Assignment #7

Daire, S.A. (2010). Celebrating mathematics all year 'round. Mathematics Teacher, 103 (7), Pages 509-513.

The authors main goal of this article is to promote, as the title suggests, the celebration of mathematics all year 'round. She talks about how her school has cultivated a love of math, in particular the number pi, all because she started celebrating pi day, March 14, every year. Through hard work, generous donations, and personal sacrifices the author was able to extend her pi day celebrations to involve a wider range of students and other faculty members. Events, such as a Monopoly tournament, Halloween themed story problems, and a Valentines Day Love of Mathematics contest, are now a regular part of the school's yearly schedule. Getting students involved in the voluntary events can be difficult, but the draw of prizes such as a graphing calculator, 5 x 5 x 5 Rubik's cube, trophies, and other gifts, especially in a lower income area, is a great way to encourage participation. The article ends stating the fact that due to the implementation of events such as these, the students at this school have had much higher scores in statewide tests and have a greater desire to participate in mathematical discoveries.

I feel that this article hit its main goal right on the head. After reading, I am now quite excited to try some of these ideas once I start teaching. I can see how there would be difficulties getting other faculty/staff members involved, but I'm sure that with time it could happen. One thing that I feel would be nice would be a follow-up article on how to get other people excited about it. What other ideas could work to get more people involved? How did she approach people, especially when she was asking for money our other resources? Can teachers give help to the students when they're trying to solve problems? These are some questions that I feel would be helpful to others when trying to implement these ideas. What a wonderful article!

Assignment #6

Otten, S., Herbel-Eisenmann, B. A., and Males, L. M. (2010). Proof in Algebra: reasoning beyond examples. Mathematics Teacher, 103(7), Page 514.

The goal of the authors of this article was to convey the idea that proofs could be utilized in beginning algebra classes, not just geometry. Through a case study of an algebra one class, the authors demonstrated how the cross-multiply rule is generally taught, and then presented an alternate version in which the use of proof could easily have been inserted into the discussion. Usually, students only encounter proofs when they come to geometry; however, the use of proofs in algebra can greatly assist students in their comprehension of difficult concepts. Also, the discovery of why rules work in algebra, will also lead students to a greater capacity to like math because it's no longer "a collection of procedures and rules" that students feel they have no ownership over, rather it provides the students the opportunity to "realize that mathematics is a state of mind characterized by inquiry and a thirst for justification."

Although I agree with what the article is arguing I feel that, due to the fact that they invented a hypothetical situtation instead of actually testing out their idea, they did not quite present as strong as an argument as they could have. It is slightly ironic to me that the article is written about proving things, as opposed to just showing examples, but then the authors present an example instead of setting out to prove their point. I do like the ease at which the teacher, in the hypothetical situation, was able to construct a proof. It really didn't take that much more time to generalize the cross-multiply situation to show that it works for all situations. Once again; however, I have a problem because of the fact that the cross-multiply rule is an easy situation to come up with a proof for. What about more complicated situations such as solving systems of equations or how we came up with the quadratic formula? In these situations it could take quite a bit more effort to show why they work. In summary, proof of things in algebra is a good thing that teachers should look into, but at the same time, it might just be better to provide students with examples.

Assignment #5

In her article, Warrington mentions many advantages of allowing children to figure things out on their own, such as children feeling a greater ability to exchange ideas and giving children the capacity to assign greater meaning to numeric problems by creating story problems for them, the biggest advantage put forward is the fact that children, if placed in the correct setting, "can construct knowledge about sophisticated and abstract concepts in mathematics without algorithms." In other words, children in Warrington's class were able to solve quite complex problems, such as 4 2/5 divided by 1/3, in their minds without the use of the 'invert and multiply' rule. Warrington makes the claim that because the students had been actively involved in the discovery process they were able to solve math problems faster and better than students who learn just the algorithms themselves. This ability for children to learn how to solve problems without algorithmic methods is a valuable skill for students to obtain because of the fact that it allows them to delve deeper into the world of mathematics.

The process of constructing knowledge on their own, children truly can learn many things about math. However, there are a few problems with this method of learning. The first is the fact that if a teacher doesn't know the correct way to approach this method of teaching, how can they be expected to "provide learning environments that allow children to be successful"? Another problem is that it would be difficult for a teacher to ensure that every student is learning the necessary skills to allow them to progress on to the next 'step' in the mathematical process. Math is all about utilizing past knowledge to delve deeper into mathematical solutions, so if a child doesn't understand a building block of math, they will find it difficult to advance in math. While I do see the advantages of this style of learning, I do think that it might not be as wonderful as it is a first look.

Assignment #4

In von Glasersfeld's paper on constructivism, rather than making the claim that we acquire or obtain knowledge, he states that we "construct" knowledge through our experiences in life. His use of the word construct helps us see what he is meaning. Just as a construction worker uses a blueprint, or knowledge gained beforehand, to build a house, so do we 'construct' our view of the world through experiences we have had in the past. Continuing the analogy, one does not build a house starting with the roof. A solid foundation is necessary for the walls, doors, windows, support beams, and roof to stand. So it is with our knowledge, we only know the things that we know because of the foundations that we've had. Also, just as a worker has his own 'best' method of construction, so too do we form our own 'best' ideas of what the difference between truth and error are. And, just like the construction worker, we don't change our ideas of truth until we're presented with something that provides a clear counter-example to what we believe. Thus, von Glasersfeld is making the argument that the only way we know things is through experience; hence, it's impossible for us to know the 'truth.'

One important implication that constructivism has on mathematics is how it points out the fact that just because I view a topic one way does not mean that everyone views it the same way. I know that as I have tutored, subjects that I have found to be quite simple to understand, such as rules of exponents, others have really struggled with. After reading this article I've come to learn the importance of trying to see through my student's eyes. Trying to really see where they're coming from and what 'baggage', whether good or bad, they're bringing with them to the proverbial table. As I try to learn more about how people view various aspects of math, I can then modify my teaching to fit into their world views and ideas. By doing this, I can help them create a more accurate view of math and how it works.

Assignment #3

The goal of Erlwanger's paper is to show the danger of learning math strictly on your own, the problems of not teaching the 'why' behind the 'how', and the faulty conclusions that students obtain from not learning correct principles. According to the IPI system of mathematics, Benny was one of the more advanced students in the program, but upon investigation, Erlwanger discovered many false conceptions that Benny had with regards to math. The fact that Benny in effect 'worked the system', or put down the answer he knew was wanted as opposed to what he felt was correct is a good example of this. By working by himself, Benny had come to many false conclusions with regards to many fundamental parts of math such as fractions and decimals. Because Benny didn't understand why fractions worked the way they do, he made many mistakes. Also, the fact that he didn't have someone checking his work was a large contributor to his coming up with incorrect ideas of math. Had someone been more involved in the learning process, they could have corrected many of his erroneous assumptions.

A main argument that is still valid today is that of the danger of learning math strictly on your own. It's still a valid argument because of the fact that math is a difficult subject to learn and if you have to teach yourself it is easy to create false ideas of how and why math works. I currently am a tutor for some students who are taking some online math classes, and I really don't know how they are expected to learn all the material without someone to explain it to. The language of the text is difficult to understand, the homework is challenging and tests the limits of the concepts, not the basics, also, it's easy to get rules confused and use the wrong rule in a situation. Having witnessed first-hand what not having a teacher does to a student, I know it's SO important to have someone involved in the learning process.

Assignment #2

In Skemp's article, he discusses the pros and cons of two types of mathematical learning; relational understanding, or the understanding of why something works and it's application, and instrumental understanding, or the understanding how to solve a problem. He says that both are valid means of teaching children math, but that the advantages of relational understanding are greater than those of the instrumental approach. The first of the four advantages of relational understanding is the fact that it can be applied to a broader range of problems. The importance of this fact is seen when looking at a disadvantage of instrumental understanding; that of if the problem is slightly different than the norm, the student will get it wrong because they only know how to solve things a certain way. The next advantage mentioned is that it is easier remember things taught relationally as opposed to instrumentally. This may seem contradictory because of the fact that with instrumental learning all you need to do is memorize a formula, but you have to memorize a formula for EVERYTHING instead of just learning how we obtain the formulas in a certain field of math and applying it. Moving on to the third advantage of relational understanding, and that is the fact that this form of learning can be a goal in and of itself; i.e. the reward of figuring out why a certain way of solving works is satisfying to the solvee. This point also leads us to one of the disadvantages of relational understanding, that of the time it takes to achieve the necessary understanding to apply said way of solving. The final advantage of relational understanding talked about is the fact that the satisfaction of discovery will lead others to more discoveries on their own. Now, the advantages of instrumental understanding such as simplicity of understanding and the immediate satisfaction of solving a problem are good, but when held up to those of relational understanding it is clear that relational understanding is the better way of learning.

Assignment #1

• What is mathematics?
  

Math is the study of numbers and how to manipulate them for use in the world. It really is what makes the world we live in work. Without math we'd have no computers, cell phones, cars, amusement parks, movie theaters, plumbing, the list is endless. That's what math is.

• How do I learn mathematics best?

I best learn by example. If I can see how something works it makes doing that thing SO much easier for me. I'm a big fan of the 'plug and chug' method where you're given a set way to solve something and you take that way and 'plug' the various parts in and 'chug' away at solving the problem. I really struggle trying to figure out wordy theorems and definitions. I like knowing what the definitions and theorems are, but then I need to see how they apply to what we're talking about. Learning by example is what works for me.

• How will my students learn mathematics best?

I believe that my students will best learn the same way I learn, by example. It makes sense for this to be true because of the fact that you teach best what you're best equipped to teach, and I'm best equipped to teach by example. As evidenced by these posts, I'm not the most adept person when it comes to words so using examples will best assist me in teaching my students important concepts. I've had quite a bit of tutoring experience and the people that I help have always understood the best when we actually work through a problem instead of them trying to figure it out by me talking about it.

• What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics?

One thing is the obvious fact that it's being taught. You can't learn what you're not taught. :) One practice that I think is very beneficial is group studying. When you are learning a subject with your peers I believe that you can explore more aspects of the subject because you are learning through more than one perspective. I also feel that the tried and true method of daily homework assignments is another great way for students to learn. Math is something that needs to be focused on a little bit each day as opposed to big chunks of learning only once or twice a week.

• What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

One of the things that I find most detrimental is the whole 'draw a picture' approach to math. Students in elementary school are being asked to visually represent everything that they do in math, and it's not necessary. I realize that everyone learns differently and that visual representation is one of the best ways to learn; however, I feel that the majority of students would learn math better if they learn it the way it has been taught for years; i.e. 2+2=4. I especially feel that the 'draw a picture' approach is detrimental because of the fact that when students reach higher level classes, such as Algebra 2 and Calculus, there are some things that just can't be represented visually.