I really enjoyed Zull's discussion in chapter 9 about the need for reflection in order to have an aha! moment. I am beginning to understand how important reflection is, but how little time we set aside for it in our curriculum. We are rushing through so much material with our students that when we feel like we are a little bit ahead we continue to rush on instead of just slowing down and taking some time for reflection. I completely understand the feelings of the teacher on page 161 who complained that his students have no idea if their answers are reasonable or not. So often, I forget that I wasn't always successful in reflecting on an answer and determining whether it made sense or not. I felt that I was lucky if I picked the correct formula from my hat and got an answer at all! Understanding reasonableness is a skill that takes time (in two ways). As students take more and more math courses, they will become better and better at judging and interpreting their answers. In addition, this skill also requires time in the classroom. If it is something we think is important for the education of our students, we must devote time to it in class. With the way things are currently going in the education system, I can see that teachers don't always feel that they have sufficient time to deeply focus on things like interpreting answers. Sometimes we feel successful if we are able to teach our students all of the formulas they are supposed to know for testing purposes. I think it is really sad that we are sort of taking the math out of math education. Students need to understand that sometimes you won't know the answer right away, but that does not mean you are bad at math or that you should just give up! Mathematicians dedicate their lives to solving one problem. Can you imagine the reflection process that must occur for these mathematicians? It's okay to take time to reflect on problems and let them turn over and over in your mind. During all of the time you spend thinking about the problem, your brain is trying to make connections and put together solutions. This is mathematics, and I hope that I can show this to my students.
Zull also has a great discussion in chapter 8 (similar to the discussion in Zadina chapter 6) about how math is made up of two processes: estimation and calculation. Calculation activates areas mainly in the front cortex; interestingly, arithmetic calculations most actively engage the same area that is associated with production of language. We use a different part of our brain, however, when we try to estimate answers. We use the where pathway in the upper back cortex. I hadn't thought about these two processes, estimation and calculation, in these terms before now. Estimation uses the where pathway because in order to estimate you have to be able to compare, which means you need to know where things are with respect to one another. For example, if I ask my students to estimate the number of gumballs in a jar of 100 gumballs, they have to be able to compare numbers on a number line to one another. A student knows that 2 is not the correct answer because it's too small of an estimate and they can clearly count more than 2 in the jar. So much of math depends on this ability to estimate and determine the reasonableness of our answers, but again I feel that we focus so heavily on the calculation aspect that we are taking away from our students' math experience. In my classroom, I hope to strike a good balance between the calculation and estimation aspects of mathematics, and in doing so I hope to help those students who are weak in one area or the other, or possibly both!
Saturday, April 25, 2015
Monday, April 13, 2015
Prior Knowledge is Concrete
I think that Zull did a wonderful job of summarizing the ten key points of chapter 6 on pages 108 and 109. I have heard the phrase "prior knowledge" countless times throughout the last year, and I have been told endlessly that in order to teach a concept you must first "activate a student's prior knowledge." But what is prior knowledge? If I want to teach my students how to solve linear equations, I know that they need to understand multiplication and division beforehand (which is something I feel that I shouldn't have to review before teaching them how to solve linear equations). This seems like common sense. But prior knowledge is so much more than this. Prior knowledge, as stated in Zull chapter 6, is the physical connection of the neuronal networks in our brain. Prior knowledge is not an abstract idea that we expect our students to magically have, it is a concrete connection in the brains of each of our students; and chances are, the connections and prior knowledge are different in each student's brain. Each student is bringing something different to the table. If a student has understood multiplication incorrectly, then the concrete connections in his or her brain have been formed on the basis of incorrect information. Before I can teach that student how to solve linear equations, I must first address the issue of incorrect understanding of multiplication. It would be counterproductive for me to ignore the prior knowledge of my students if there is a problem that I need to address before moving to more complex material.
I also think that Zull's discussion of "the concrete" is very interesting. Zull suggests that "the neuronal networks formed when we sense the outside world are most likely to be similar in each of us; they are created from the same source-the physical world" (p. 102). This neuronal network based on the concrete, outside world is formed prior to our interpretation. So the knowledge forms on the basis of the concrete, which is then left up to the interpretation of the individual. Zull suggests that it is important for the teacher to start with the concrete and then move to the more abstract and theoretical aspects of content. It seems to make so much sense, but this isn't always the way I have approached my lessons. I usually make the real world connection at the end, after the abstract and theory based discussions and learning. After reading this information and better understanding how prior knowledge and neuronal networks are formed, I think it is definitely worth it to try to start a new lesson with a real world example so that the students form their network based on concrete information.
I also think that Zull's discussion of "the concrete" is very interesting. Zull suggests that "the neuronal networks formed when we sense the outside world are most likely to be similar in each of us; they are created from the same source-the physical world" (p. 102). This neuronal network based on the concrete, outside world is formed prior to our interpretation. So the knowledge forms on the basis of the concrete, which is then left up to the interpretation of the individual. Zull suggests that it is important for the teacher to start with the concrete and then move to the more abstract and theoretical aspects of content. It seems to make so much sense, but this isn't always the way I have approached my lessons. I usually make the real world connection at the end, after the abstract and theory based discussions and learning. After reading this information and better understanding how prior knowledge and neuronal networks are formed, I think it is definitely worth it to try to start a new lesson with a real world example so that the students form their network based on concrete information.
Monday, April 6, 2015
A Foundation to Build On
I've discussed some of these thoughts previously on my blog; however, they are the topic of this week's course blog discussion (not to mention a topic that is very interesting to me) so I decided to post about it again... memorization in math. Every educator has his or her opinion about memorization in mathematics. Is it ever necessary? Is it sometimes necessary? Is it the only way the students will pass the course? I happened upon a blog post titled "When Not Memorizing Gets in the Way of Learning." Link: When Not Memorizing Gets in the Way of Learning (Many readers held a discussion of the post at the bottom of the page, it is also very interesting!)
The author, a mathematics teacher named Ben Orlin, discusses how necessary it is for students to learn information that must be memorized. BUT, he also discusses how useless this information is if it is memorized out of context. For example, students in 5th grade math have no business memorizing facts for trigonometry. This is useless information that they have no prior knowledge of and will not continuing building on this knowledge until much further down the road. I share the same opinion as Orlin, in a nutshell. Like Orlin, I argue that the cell phone in your pocket does not take the place of the knowledge in your brain. Without those connections in your brain, without those basic understandings of the information you are learning, Wikipedia in your pocket is useless. You can't search for information if you don't know what you are searching for. This is critical thinking - using the tools you have to find relevant information in order to solve a (sometimes more complex) problem. So if you don't have the foundation to build on, the technology and the internet are going to be useless to you.
In this blog post, Orlin states that "the problem is when memorization spreads like a weed, and begins to substitute for reason. The problem is "ASTC," "FOIL," and other mnemonic shortcuts that circumvent actual mathematical reasoning. The problem is when all of algebra or calculus is reduced to chugging through formulas whose origins and purpose you don't understand. That's when math stops being math, and becomes Following Recipes 101, a far less meaningful and worthwhile class."
I really couldn't have said it better than Orlin. In so many math classes, memorization has become the substitute for reason. And when students get to high school math courses, they have no idea how to reason. They have only been exposed to fact memorization in their mathematics experiences; therefore, they think that fact memorization will continue to prove the best way to learn math. They don't understand that all of those basic concepts they have learned along the way are all coming together in a bigger picture, such as algebra or geometry. For example, algebra is all about letting letters represent unknown numbers. We convert real life situations into numeric equations and math is the tool that allows us to find the missing pieces. We treat these letters just like numbers (they are just holding the place for a number... but we use a letter so we know the number is missing. We could use hearts and smiley faces and it would serve the same purpose, but throughout history certain letters have come to stand for certain missing quantities. We can multiply, divide, add, and subtract these letters just as we can with numbers.) It's amazing really, but an alarming number of students have no idea what algebra is or what the purpose of algebra is... even after they pass algebra. I think that memorization has a place in mathematics, but never without proper context or without conceptual understanding to go along with it. We are doing our students a disservice in math by allowing them to memorize facts and never giving them the opportunity to learn real mathematics.
The author, a mathematics teacher named Ben Orlin, discusses how necessary it is for students to learn information that must be memorized. BUT, he also discusses how useless this information is if it is memorized out of context. For example, students in 5th grade math have no business memorizing facts for trigonometry. This is useless information that they have no prior knowledge of and will not continuing building on this knowledge until much further down the road. I share the same opinion as Orlin, in a nutshell. Like Orlin, I argue that the cell phone in your pocket does not take the place of the knowledge in your brain. Without those connections in your brain, without those basic understandings of the information you are learning, Wikipedia in your pocket is useless. You can't search for information if you don't know what you are searching for. This is critical thinking - using the tools you have to find relevant information in order to solve a (sometimes more complex) problem. So if you don't have the foundation to build on, the technology and the internet are going to be useless to you.
In this blog post, Orlin states that "the problem is when memorization spreads like a weed, and begins to substitute for reason. The problem is "ASTC," "FOIL," and other mnemonic shortcuts that circumvent actual mathematical reasoning. The problem is when all of algebra or calculus is reduced to chugging through formulas whose origins and purpose you don't understand. That's when math stops being math, and becomes Following Recipes 101, a far less meaningful and worthwhile class."
I really couldn't have said it better than Orlin. In so many math classes, memorization has become the substitute for reason. And when students get to high school math courses, they have no idea how to reason. They have only been exposed to fact memorization in their mathematics experiences; therefore, they think that fact memorization will continue to prove the best way to learn math. They don't understand that all of those basic concepts they have learned along the way are all coming together in a bigger picture, such as algebra or geometry. For example, algebra is all about letting letters represent unknown numbers. We convert real life situations into numeric equations and math is the tool that allows us to find the missing pieces. We treat these letters just like numbers (they are just holding the place for a number... but we use a letter so we know the number is missing. We could use hearts and smiley faces and it would serve the same purpose, but throughout history certain letters have come to stand for certain missing quantities. We can multiply, divide, add, and subtract these letters just as we can with numbers.) It's amazing really, but an alarming number of students have no idea what algebra is or what the purpose of algebra is... even after they pass algebra. I think that memorization has a place in mathematics, but never without proper context or without conceptual understanding to go along with it. We are doing our students a disservice in math by allowing them to memorize facts and never giving them the opportunity to learn real mathematics.
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