Professional Development: “Onward and Upward!”

I love blogging and journaling. My only problem is writing out to other people and posting. Over the semester, I have gained more confidence in writing things out, not really caring about whether or not anyone actually reads or takes an interest in what I say. In a way, writing the blogs has helped me reflect on everything we have learned. It also allowed me to self-reflect and see where I can do better in regard to teaching. It also gave me ideas and got me thinking about possible solutions. If I proceed with teaching, the blog will definitely be something I keep up. It’s like an online diary in a sense, and if anybody stumbles upon it, then they can add their thoughts, and we can get a conversation going.

Discussion and sharing ideas, at least for me, is really helpful. And I would like to see more efforts be directed toward self-reflection and sharing ideas. Everyone has their own experience, but no teacher is alone in their experience. That should be one of the first orders of business: creating a network of people you can talk to. I think schools should make those groups. I know time and money is a factor, but I think it is worth having teachers come together and share teaching styles, problems, and so on. As a new teacher, that would be something I would like to start; having all of the teachers communicate and be on the same page. Having a stronger community of teachers can ensure that everyone gets included. There won’t be old man Jones in Room 353 at the end of the hall that everyone knows, but nobody talks to.

Getting better at something doesn’t happen overnight. It takes small steps of getting better to see a difference. So, keeping track and planning out little goals can allow you to see how far you have come. When I was putting together all of my materials in summary, I saw everything I actually accomplished. I was thinking that I had gone nowhere. However, after reflecting, I saw how much I have changed and how much I have taken away from keeping this “journal”. I tend to keep an open-mind and try to see the other sides of the x-sided figure. I will hear thoughts out until the end, but I will sort out what I think is bad or good based on my experiences. If I see something I like, I will not be afraid to try it. This allows for a transition into what I thought was useful and not useful.

Although I agree with the things NCTM says, that’s pretty much where it ends. I just agree. There is no real room for debate. However, I do think the materials can be used and modified in ways we have learned over the semester. So, in that, I can definitely find some use. So, I could potentially take some topic and transform it into a game by tweaking based on materials and the lesson. There is definitely some promise in the materials, so long as careful consideration is taken into account when forming lessons.

Another thing I found useful was video-taping the lesson and reflecting on it. I think that is a good idea, though there are many issues regarding the use of video in a classroom. Now, to expound on this: We have technology. Since we have technology, most of what we do can be online. We can, potentially, speak to people thousands of miles away in different countries. There are always technical difficulties, but the point is that was can talk with people and we don’t have to be left in the dark. We can have web seminars and include everyone. We don’t have to have people flying in from whichever country they come from. There can be a live feed we log into and watch, being able to send in questions through the feed. (It’s like the Twitcam? Thing, if that’s what you call it.) Something like this would be just as useful. Communication, in general, is useful. And, now, there are so many avenues of communication where you can take your pick.

In the future, I will definitely try to make my blog bigger and broadcast it. By sharing my experiences, I will be able to offer others some insight that they can agree or disagree with. And I think that will be the most useful to me. In the moment, you think that your views are the only views, but everyone has different thinking styles. People surprise you by thinking in different ways, and we should keep that in mind when teaching students. They think in different ways as well, and we need to be able to give them multiple ways to approach a problem. That is another thing I will take into a future classroom. The students need to be the top priority when coming up with lessons.

In all, as a future teacher, I will keep an open mind and take in other ideas. I will continue to build a network of people that I can talk to about experiences and problems in the classroom. (As well as talking about the things that work.) I will do that by creating multiple outlets and sharing the blog posts relentlessly until some unsuspecting soul stumbles upon it in their searches. Sooner or later, more and more people will see it. I will use NCTM as more of a springboard for information and materials, but I will try to broaden my horizons with other groups that I find. To close, this semester was very eye-opening, and I learned how much technology can help with communication and how much building a network of people can help you not feel alone in your experiences.

The List Goes On: Tweets, Replies, Blogs, and Comments

I believe this post contains all of the social media part for the semester. I could be missing a few here are there, but this is a majority of the reaching out I have attempted. Also, as a part of my training for the Math School I work at, I interacted with teachers across the branches in order to go over lessons and exchange teaching strategies and experiences, which was sort of like the last assignment we watched with the teachers reflecting and reviewing their lessons, then revising their plans and trying something new. We did these “webinars” once a month, the last one being a few days ago. It was actually really helpful when I needed help teaching a particular topic because the kids weren’t responding well or what have you.

Sam helped me find the snipping tool, and WordPress did the rest in putting the images together! So here it is:

Education as a Business

This week, we watched a video regarding the use of video technology as a means of self and peer evaluation. Essentially, a pre-service teacher would be on camera teaching, and they would be able to view themselves after. Another teacher would comment and offer constructive criticism. The goal is to build a network of communication between new and more experienced teachers.

The whole point is for teachers to reach out and connect with one another. With all of the technology available at our fingertips, we could speak with teachers around the world in seconds. It’s easy to make connections with people and start communities where we can all help each other and share experiences. Let’s face it, we are never alone in our experiences. There is always a solution and someone who has been in a similar situation. Not every solution works, but by reaching out, we have given ourselves more opportunities to solve problems. Most people want to help you if you ask for help, so why not ask for it?

Communication saves lives, but not enough people communicate. Or, at least, in the right ways. That’s how wars are started; miscommunication and white men taking land away from other people. So, the first step would be to put yourself out there like we have been doing all semester. My main issue is that I tend to disconnect from social media and stand on the fringe, watching all of the goings on. I want to be able to gather more courage to post frequently about the issues I care about. I just don’t like unproductive discussions where one side is so stuck in a particular mindset that all other possibilities are dropped by the wayside. I feel that, on the internet, people are more aggressive as opposed to encouraging. Though I have learned through my attempts at communicating with others that if you find the right people to communicate with, you will make some progress and gain some insight. (As long as you don’t throw yourself into an echo-chamber of bad company, you should be good to go.)

To get back on topic with the video we watched, I wanted to mention that I had mixed feelings about it. So many things could go wrong when using video in a classroom. While I think video is a brilliant means of self-evaluation, there may be some issues such as focus in the classroom and creating an artificial environment. There are other ethical issues as well that may be brought up. However, I think that if we fine-tuned that idea, it could work in our favor.

As a former member of color guard and chorus, we always video-taped our performances and played them back. In rehearsals we needed to practice as if we were actually performing. We were to bring everything we had to the floor. Our instructor would replay our parts and tell us to watch what we were doing incorrectly. It was eye-opening to see the mistakes that we were making by viewing ourselves. In the moment, we were focused on getting through the routine. After knowing the faults and looking back on ourselves with an objective mind, we can pick at those things that we missed. The same thing would go for teaching. When you are teaching for the first time, you are in the moment, worrying about teaching and getting through the material. However, when given the chance to review that “performance” you can see where you need to improve so that the kids will understand you.

Teaching is an art, a performance. You have to step outside of yourself and put on a show for those who are watching. So, being able to view yourself would be a good form of evaluation as a sort of self-reflection, bringing awareness into the equation. This is something I would support as a means of using for your own benefit.

On the other hand, the video was plugging the technology used. EDTHENA was the product being marketed. While I understand that it is helpful technology, I hate that everything in education is being treated like a business. Education is more politics than learning, it seems. That is another problem that needs to be “solved”. We need to care more about the students than selling products. Students are like guinea pigs in the equation, testing all of this technology. (I’m not saying to eliminate technology here. I think that we need to show students how to use it in a smart manner where it will benefit them. I just hate that education has become a market for profit.)

I think the video made grant points and it has merit in helping teachers and building a thread among them, but it should be fine-tuned. It can’t be another fad in teaching, only in place to be sold and forgotten when marketability reaches an all-time low. Anyway, these ideas are a step in the right direction even if the presentation may not be.

Let me know your thoughts on film in the classroom and using it as a means of evaluation! Or let me know any thoughts you may have on this post. Feel free to disagree. I look forward to reading your thoughts and opinions.

Next Week on Blog Posts with Rebecca: Wrapping Up and a Step Forward

This week on Tweets with Rebecca: I did tweet, but the response was nonexistent. And I saw a blog I enjoyed reading. The posts were interesting, but not as helpful as I would have liked. It dealt more with math and general and what a math degree could offer. In review, interesting, but not completely teaching related.

Until next week! Have a beautiful day wherever you may be!


What’s Proof for Some, Is Not For One

This week’s article is a continuation on the fact that the NCTM is writing under the best possible conditions. There are always cases where students will not be on the same page. They will drown if you don’t give them something to keep them afloat. We hope that if we just throw them in, they will swim. However, in the case of mathematics and non-life-threatening situations, they will need to be thrown some kind of driftwood to keep their heads above the water.

If we throw our kids in, we can’t expect so much from them. Sure, you might have some kids who will grasp onto something, but you can’t expect perfection from young students. This is the age where they get to experiment and explore. They need to gain the skills and confidence demanded by proofs. They also need to be exposed to various proofs. Again, as teachers, we should pick and choose the best ways to present proofs to students. We don’t want to overwhelm students, but we also don’t want to coddle them. (Kids get enough of that now without teachers adding to it).

Generally, we tend to teach to one type of student, not keeping other learning styles in mind. In class, we always talk about the frog vs. bird approach where frogs hop around until they grasp at something and birds tend to look at the bigger picture from a distance. I would like to think of myself as a bit of both, but I know I’m more of a frog. If I can’t picture something, I’ll start making the picture and see where it’s going. If you can’t think of something right away, then DO something. You won’t go anywhere if you don’t start. Though, starting is the hardest, and you need some kind of entry point if you are to ever get moving on a problem. Kids need more exposure to math concepts, computation, and strategies if they are going to throw themselves at a problem.

As mathematicians, our methods of proving can be explained by truth tables in classical logic (which might be a little faulty…that may need some revision. I mean, just take a look at the conditional, or implies, operator…that brings along with it a whole mound of problems). Logic would be a great addition to introducing proofs. I wasn’t introduced to truth tables until my first year of college, and even then I didn’t make the connections between proofs and logic until I took a philosophy course in logic. Taking that course allowed me to make the connections required to understanding proof and why certain proof methods work. Logic allows you to better see the inner workings of things.

Overall, kids need more tools if they are going to have any hope of understanding proofs. The article we read doesn’t seem possible for every combination of classroom. It assumes prior knowledge that students may or may not have, and it’s a bit more sophisticated than an introductory proof. In addition, the proof was also very directed to a certain way of thinking and a certain answer. Like mentioned above, students do not think in the same way, so open-ended questions would allow for students to think in their own way. By closing off the questions, we are closing off students that think in other ways. We need proofs that will allow students to give more ideas and bounce their thoughts off of each other. Maybe, a good way to get kids thinking would be to have an open discussion where the kids offer their ideas and try to explain their ideas to their peers.

Proofs seem to rely on some “mastery” or prior knowledge. I believe that proofs should follow the skills that students have by high school and build on them. I also believe that as students continue to grow their skills, more proofs should be introduced to them. They need an idea as to what proofs look like before they can continue to write their own. It’s like writing a paper. You need vocabulary. You need to know sentence structure. You need to have your own style and voice. You need to know how to manipulate the words in order to sound cohesive. You need to make your points clear and focused and formulate good arguments based on evidence. (I would even go as far as comparing proofs to sonnets, having a particular structure, following certain rules and stringing together words strategically to make a point.)

We don’t want to give students the wrong idea about proof. We want to be clear about what proofs are. They are not just examples and solving problems. The best proofs offer a bit of frustration and a hint of adventure.

This week on tweets with Rebecca: I haven’t been great with Twitter. I’m going to stop fighting with my computer and use it more.

Anyway, I want to know: How would you present a proof to a high school student? How would you determine what would be the best proof for a given class?

Thanks for reading!

Sr Argi rw Vrs sr Argi

If you haven’t already guessed, this week’s reading and assignment involved a bit of cryptography. Can you figure out the message? (It might be too short, but I maintained word length. Hint: Look at letter frequencies, though you probably already knew that.)

Today’s world offers many tools for us to use when solving problems. Such tools are widely available, and it would be a shame not to use, or learn how to use, the technology that is ever present in the world we live in. What are we so afraid of? Well, it isn’t a perfect world. Kids abuse the system. They use technology for evil things like looking up answers in Google. In a world where information is available at the click of a button or a swipe of the screen, many students have little patience for figuring things out on their own if it takes more than five minutes.

Though, that doesn’t mean we shouldn’t use technology as a resource. We need to expose students to technology that will be useful to helping them solve problems and see the connections. It might allow for interesting activities. Kids get bored easily. Keeping their attention and coming up with an engaging activity requires more than a few manipulatives. (Especially when it comes to middle school and high school students.)

Teaching students to use computers and programs to assist them would be a great asset to the student’s education. They might even find class more enjoyable when they are given something they believe they will use in the future.

The only real issue seems to be the access to technology. Some schools and students simply do not have the means to provide such materials. If that is the case, then trying to do certain activities may be unfair to students who do not have the means of obtaining certain materials. I still believe technology should be utilized when given the opportunity. When analyzing a paragraph and gaining data to crack a message, you don’t want to spend hours gathering the information by hand. (It’s definitely possible, but I know first-hand how tedious it can be. I tried decoding the message given in class by hand because I didn’t have my computer with me.) For class, we discussed that some kind of clue or prior knowledge helps you get started, and that is the general idea with most problems: existing knowledge gives you the direction. Now, use those tolls to get you to the next step.

Technology is capable of helping students, and they might even be able to show us things they have learned. These are times when the teacher may become the student. Technology will free the student to think about problems in a different way. However, we need to show them how to properly use it. If you need to go next door to ask for a cup of sugar, you wouldn’t take your car. (Unless, of course, your closest neighbor happens to be 8 miles away, and walking there would be impossible. Let’s pretend your neighbor is less than 100 feet away.) Now, if you are forced to go to the market (which happens to be 15 miles away), you might want to use your car for efficiency. A computer it the same thing. It is a means to get you from one place to another. Your knowledge gives you the direction and the computer offers you a way to get there efficiently.

We need to allow the real world to collide with school. Too often, schools are trying to keep technology out. Students find school irrelevant because of the tools they have at their disposal. They don’t understand why memorization of facts is necessary when they can just look it up. Students may have a point, and they should be allowed to argue their case.

What do you think about technology and engaging students through use of technology? Let me know what you think, until next time!


Different Paths to the Same Answer

Week 9 Response:

Initial Perception:

How do we start? What do we do? What is the question? It’s all very relative. This week, we were posed this problem: The answer is 10. What is the question? Well, what is it? It can be anything really. The whole point is that the student will give an answer based on what they know. By asking questions like this, we can see what the student already knows. Possible answers could be word problems, equations, a phrase in Spanish, and so on. It depends on where you are coming from and what your background is.

This question was to introduce us to the concept of the initial perception of the problem. Before, we spoke about multiple entry points into a problem, and this is the same idea. How students perceive a problem depends on how the problem is presented to them. Some problems are set up in order to push a student’s thinking in a certain direction, one that might be different than the intended finish line. The question posed about the stickers in the article forced students into making some kind of chart or table rather than thinking of the problem in a more abstract manner, which should have been the idea since we were looking for a formula to generalize the pattern. However, the students will learn to separate the pattern from the thinking.

It is not easy to tailor your problems toward a general classroom because, as I’ve mentioned before in previous posts, no classroom is the same, and you have a diverse group of students in nearly every class. I teach a particular curriculum to three first grade classes. It is the same material, but each class is different. One might easily grasp the concept and questions, but the other class will stare at me like I’m speaking in ancient tongues. I have to adjust for every class because they each approach the problems differently. I’m sure that is no different than other classrooms across the board.

In class, we discussed where teachers want their students, what their goals are, and what to expect in the classroom. The goal of patterns and expressing patterns is supposed to go something like this: obtain the data, see the pattern, explain the pattern, express the pattern, and explain the general case in terms of the variables. Many students seem to tumble and fall when it comes to explaining and expressing their pattern. Most times, they can see the pattern and they know what it is, but they have a hard time putting it together. If they get through that obstacle, then they will fight against the dragon that is explaining the general case in terms of the variables. The problem in most classroom is that there is not much headway in pushing explanations. Once we got the pattern, most are pleased with that. Why would we need those silly letters anyway? Well, the whole point was the generalize. So, maybe we should generalize.

The issue is trying to explain variables to students. What are variables? Well, in this case, we don’t need just letters. We can use the names that we gave to our lists. For example, the problem dealt with # of cubes and # of stickers. Why not write it like it is said? Students may not draw the connection between what they were looking for and some random x and y. So if you are looking for a general equation, might as well say what you know. For example, # of stickers = (# of cubes)4 + 2. This needs to be the goal. Still, people will have a hard time getting to this point. In any case, our new goal might be to use names in the beginning so that later on, students are more comfortable with using the letters to replace the names. This way, they can attach meaning and reason to the names.

There is no set solution to the problem. All we have are ideas and ways to rid ourselves of the endless abyss of problems. There will always be issues, but duct tape works for those smaller fixes.

“Algebra is arithmetic on names,” was another idea we spoke about. Many people just have a hard time associating names and math. Math is in the domain of numbers to most, and bringing in names and letters will throw people off. It is a very odd thing to people. Though, I kept thinking that it is not completely strange. I mean, take a look at technology and computer programming. Taking a jump into an introductory course in Java, one of the first things we learned was storing numbers in names. Essentially, we could code something like: Jonathon + Apples =. Of course, we would need to assign numbers to those names, but the idea is the same. Kids are learning computer languages as early as middle school. Why can’t we expose this stuff to them? The world is changing, and soon enough technology will completely take over. So, combining such technology with mathematical ideas could be quite beneficial since they go hand in hand.

I also think we should focus on offering explanation and more logical ways of thinking. Thinking your way through a problem will make the ideas easier to explain. Yes, it is good to see numbers and exhaust all methods, but it should not be the only way to enter a problem, and problems should be presented as open-ended to allow for different initial perceptions and entry points.

Of course none of this is easy to implement, but the ideas are there. We want to know what is happening when people make connections between the steps in expressing and generalizing patterns. Why can’t people make that jump? Maybe there is more to it than just doing it. There seems to be different processes going on when thinking about a problem in a particular way. If we output a series of numbers, we don’t always reason and make connections between those numbers why they are changing like they are. However, if we take a more logical approach and reason through getting some expression for a pattern, we are revealing to ourselves these connections between the numbers and why we are doing what we are doing within the pattern.

In all, we need to present problems in a way that allows for different approaches so that students don’t get glued to one particular way of thinking and representing things. We also need to take into account the psychology of the student. Some students will not be able to make the jump, and that is the reality. The only thing we can do is give them the tools and the opportunity to do so.

This week in Tweets with Rebecca, I only tweeted out an article I liked. Other than that a few titles caught my eye, but I have yet to read them. They are still saved in my bookmarks.

Let me know what you think about on the initial perception of the problem and how we should present problems to students. I think that seems like a good discussion.

Until next time!

Discussion, Reflection, and Questioning Authority

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!

Empowerment and The Ideal Problem

Week 7: Effective Mathematics Instruction

The reading/video for week 7 dealt with effective mathematics instruction. The main idea defined low level and high level tasks and what constituted as a high or low level task. One part mentioned how we should start with high level, cognitively complex tasks, which I disagree with. You need to gauge topics based on the students in your class. Levels within the class vary. Yes, all students should be given the opportunity to reason and think. However, the level of the students’ needs to be taken into consideration. You don’t want to start with something that will go over their heads. They will get frustrated and give up. Also, you don’t want to give them something that is too easy, because the students will get bored. Rather, testing the waters and giving them a problem with multiple points of entry might be the best course of action. Most classes will have varying levels of students and it is up to the teacher to determine how far to take the challenges. You want to be able to offer something for every student in the class.

The PowerPoint defined low-level tasks as (rote) memorization and procedures without connections. High-level tasks were defined as procedures with connections and “doing” mathematics. The problem with using high level tasks is that the teacher does not completely follow through with its implementation. Student learning will be at its highest when a high-level task is carried out consistently, which seems to be the greatest challenge. In all, I believe that while we should provide different opportunities for students to learn math, we should also consider the level of the students within the classroom setting. It is not as simple as giving the students a challenging problem and expecting them to reason.

In class, we spoke about empowerment, which seems to be our greatest weapon. Essentially, it is giving the ability, or opportunity, to do something that you couldn’t do before. We also touched upon gaining investment in the problem. Some students, especially young ones, don’t care about learning things that deal with taxes. It does not interest them nor is it prevalent in their lives at that point in time. The best type of problem a teacher can offer is one that is engaging, empowering, and demanding. As a student and teacher, I have to agree with that. Most students lose interest in problems that they cannot connect with. Students in my classes have asked why they need to know certain problems, and sometimes, I can’t provide a concrete answer. To me, they are important because I need to know information about taxes and personal finance. As a twelve-year old, they are more concerned with who is going to the dance next week with Johnny or Susan.

Getting through to students is like going through a medieval city. You have to go over every wall before you can reach the center, and it’s not as easy as knocking and asking for entrance. The first wall is grabbing the attention of the students, peaking their interest. The next wall is presenting the complex problem in a way that they are able to approach it. Further into the city, you have to offer them something in order to pass through the next wall. You must give them something that allows for them to be empowered. You need to offer them something they were not able to do before while demanding that they try to figure it out on their own. You need to be sneaky enough to demand more while making it seem like you are giving them more. Only then can you imagine getting through to every layer of the city.

What do you consider to be the ideal problem?

It Has Been a Long While

I keep meaning to post. In fact, I have partial word documents with thoughts of weeks long gone. I was going to make a massive post, but I decided to split them up so as not to overwhelm anybody looking for a quick update.

Week 6: Hacker Gets Hacked

This has to be one of my favorite combination of readings, not only because of the content, but because of the writing. Dr. Keith Devlin’s argument was beautifully executed. Also, his argument reflected many of my thoughts on “Hacker’s Plan” for getting rid of Algebra II. As a mathematics major, getting through Hacker’s article included about fifty eye-rolls and thirty huffs of frustration. He was playing into a common held idea that algebra is completely useless. I can understand and see where he is coming from, but his argument was plagued with incorrect information and half-truths. Also, the part where he says we should replace algebra courses with a statistics course because 47% versus 55% fail…well, that’s about half for both… and that is also quite precise to makes that 47% look better to the general public. There are issues with how algebra is taught, but getting rid of it doesn’t seem to be the best course of action.

Just as I was close to losing it over Hacker’s article, I read Devlin’s article regarding Hacker’s. I went from a ripping-my-hair-out mentality to one of elation. He carefully, and respectfully, deconstructed the layers of Hacker’s argument for ridding the earth, and nearby planets, of algebra. The best part of the argument was that Devlin used Hacker as a springboard to say that Hacker has a point under all of the absurdity. Devlin gives us the real argument regarding algebra, which lies in the skewed idea of what algebra really is. It isn’t just a group of letters without meaning that we have to solve.

In Hacker versus Devlin, we see a mathematician defending his field against an outsider holding common societal views on the meaning of algebra. He mentions the real “villain” of algebra, who happens to be a Frenchman by the name of François Viète. He brought to us the mechanical version of algebra where we are given a set of formulas with which we are to evaluate and enter numbers. (He seems to be the reason why we can’t have nice things.) Instead of thinking logically about problems, it has become a game of meaningless alphabet soup.

I believe the most important part of the article is the fact that people should question information and understand the reasons why we are learning what we learn. On the internet, most people will read an article and run with it, especially if the source is credible. Questioning authority is not commonplace anymore. It is like the vaccination debate. Reading a couple of articles seems to be the equivalent of a doctoral degree in an unrelated subject matter. Andrew Hacker is not a mathematician. He raises some issues, but misses the mark when it comes to the conclusions. As a mathematics major, my trust would lie in a mathematician and my own experiences to formulate an opinion. For non-mathematics majors, I would suggest taking math advice from a person who is immersed in the subject. (Not someone who is playing on the common ideas and negative views that the everyday person has about math, especially algebra.) Sometimes, what you want to hear is not always correct. Open minds and open ears can lead to fields of change. We can’t keep throwing ourselves into echo chambers.

I say that, knowing it won’t happen. Algebra is not the problem. The problem lies in the system and its requirements for how it should be taught. Algebra was intended as a way of thinking, not just the symbols on the page. Here is where I throw in hope of changing the system and that we should be the next generation of teachers to change it and all of that jazz. The reality is that the brunt of changing the system is placed on new teachers and unless we want to start WWIII or the Teacher Revolution (which doesn’t have a nice ring to it), we will not be able to do much. I think something more needs to happen. However, I’m not certain what this something is. Students vary in regard to learning styles and time it takes for understanding. Trying to generalize any specific strategies is nearly impossible. A possible solution is to make changes at the classroom level. It takes time, but if teachers stir in everything they have learned and work around what they are given, then change can potentially happen on a small scale, at least at the beginning. It might not be possible to completely reconstruct the education system, but patching up the frayed pieces seems to be the best option. We have a shell that can be used. We can take parts that work and glue them together to stabilize the fragile framework.