NASA laboratory. It proves why certain things are the way that they are.

I learn math best if I understand why and if I can see all the different reasons why. Math has never been an easy thing for me and I usually have to see different ways to approach a problem before it really starts to make sense. I think that my students will learn best this way as well. By asking why, reasoning skills are developed that will help them to make connections for themselves. If students are just told how to do something, they'll choke when something out of the ordinary shows up because it didn't follow the pattern that they were shown how to do. It's also important for them to see all the different possibilities and angles that can be taken to find a solution. Each student will think different and one way might be great for one student while another doesn't understand. I think they'll learn best if they understand why and if they can see many possibilities to a solution.

Math assignments are great. You can't just learn math by listening to a teacher lecture, there has to be hands on experience that is varies within a topic so that they can see the application. I also have always thought that it was good that I had to memorize my times tables back in second grade.

I think that teachers that don't know how to teach are detrimental to students learning math. Teachers that don't care, aren't excited, don't show the application in every day life, and only show one way how to do something and can't explain why.