Week 9 Response:
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!