Week 8 Responses: Using Multiple Representations in Algebra

Words versus Actions:

Everything sounds good on paper. Teaching and learning multiple representations sounds like a great idea. Actually implementing such a strategy? Not so much. In general, we are glued to our own ideas. It takes a lot to change beliefs that have been hammered into our skulls from day 1. Once we address the root of that problem, we can find a way to present multiple representations to students where they might actually be able to grasp the idea that their way is not the only way.

This also brings up the issue of communication within the classroom. The traditional vision of students sitting at their desks, backs straight, eyes forward and attentive is not, and should not be, expected. Instead of explorers, students have become robots, treating the teacher’s word like the word of God. While that may sound like a grand old time, we have left our students to rely on what we tell them. They no longer care about learning for themselves. They learn for the test and forget about it after, which is another issue that needs to be addressed. Before we can think about implementing certain strategies, we have to think about if such strategies are possible given the condition of the students.

Another point the PowerPoint makes is about having problems with access points. This, I find intriguing. This, I think we can work with. In a classroom, we have varying levels of students. Going beyond that, no classroom is alike, and that is due to the mix of students. So, having problems that every student can start, to see where everyone is, is a great idea. That way you don’t discourage the weaker students and you can add a bit of challenge for the stronger students.

In the classroom, discussion should be encouraged among the students. As a student, I find it helpful to see how my classmates approach a problem. If I can explain how I completed a problem to them, and have them question me, I understand the material better. It is also beneficial to see other strategies that may be better than how I went about the problem. If I see something I like, I might adopt that strategy into my existing knowledge. I find it easier to keep an open mind about problem solving, but many others are very closed to different ideas. For example, I was in a class where we were supposed to prove something using induction. I wrote every step out, which was the way I learned how to prove by principle mathematical induction. I was working with two other classmates, and one of them looked at my proof and criticized how I went about it. There was nothing constructive in his criticism, just the fact that induction was not done in that manner and that I should rewrite the whole thing. I tried to defend my case, but it fell on deaf ears. Naturally, I handed it in without changing anything and all was well with the world. From then on, I decided to work with people that would be more constructive, which was the best decision for that class. We were able to bounce ideas off of each other and fix mistakes without being rude to one another.

My point here is that bouncing strategies and ideas off of those people around you is beneficial, especially when those people are in your peer group. When a student does not understand something, they tend to blame themselves and think they are not smart. They think they are alone in their inability to understand. If they are able to connect with other students, they will be able to see that they are not alone and they can help each other. It’s like computer programming. Say Johnny is an expert in forming arrays, and Julie is an expert in lists while Dan is great with string concatenation and math. They can all work together to solve a problem that involves combining their strengths. One student may specialize in a certain strategy and we just have to unlock those strategies and combine them.

We want students to feel comfortable and confident. It will not be easy to put together all of the representations, but they can be exposed to them. That might be the best we can do given the attitudes students may have. They need to be able to explore and question. They also need to read carefully and listen. Students have a hard time separating information and determining what is given and what is being asked. Once they see what is being asked, they need to be able to think about it logically and ask themselves questions that will lead them down different paths. They need a direction to go in. As teachers, there isn’t always time to ask probing questions, but trying to do so will help avoid the blank stares and mechanical actions of students who look for answers.

I liked the idea of having students reflect on their thinking. Maybe we can give students thinking journals. We can give them a weekly problem that they can work on in their spare time. They don’t have to answer the problem correctly, but they have to show their thinking process. The whole point would be to get them to explain themselves and their thinking.

If we want to use multiple representations, we need to be able to phrase the questions in a manner that allows for the opportunity to explore a new way of modeling the problem. Otherwise, students will stick to what they are used to. Forcing students to see things in a new way might be one of the only ways to ensure they don’t cling to the same idea. (Sadly, we cannot force a way of thinking.)

Another topic brought up was technology. Technology is something new for students. Growing up, I did not have half of the things this generation has. These students have so much power at their fingertips. They can use computers to help them visualize problems. However, as teachers, we cannot have students relying on such tools without understanding concepts. We need to show them when it is okay to utilize technology and when it is unnecessary. Students need to learn how to use technology to help them rather than just getting an answer. They should be able to reason and understand the answer that a computer might spit out.

The last topic covered involved equity. I am all about learning for all and making sure every student gets the same opportunities. However, that’s just not possible. We cannot accommodate every student. The range of levels in a single classroom is never the same. One student could multiply large numbers easily while another has problems adding large numbers. You never know what to expect in a classroom and how much you will have to change your style of teaching to fit that particular class. While all students should be granted the access to a math program that plays to their strengths and existing knowledge, it is not always a requirement that can be met, and the reason combines many factors that cannot be controlled.

Here is the part where I will share my unpopular opinion. You can skip this if you would like. Some students require more attention than others. Some require more challenge. Some require extra help. While it would be great to have different levels in the classroom, there should be exceptions. I know some schools have honor classes and base level classes, but there needs to be more than that, especially in grade school. To many students, especially those with disabilities, get left behind, and it is not fair to them. We need to be able to accommodate every type of student. Mixing students with learning disabilities in a regular classroom setting should not be the answer. As a society, we need to offer opportunities for everyone and rid ourselves of this stigma that we all learn the same way. The whole point in presenting the multiple representation idea is the fact that students learn and approach problems in different ways. We can’t expect everyone to succeed if they aren’t given a chance to succeed from the start.

This post is now massive, so I think I’m going to conclude here! This is all for now! There wasn’t much in the realm of tweeting, but I can across a few articles that I have bookmarked for later. I’ll let you all know what I think about them! Anyway, let me know your thoughts!