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.
Thursday, March 26, 2015
"Why do I have to learn this?"
The all too familiar question that I so often experienced from the students over my last two student teaching experiences... "Ms. P, I'm not trying to be rude, but why do we have to learn this?" Looking back on this experience, I realize that my students asked this question because, as Zull (2002) states, "we will always be motivated to learn things that fit into what we want and to resist those that don't..." (p. 65). If students don't see the material as information that aligns with what they want in their lives, then they most likely have little motivation to learn the information. This all makes sense, but I have never thought of it in terms of cognition before. This ties together so many aspects of, in my opinion, an effective lesson. The need for real-world application and the need to incorporate student interest isn't simply because it appeases the administration, these things are necessary for student motivation. And once the students are motivated to learn, there is no telling what potential the lesson has! In addition, providing opportunities for students to use the content, in my case- math, to find errors in each other's work is necessary. This simple task is a real world application of math... you can use math as a tool to make sure you aren't being taken advantage of at the store, or by the plumber, or by the car salesman. Would you rather pay the $23 you expected, or the $46 you were accidentally charged by the salesman? Without math, calculating this error would be impossible. We participate in these tasks all the time, but we might not think of them as a direct application of math until we stop and reflect.
In my classroom, I hope to spend a lot of time discussing the fact that math is a tool that we can use to understand the world around us. Math isn't just a set of procedures you should memorize to move on to the next course, math is a set of strategies that can help us make brilliant findings and invent amazing technology. Without math, you wouldn't have the cell phone you use so often, or the laptop you play on instead of doing your homework, or the medicine that you take when you get sick. Without math, we would know very little about the world around us. And perhaps educators hesitate to have these discussions, "because what if the students ask a question that I can't answer?" I have definitely thought of this scenario and fear that my students will lose respect for me. They will wonder how I can expect them to learn the material if I can't even answer their questions. I think that it's okay for teachers to not know the answer sometimes. Especially when trying to help the students relate the material to the real world. I also think that an appropriate technique for handling the situation would be to take a little time to help the students find the answer. Use the opportunity as a learning experience for both myself and my students, since "the learning cycle is about life itself" (Zull, p. 48).
Interesting article on why we learn math (other cool resources linked at the bottom of the article)...
Why Do We Learn Math?
In my classroom, I hope to spend a lot of time discussing the fact that math is a tool that we can use to understand the world around us. Math isn't just a set of procedures you should memorize to move on to the next course, math is a set of strategies that can help us make brilliant findings and invent amazing technology. Without math, you wouldn't have the cell phone you use so often, or the laptop you play on instead of doing your homework, or the medicine that you take when you get sick. Without math, we would know very little about the world around us. And perhaps educators hesitate to have these discussions, "because what if the students ask a question that I can't answer?" I have definitely thought of this scenario and fear that my students will lose respect for me. They will wonder how I can expect them to learn the material if I can't even answer their questions. I think that it's okay for teachers to not know the answer sometimes. Especially when trying to help the students relate the material to the real world. I also think that an appropriate technique for handling the situation would be to take a little time to help the students find the answer. Use the opportunity as a learning experience for both myself and my students, since "the learning cycle is about life itself" (Zull, p. 48).
Interesting article on why we learn math (other cool resources linked at the bottom of the article)...
Why Do We Learn Math?
Thursday, March 19, 2015
Seeing Patterns
This week I covered some basic statistics with my Algebra I students. I love the brief discussion on week 6: day 4 of Dr. Zadina's workbook. She addresses the fact that the brain is designed to see patterns and to see differences in patterns. This is math in a nutshell- the study of hidden patterns all around us. (Cool article here). So at what point does math get so hard for some students? If the brain literally wants to see patterns and give us a red flag when there is a deviation from the pattern, why is math content sometimes so difficult grasp? All of this made me start thinking about ways that I can present the content to my students in a way that allows them to see patterns and develop the relevant processes from the patterns that they are seeing. Keeping in mind that I simply don't have time to allow my students to "reinvent" concepts on their own, I decided to try a few things that would allow them to come up with at least come up with a process on their own. I know that so often we just hand our students a process or a list of steps and show them how to work the problems, and let's face it- it's the easiest way for us to make sure that we taught the material. But this week I was determined to try.
We started the week by quickly reviewing mean, median, mode, and range. The majority of my students remembered these concepts with no problem. So then we moved on to discuss outliers- what are they and why are outliers relevant to a set of data? More importantly, how do they affect the data? I would show the students a set of data and tell them the outlier. It was up to them to tell me why the particular value was an outlier. I thought this was a great way for their brain to recognize a pattern- the outlier is much less or much greater than the rest of the data set. For example, if all of the numbers in the data set are around 30 and one value is 1,000 this is a deviation from the rest of the data. The students quickly picked up on this and they understood that this value was the outlier. They also realized that this value sometimes greatly affected the mean of the data. If all of the values are near 30, then adding that 1,000 to the data would make the "average" appear higher than it really is.
On the day that we covered standard deviation, I made an effort to give my students the chance to determine the process we needed to use to calculate the standard deviation. I gave the students a handout with a couple of completely worked out examples. I gave them some time to work in a group to try and figure out what was happening with the data set in order to determine the standard deviation. It would have been so easy for me to give them a list of steps and show them exactly what to do. I just felt like it would be much more meaningful for them to try to come up with that process on their own. So after 15 minutes or so of discussion, many of the students were coming up with various processes. I gave them a new example that wasn't worked out, it simply had a data set and the answer to the standard deviation. I asked them to try their steps out on a new example to see if they came up with the correct standard deviation. After a while, we came together as a class to fill out a six-step graphic organizer.
As we filled in the steps, I had the students explain things like "we figured out that you take the square root of 16 because 4 squared gives us the value in the previous step... so then we made sure it worked for both examples." Doing it this way probably took more time than if I had just handed them a set of steps at the beginning of class, but they were SO excited that they figured this out on their own. There was discussion, friendly competition, some disagreement, and it required the students to justify their answers and analyze patterns.
We started the week by quickly reviewing mean, median, mode, and range. The majority of my students remembered these concepts with no problem. So then we moved on to discuss outliers- what are they and why are outliers relevant to a set of data? More importantly, how do they affect the data? I would show the students a set of data and tell them the outlier. It was up to them to tell me why the particular value was an outlier. I thought this was a great way for their brain to recognize a pattern- the outlier is much less or much greater than the rest of the data set. For example, if all of the numbers in the data set are around 30 and one value is 1,000 this is a deviation from the rest of the data. The students quickly picked up on this and they understood that this value was the outlier. They also realized that this value sometimes greatly affected the mean of the data. If all of the values are near 30, then adding that 1,000 to the data would make the "average" appear higher than it really is.
On the day that we covered standard deviation, I made an effort to give my students the chance to determine the process we needed to use to calculate the standard deviation. I gave the students a handout with a couple of completely worked out examples. I gave them some time to work in a group to try and figure out what was happening with the data set in order to determine the standard deviation. It would have been so easy for me to give them a list of steps and show them exactly what to do. I just felt like it would be much more meaningful for them to try to come up with that process on their own. So after 15 minutes or so of discussion, many of the students were coming up with various processes. I gave them a new example that wasn't worked out, it simply had a data set and the answer to the standard deviation. I asked them to try their steps out on a new example to see if they came up with the correct standard deviation. After a while, we came together as a class to fill out a six-step graphic organizer.
As we filled in the steps, I had the students explain things like "we figured out that you take the square root of 16 because 4 squared gives us the value in the previous step... so then we made sure it worked for both examples." Doing it this way probably took more time than if I had just handed them a set of steps at the beginning of class, but they were SO excited that they figured this out on their own. There was discussion, friendly competition, some disagreement, and it required the students to justify their answers and analyze patterns.
Wednesday, March 11, 2015
Focusing on Improvement
In Chapter 8 of our text book, Dr. Zadina discusses the need for talking about results with students rather than only focusing on "number correct." This discussion reminded me of math Sprints that have become an increasingly popular way to work on pattern recognition as well as fluency in basic math skills. I tried a couple of sprints with my classes this week, and they really enjoyed them!
The students have one minute to complete the first page of problems. It begins with "read, set, go" much like a race. After one minute, the teacher calls out the answers to the problems; the students follow along on their paper and say "yes" if they wrote the correct answer. If they wrote the incorrect answer, they simply draw a line through their answer but they don't respond orally. Next, the teacher asks how many students got 1 correct, 2, 3, etc. until only one student's hand remains. The class claps or snaps or gives some other acknowledgement to the winner of that road. The students then take a little time to complete the remaining problems on the page to continue working on the particular skill. The students then get another page of problems and one minute on the clock. After one minute, the teacher again calls out the answers as students respond with a "yes" if they wrote the correct answer. Students add up the number of problems they answered correctly, but instead of writing how many they got correct at the top of their paper they should calculate and write their improvement. The sprints are designed so that students will more quickly recognize the pattern on the second page of problems, which leads to an increased number of correct problems. Instead of recognizing the student who got the most problems correct, the class recognizes the student who showed the most improvement. Chances are, this is a student who wouldn't normally be recognized for a strong performance in math class. Not only did my students get a chance to work on their math skills, they had the opportunity to engage in a friendly competition with their classmates as well as the opportunity to recognize students in the class who sometimes just slip by without being heard.
Here is a link to a more detailed explanation of Sprints:
Friday, March 6, 2015
Activating Multiple Pathways
In week 4 of Dr. Zadina's workbook, she discusses ways in which teachers should try to activate multiple pathways for students so that they can create a more intricate network of the information in the brain. She suggests selecting a lesson and composing a list of various assignments that the students are free to choose from. Keeping in mind her discussion of executive functions in chapter 7 of our textbook, I thought it would be interesting to also incorporate some of those ideas into the lesson.
The topic I chose to work with was box plots. I wanted my students to understand how to read, create, analyze, and compare box plots. The lesson consisted of a brief introduction and a teacher led example of creating a box plot. During this time, we had a whole class discussion about how to read a box plot so that the students would be familiar with this before completing the other activities. After discussing how to read a box plot, we completed an example of creating a box plot. I didn't feel as though this was something that the students would be able to "discover" on their own, which is why I decided to lead an example. I then gave the students a list of assignments that involved reading, creating, analyzing, and comparing box plots. This page outlined their assignments for two class periods. At the end of the second class period, the students understood that they would be responsible for the concepts and would be completing a quiz. Some of the assignments involved worksheets. Some involved task cards in which the students receive 10 or so cards; on the bottom of each card is a problem, on the top of each is an answer. The students choose a card from the stack and work the problem presented, they then look for the answer on the top of another card. Using these cards, they are able to self-assess because they know that if their answer doesn't appear on any of the cards, they made a mistake and need to re-work the problem. We recently made accounts on Khan Academy, so some assignments involved watching a video and completing the assignment using their laptop. Some of the assignments involved creating a foldable in order to learn more about box plots. I asked the students to complete 6 out of the 12 or so assignments on the paper. The page also outlined the focus of each assignment; for example, task cards - reading box plots, Khan Academy activity 1- comparing box plots.
I explained that they were responsible for choosing which assignments to complete and making sure that they kept an eye on time. After the quiz at the end of the second period, we had a discussion about which activities they enjoyed the most and which activities helped them understand the material the best. My students made comments like "well since I really understood how to read box plots right after you did an example, I thought it would be better to spend my time on making the box plots and comparing them instead of doing the assignments that just wanted me to read them." It was awesome! So the activity even involved metacognition; I felt like my students were thinking about their own learning and getting to know the ways that they learned best. Is it by working basic problems and leading to more difficult ones, or is it working difficult problems and figuring out the mistakes they are making? I felt like it was a great assignment and I would definitely try it again!
The topic I chose to work with was box plots. I wanted my students to understand how to read, create, analyze, and compare box plots. The lesson consisted of a brief introduction and a teacher led example of creating a box plot. During this time, we had a whole class discussion about how to read a box plot so that the students would be familiar with this before completing the other activities. After discussing how to read a box plot, we completed an example of creating a box plot. I didn't feel as though this was something that the students would be able to "discover" on their own, which is why I decided to lead an example. I then gave the students a list of assignments that involved reading, creating, analyzing, and comparing box plots. This page outlined their assignments for two class periods. At the end of the second class period, the students understood that they would be responsible for the concepts and would be completing a quiz. Some of the assignments involved worksheets. Some involved task cards in which the students receive 10 or so cards; on the bottom of each card is a problem, on the top of each is an answer. The students choose a card from the stack and work the problem presented, they then look for the answer on the top of another card. Using these cards, they are able to self-assess because they know that if their answer doesn't appear on any of the cards, they made a mistake and need to re-work the problem. We recently made accounts on Khan Academy, so some assignments involved watching a video and completing the assignment using their laptop. Some of the assignments involved creating a foldable in order to learn more about box plots. I asked the students to complete 6 out of the 12 or so assignments on the paper. The page also outlined the focus of each assignment; for example, task cards - reading box plots, Khan Academy activity 1- comparing box plots.
I explained that they were responsible for choosing which assignments to complete and making sure that they kept an eye on time. After the quiz at the end of the second period, we had a discussion about which activities they enjoyed the most and which activities helped them understand the material the best. My students made comments like "well since I really understood how to read box plots right after you did an example, I thought it would be better to spend my time on making the box plots and comparing them instead of doing the assignments that just wanted me to read them." It was awesome! So the activity even involved metacognition; I felt like my students were thinking about their own learning and getting to know the ways that they learned best. Is it by working basic problems and leading to more difficult ones, or is it working difficult problems and figuring out the mistakes they are making? I felt like it was a great assignment and I would definitely try it again!
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