NCTM Article Number TWO

Hudnutt, B.S., & Panoff, R.M. (2002). Mathematically appropriate uses of technology. On-math, 1(2), 1-5

The main idea of this article is that technology should be used in classrooms because it can help students accomplish tasks they couldn't do otherwise. Students can use technology to see what happens to equations when algebra gets complicated and when it comes to the point that the arithmetic is hindering the teaching. When studying probability the computer can compute thousands of cases in seconds, when it would have taken a student hours to do the same thing. Students can explore lots of graphs and observe how changes in an equations change a graph very simply on a computer. The same effect may have been muddled in having to make so many graphs by hand had the students not had access to such technology. The author acknowledges that technology can have it's downsides with technical difficulties, but overall it is beneficial in the classroom.

I think technology is beneficial in the classroom. I think it allows students to explore complex ideas in a shorter amount of time. As was stated in the article, sometimes students forget what they are actually looking for and get lost in complex algebra. I am not saying that technology should replace arithmetic, but I definitely think it should be used in the classroom for exploration and for teaching the big ideas of what is going on at that point in the course. I have observed lots of classrooms with advanced technology for my 276 class and it is amazing the things we can do in with math and technology. The teacher can make perfect shapes that can show exactly what they were trying to show. This technology can also help with students who have IEP's and need the notes from class. The teacher can simply print off a copy of everything shown in class. All this and more are reasons I know technology is beneficial in today's classroom.

NCTM article

Switzer, M.J. (2010). Bridging the math gap. Mathematics teaching in the middle school, 15(7), 400-405.

This article had two main ideas, first there is a gap between elementary and middle school learning, and second there are ways to bridge that gap. The author addressed the issue of a gap by stating that middle school experiences don't connect with elementary experiences. Often times middle school teachers don't quite know their students' prior knowledge. He did recognize that even though middle school teachers will often have a basic idea of what the students have or should have learned in elementary school, the school structure differences and other factors make it hard to communicate between elementary and middle school teachers. One of the ways he mentioned that can help fill this gap is for middle school teachers to become better acquainted with their students' prior knowledge. A way to do this is to encourage middle and elementary school teachers to work together and communicate students' learning. Much of the paper consisted of algorithms and methods to teach these algorithms that would help students better transition into middle school, thus helping to bridge the gap.

I agree with the author that there is a gap between middle and elementary school, and there are multiple ways to bridge that gap. More than just a gap between middle school and elementary school, there can be a gap between middle and high school learning of mathematics. From my own experience going into middle school is hard enough but sometimes the teachers presented material in a way we had never seen before in elementary school. Had they started with the math we knew and built off of that I would have felt much better equipped to learn. I found that to be the same when going into high school. For example I remember getting into high school and feeling as though I had never learned slope before. However I had learned it in middle school, it just wasn't taught the same way as it was in middle school so I had a hard time making that connectoin. Some of the ways this could have been fixed were mentioned in the article. It would be beneficial for the middle school teachers to communicate with the elementary teachers. This could have both schools working to better student learning in order to have a smooth transition between schools. Along with that I think that teachers should make an effort to figure out what their students know, and then build off that knowledge.

Haley Bly

Warrington Paper

One advantage of teaching mathematics without telling the students the procedure/answers is that it promotes discourse among the classroom. For example when the students were stuck on 4 2/3 ÷ 1/3 if the teacher had told them straight away that the lone girl was correct, there would be no point in the other students arguing their ideas any longer. Also because she had to explain it against the whole classroom they all listened to her reasoning. They didn't simply wait for the teacher to back one answer. Another advantage I saw was that the teacher used the previous topic in order to introduce the new topic. She started with whole number division and worked her way to compound fraction division. It seemed like the students followed her transitions pretty well.

Some disadvantages of not telling the students the procedure/answers is that some students get lost with such little direction, and the material may get confusing without concrete terms and procedures. The teacher is obviously running the discussion in this classroom, but the students seem to dominate the discourse. Its hard to learn from peers because they aren't trained to teach, thus some students have an easier time listening to the teacher who can explain it with a confident voice. I know that when I'm in classes sometimes I disregard students telling me what they do because it just confuses me more. Oftentimes I need to have direction straight from the teacher and these students don't seem to get that firm direction. The other disadvantage I mentioned is varying terms. If there are not set names and rules math can get really confusing when moving into a different class. If the students in this class discover things such as slope or trig functions they will have a difficult time taking all their procedures and realizing that maybe what they do is less efficient than the new class, and they may take a while to relearn the real names. I didn't see examples of these disadvantages in the reading, but I'm not convinced that her class is always this perfect.

haley bly

TRYING to make sense of von Glasersfeld and Constructivism

When von Glasersfeld said constructing knowledge, he means the students have to create what they come to know. When a student listens to a lecture they do not take in exactly what the teacher said. The student filters the information and constructs it into knowledge based on how they saw the information through their OWN lens. Von Glasersfeld did not call this aquiring knowledge or gaining knowledge because the student doesn't actually take in what the teacher said. They take in what they thought they heard. It's usually wrong...or as Dr Seibert suggested, viable; however it is not correct knowledge because it is simply how that particular student interpreted what they heard.

If I believed in constructivism I would have a hard time seeing what my students gained from my class/lectures. So in order to work with that problem I would have my students write in math journals. At the end of each class period the students will take about 5 minutes to write what new things they learned and explain to me how they understood it. They can also ask questions etc. As a constructivist I don't believe students actually "aquire" what I have tried to teach. They construct their own knowledge so in order to better teach them I would use journals to properly assess what the students have constructed about the material. This way I can reclarify what was misunderstood and I can keep tabs on my students' knowledge.

Haley

Erlwanger's Paper in a Nutshell

Erlwanger's main point is the necessity for reasoning behind mathematical rules. He stresses the need to understand the concepts underlying the procedures. Benny's main problem was that he was not able to derive his rules from mathematical concepts. When Erlwanger questioned Benny's rules, many of them had shaky reasoning. Benny had no idea why his own "rules" were correct, he simply used them. Erlwanger's paper is pointing out that had Benny been taught mathematical concepts, his rules would have made sense to him, and Benny could have enjoyed mathematics.

This goes back to the difference between relational and instrumental understanding. Benny thought math had no reason. He called it magic and he said it didn't make sense, however he was doing very well on his exams and homework assignments. This is a little warning sign to teachers that good grades do not always mean the student understands. Sometimes in math classes we are pressured into working more for the grade and less for the understanding. This is a problem teachers need to try and avoid because as we were shown in Benny's case it is detrimental to actual learning.

Two Types of Understanding

Students can learn mathematics in one of two ways, namely relational understanding and instrumental understanding. Instrumental understanding simply teaches the students how to find the correct answer, no more no less. Relational understanding also teaches the student how to find the correct answer, but it goes beyond that and teaches the student why that is the way to find the correct answer. Thus we see they are both effective in getting the student to pass the class, or maybe just a particular assignment, but in the long run relational understanding is more beneficial to the student's learning. The positive sides to instrumental understanding is that the results are seen quickly, and the rules are usually easier to understand than are the concepts behind those rules. Because of these benefits, students are able to find answers more quickly. However, with relational understanding students are able to adapt to new tasks more easily. They can see a new subject build off a previous one. Though relational concepts are not easier to understand, they are easier to remember because the student learns the big picture, and is then able to apply that bigger idea to smaller scenarios. It is easy to see issues with instrumental understanding; the students forget more easily, they do not know how to evolve what formulas they learned, etc. However there are several disadvantages to relational understanding. It takes a long time to achieve relational understanding, sometimes the level of difficulty exceeds the students' abilities, and often it is hard for teachers to teach relational understanding when so many of their colleagues are not doing so. However, it is still a good procedure to help students understand the fundamental ideas behind the method for solving a problem. Instrumental understanding is embedded within relational understanding, we simply need to broaden what we teach so that students can truly understand what they learn.

Haley

1. What is mathematics?
Mathematics is finding a concrete process to solve a problem. It is not very debatable that is what makes it so perfect.

2. How do I learn mathematics best? Explain why you believe this.
I learn mathematics best when a teacher can explain a process and then give examples on how to solve related problems. Since most mathematics is a process with little opinion involved, it is necessary to learn a lot by example. Once I can to a few examples it is easy to do almost any similar problems.

3. How will my students learn mathematics best? Explain why you think this is true.
I would like my students to be able to learn mathematics best through asking questions when they don't understand. I would like them to also learn by working on guided problems in class and then being able to test their own abilities as they go home and try to do similar problems on their own. I think that if they can follow and participate through suggestion or question they will be better able to complete homework problems which will help the ideas stay strong in their mind. It is an invigorating feeling to be able to do a homework problem all by yourself so that is what I would like to see my students enjoying.

4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
When students are encouraged to give their suggestions on how to solve a problem, it strongly promotes their learning and confidence. When they give a correct solution, or even if they give a mostly correct process to find a solution it helps to reassure their studies and knowledge. Another current practice is splitting students up into small groups or partners to work on harder problems. It helps the students to bounce ideas off of each other and using their notes and textbook they are able to find solutions without very much help needed from the teacher. A valuable tool for students is their text and it is so important for them to realize this is more than just a spot to find homework.

5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
A major detriment to students' learning is when teachers assign homework that they don't intend to collect. This puts the students in a situation where most of the time they don't actually do the assignment. I find homework to be one of the most important ways to learn math. When students don't diligently try the homework on their own, it puts them at a disadvantage for the exams.