When the administrators at my school suggested that I adopt a more traditional "spoon feeding" approach to instruction I was taken a bit by surprise. It is one thing to say that my approach is not effective at helping students learn problem solving skills. But that isn't what they said. They said that focusing on these skills would harm the students. Why would anyone think this?
It is important to recognize that people don't always say what they mean. In this case, there are a number of factors that the administrators are concerned about that motivated them to talk to me but were not the focus of the conversation.
First, there are several students that don't like how my class is run, and they complain to the administration. Their parents complain, too. This makes life more difficult for administrators.
Second, the administration is worried about core tests. Schools are judged to a large extent based on how students perform on core tests, and they know that core tests aren't designed to test problem solving ability. If I spend class time teaching problem solving skills there is less time to practice formulas and algorithms.
Third, they don't actually think that teaching problem solving skills works. One of the beautiful things about mathematics is that you don't actually have to memorize a lot of stuff. You have to understand the basics and the rest can be derived. But the process of using basic skills to solve complex problems is not easy. In my conversation with the administration they mentioned that another teacher who used to work at the school tried to teach from a problem solving perspective for 10 years, but that teacher's scores were always lower than average. What if teaching problem solving simply doesn't work?
I tend to think that the first issue was the biggest motivator behind my administration's suggestion. They get worried when parents complain. But the last question is the one I am most interested in. What if efforts to teach problem solving aren't effective. What if I am sacrificing time practicing formulas and I am not getting anything in return?
Unfortunately, I can't give a definitive answer to this question. It is more a matter of faith for me at this point. I think that a lot of the formulas and algorithms we learn are dispensable, so I don't feel too bad about spending less time memorizing them. But I have little hard evidence that the time is better spent on problem solving. It would certainly be depressing if this was something that couldn't be taught.
There are a few studies indicating that problem solving instruction does work (this paper gives an overview of some of the research). But I wouldn't say there is enough evidence to say that we know for sure. It was interesting that the quality of their thought process improved on my last quiz simply by introducing a rubric that emphasized problem solving. But all this shows is that students have some inherent problem solving skills that we aren't really requiring them to access. What I would really like to know is whether their ability will improve given some focused instruction.
Sunday, January 29, 2012
Monday, January 23, 2012
Spoon Feed or Die
Our quarter ended last week and I had the opportunity to grade my first set of exams based on a new rubric I introduced to the students to get them to feel comfortable approaching challenging problems. I gave them points for things like asking good questions, guess and check, making conjectures and testing them, drawing good pictures, etc.
The results were pretty remarkable. Students in geometry that had previously left any question that required more than one step blank worked 45 minutes to an hour on a very challenging problem. They experimented, followed through on their conjectures, and came up with a variety of creative approaches to the problem. It gave me a lot of insight into how they think, and what their strengths and weaknesses are. I actually enjoyed grading the papers, even though it was a lot more difficult than passing an answer sheet through a scantron.
In my Algebra II classes I gave the students the option of taking a challenging open ended exam with three questions or a traditional multiple choice with 45 questions. In one class, about a third chose the open ended exam and in my other class about 2/3rds tried it. Again, I was very happy that a number of students who struggled to understand quadratic equations came up with creative ways of approaching some very difficult problems. They didn't necessarily get them all right, but they opened up and engaged to a very high degree.
This doesn't mean that everything is going peachy. Today I had three classes, and one was feeling very reluctant to participate. If I expect the students to engage in a problem solving process that involves them overcoming feelings of confusion, a little opposition can kill the atmosphere.
As if the challenge of trying to convince my students and their parents that problem solving is a good idea, it seems that I also have some convincing to do with regard to my administration. I was called in to talk to the principal and the VP today, and the message was basically an ultimatum: spoon feed or else. Some students are not happy with the idea that they have to learn a new way of doing math, and their parents go straight to the principal.
I think it is part of the responsibility of a teacher, especially at my school, to help make the students and their parents feel good about what is going on the classroom. I spend a lot of time trying to make things work for those students that are having trouble but want to learn. Should I give in to the pressure and revert to a more traditional mix of direct instruction and worksheets? Not really my style.
The results were pretty remarkable. Students in geometry that had previously left any question that required more than one step blank worked 45 minutes to an hour on a very challenging problem. They experimented, followed through on their conjectures, and came up with a variety of creative approaches to the problem. It gave me a lot of insight into how they think, and what their strengths and weaknesses are. I actually enjoyed grading the papers, even though it was a lot more difficult than passing an answer sheet through a scantron.
In my Algebra II classes I gave the students the option of taking a challenging open ended exam with three questions or a traditional multiple choice with 45 questions. In one class, about a third chose the open ended exam and in my other class about 2/3rds tried it. Again, I was very happy that a number of students who struggled to understand quadratic equations came up with creative ways of approaching some very difficult problems. They didn't necessarily get them all right, but they opened up and engaged to a very high degree.
This doesn't mean that everything is going peachy. Today I had three classes, and one was feeling very reluctant to participate. If I expect the students to engage in a problem solving process that involves them overcoming feelings of confusion, a little opposition can kill the atmosphere.
As if the challenge of trying to convince my students and their parents that problem solving is a good idea, it seems that I also have some convincing to do with regard to my administration. I was called in to talk to the principal and the VP today, and the message was basically an ultimatum: spoon feed or else. Some students are not happy with the idea that they have to learn a new way of doing math, and their parents go straight to the principal.
I think it is part of the responsibility of a teacher, especially at my school, to help make the students and their parents feel good about what is going on the classroom. I spend a lot of time trying to make things work for those students that are having trouble but want to learn. Should I give in to the pressure and revert to a more traditional mix of direct instruction and worksheets? Not really my style.
Monday, January 16, 2012
Relational and Instrumental Understanding (and the role of objectives)
I am starting a new semester at the U. I have one class, and I am not quite sure what the name of it is. Something about mathematics curriculum.
But one of our first readings this semester is an article which distinguishes between two kinds of understanding: relational and instrumental. Relational understanding means being aware of connections between ideas, basically what you think the word 'understanding' should mean. Instrumental understanding means knowing a formula or process that should get you a result in a specific kind of situation.
The author contends that problems can occur when teachers want to teach relational understanding and students expect instrumental understanding, or vice versa.
Hey, that sounds familiar. The students who complain about my new approach in geometry are generally saying: we want instrumental understanding.
The fact that they have been taught that way for so many years and they have come to be good at it means that I have to make a special effort to get them into a new mindset. Not only do I have to tell them that problem solving and real understanding are the goal, I have to send them messages with the way we spend time in class, the assignments I give them, and the way they are tested.
The author mentions that one of the advantages of instrumental understanding is that it produces more "results" more quickly, and that one of the challenges to teaching to relational understanding is that the curriculum is too packed in many courses. We don't have enough time.
Something related that has been on my mind recently is the effect of having teachers state objectives for each lesson. On the surface, stating your objective is unequivocally a good thing. But it can have some unintended side effects. What if the real objectives are hard to quantify? It is easier to state instrumental objectives, so focusing on objectives can make it more likely for teachers to plan their lessons around them.
Of course, it is possible to create better objectives, and not having objectives may lead to less focused teaching. But what if less focused teaching is actually a good thing? What if it is okay to be unsure of exactly what students are going to get out of a lesson?
But one of our first readings this semester is an article which distinguishes between two kinds of understanding: relational and instrumental. Relational understanding means being aware of connections between ideas, basically what you think the word 'understanding' should mean. Instrumental understanding means knowing a formula or process that should get you a result in a specific kind of situation.
The author contends that problems can occur when teachers want to teach relational understanding and students expect instrumental understanding, or vice versa.
Hey, that sounds familiar. The students who complain about my new approach in geometry are generally saying: we want instrumental understanding.
The fact that they have been taught that way for so many years and they have come to be good at it means that I have to make a special effort to get them into a new mindset. Not only do I have to tell them that problem solving and real understanding are the goal, I have to send them messages with the way we spend time in class, the assignments I give them, and the way they are tested.
The author mentions that one of the advantages of instrumental understanding is that it produces more "results" more quickly, and that one of the challenges to teaching to relational understanding is that the curriculum is too packed in many courses. We don't have enough time.
Something related that has been on my mind recently is the effect of having teachers state objectives for each lesson. On the surface, stating your objective is unequivocally a good thing. But it can have some unintended side effects. What if the real objectives are hard to quantify? It is easier to state instrumental objectives, so focusing on objectives can make it more likely for teachers to plan their lessons around them.
Of course, it is possible to create better objectives, and not having objectives may lead to less focused teaching. But what if less focused teaching is actually a good thing? What if it is okay to be unsure of exactly what students are going to get out of a lesson?
Sunday, January 15, 2012
Problem Solving Rebellion
In addition to making my new website, over the break I did a lot of thinking about my teaching practice. Basically, I was frustrated. I felt like I was fighting the students. I wanted them to be problem solvers and they wanted me to spoon feed them information.
I decided that if this teaching thing is going to work, I have to make some changes. So with the new year I decided to try out some new methods. I started by having a little bit of a talk with each of my classes. I told them that what I really care about is not that they memorize a bunch of formulas and steps, but that they learn how to solve problems.
I told them that as teachers we send them the wrong message. We give assignments and tests with 30 questions on them, knowing that the student only has an hour or so to work them. This sends the message that each question should be done in 2 minutes or less. We give them worksheets with a half inch of room to work problems. We hurry through lessons, answering every question as quickly as possible so we can cover all of the material.
All of this tells the student that math should be quick and automatic. But that isn't really math. Math is about solving tough problems, making connections and asking questions. It is about making mistakes and recognizing that our answer can't be right. It takes time.
So, I offered my students a deal. If they agreed to engage in the process of problem solving I would give them fewer problems on tests and homework. If they don't want to engage on a given day, they can sit at a table outside my class where I can see them and work independently. But when they come into my class, they will be expected to exhibit certain behaviors that I wrote on the board:
1) Draw pictures
2) Write down formulas that might be relevant
3) Write down questions and conjectures
4) Guess and check
5) Look in their notes/book/online for answers to questions
6) Make tables of data
7) Compare results and questions with friends and discuss differences.
I also told them that from now on they are not allowed to do more than one question on a page, so that we are not tempted to cram our work. For the rest of the year my goal is not to have them learn every formula. My goal is to learn how to implement these problem solving strategies.
The response was pretty awesome. A number of students openly questioned my proposed changes:
"Math is about learning formulas and applying them," one said. "You should just teach us the formulas and the steps."
"That isn't how we are going to do math in this class anymore," was my response. "If you choose not to participate, you can sit outside at the table and work independently."
Two or three students from each class left. Three of them were sitting out there when my principal walked by and started talking to them. When I got a chance, I went outside to hear what they were talking about. The principal was trying to help them resolve concerns about the problem solving methods I was proposing, so I just nodded and went back inside.
The classroom discussions were amazing. I had the students work on a question for about 10 minutes on their own. When they had questions I would simply try to help them phrase their questions in a way that they could write down on their paper. After a while I had them get up and compare results with their classmates. They compared results, answered each other's questions, argued about different methods, and tried to come to a consensus as a class. One student that literally failed the last test led a class discussion about whether we should choose sin or cos to solve a certain question.
I tried to emphasize that the things I had written on the board were not a formula. They were skills that we had to develop. How do we draw pictures that can focus our attention and get us closer to our goal? What are examples of useful and not useful questions? (e.g. "how do I solve this problem?" is not useful) A number of students in each class came up afterward and said that they really enjoyed the new approach.
I am pretty energized about teaching right now, and I hope that the newfound energy in me and my students doesn't wear off too soon. The idea that I may have found a teaching style that really works for me feels great.
I decided that if this teaching thing is going to work, I have to make some changes. So with the new year I decided to try out some new methods. I started by having a little bit of a talk with each of my classes. I told them that what I really care about is not that they memorize a bunch of formulas and steps, but that they learn how to solve problems.
I told them that as teachers we send them the wrong message. We give assignments and tests with 30 questions on them, knowing that the student only has an hour or so to work them. This sends the message that each question should be done in 2 minutes or less. We give them worksheets with a half inch of room to work problems. We hurry through lessons, answering every question as quickly as possible so we can cover all of the material.
All of this tells the student that math should be quick and automatic. But that isn't really math. Math is about solving tough problems, making connections and asking questions. It is about making mistakes and recognizing that our answer can't be right. It takes time.
So, I offered my students a deal. If they agreed to engage in the process of problem solving I would give them fewer problems on tests and homework. If they don't want to engage on a given day, they can sit at a table outside my class where I can see them and work independently. But when they come into my class, they will be expected to exhibit certain behaviors that I wrote on the board:
1) Draw pictures
2) Write down formulas that might be relevant
3) Write down questions and conjectures
4) Guess and check
5) Look in their notes/book/online for answers to questions
6) Make tables of data
7) Compare results and questions with friends and discuss differences.
I also told them that from now on they are not allowed to do more than one question on a page, so that we are not tempted to cram our work. For the rest of the year my goal is not to have them learn every formula. My goal is to learn how to implement these problem solving strategies.
The response was pretty awesome. A number of students openly questioned my proposed changes:
"Math is about learning formulas and applying them," one said. "You should just teach us the formulas and the steps."
"That isn't how we are going to do math in this class anymore," was my response. "If you choose not to participate, you can sit outside at the table and work independently."
Two or three students from each class left. Three of them were sitting out there when my principal walked by and started talking to them. When I got a chance, I went outside to hear what they were talking about. The principal was trying to help them resolve concerns about the problem solving methods I was proposing, so I just nodded and went back inside.
The classroom discussions were amazing. I had the students work on a question for about 10 minutes on their own. When they had questions I would simply try to help them phrase their questions in a way that they could write down on their paper. After a while I had them get up and compare results with their classmates. They compared results, answered each other's questions, argued about different methods, and tried to come to a consensus as a class. One student that literally failed the last test led a class discussion about whether we should choose sin or cos to solve a certain question.
I tried to emphasize that the things I had written on the board were not a formula. They were skills that we had to develop. How do we draw pictures that can focus our attention and get us closer to our goal? What are examples of useful and not useful questions? (e.g. "how do I solve this problem?" is not useful) A number of students in each class came up afterward and said that they really enjoyed the new approach.
I am pretty energized about teaching right now, and I hope that the newfound energy in me and my students doesn't wear off too soon. The idea that I may have found a teaching style that really works for me feels great.
CongruenceProof.com
Over the Christmas break I began a little project as an extension of something I had done with my students to learn the concept of proof. I had the students make cards representing the theorems they know and color coded the conditions and consequences so they could make proofs by matching the cards. My project was to make the idea into a website. Here it is:
CongruenceProof.com
It is still in "beta", but it is functional (I hope). Go check it out and do a proof yourself.
The idea behind the site is that students don't actually understand the idea that theorems represent "if-then" statements. A lot of math is about making connections between things we already know, so I wanted a very simple visual representation of those connections (matching colors). See if you think it is intuitive. There is also an instructional video if you need a bit of help interpreting the site.
CongruenceProof.com
It is still in "beta", but it is functional (I hope). Go check it out and do a proof yourself.
The idea behind the site is that students don't actually understand the idea that theorems represent "if-then" statements. A lot of math is about making connections between things we already know, so I wanted a very simple visual representation of those connections (matching colors). See if you think it is intuitive. There is also an instructional video if you need a bit of help interpreting the site.
Sunday, January 8, 2012
Update on Puerto Rico
It is still part of our country.
Anyway, after my dispute with the Wal-Mart staff regarding whether Puerto Rico is part of the U.S., I submitted a complaint online outlining the situation. Within 24 hours, I got a phone call from the local store apologizing for how the incident was handled. So, I asked, what is the status of your policy?
The response: We still can't accept ID's from Puerto Rico, but we have requested an update to the policy because we get a lot of foreigners here wanting to purchase cigarettes and other items.
Sigh. I told them to get back to me once they heard back from corporate. I figured that continued resistance is futile. Still, I am impressed by the quick response.
Also, I have decided that this gives me even more reason to shop at Wal-Mart. Clearly they do not hire employees who are overqualified for their position. If someone doesn't need to know history or geography to function at an acceptable level of efficiency in a given position, they hire someone who doesn't know much about history or geography. Even more reason to trust that they are doing their very best to give me the lowest prices.
I wonder what would happen if I asked the staff at Whole Foods whether they accept ID from Puerto Rico?
Update to the update: the Wal-Mart regional supervisor has directed the local store to add Puerto Rico to the list of acceptable ID's. Time between complaint and resolution: 48 hrs.
Anyway, after my dispute with the Wal-Mart staff regarding whether Puerto Rico is part of the U.S., I submitted a complaint online outlining the situation. Within 24 hours, I got a phone call from the local store apologizing for how the incident was handled. So, I asked, what is the status of your policy?
The response: We still can't accept ID's from Puerto Rico, but we have requested an update to the policy because we get a lot of foreigners here wanting to purchase cigarettes and other items.
Sigh. I told them to get back to me once they heard back from corporate. I figured that continued resistance is futile. Still, I am impressed by the quick response.
Also, I have decided that this gives me even more reason to shop at Wal-Mart. Clearly they do not hire employees who are overqualified for their position. If someone doesn't need to know history or geography to function at an acceptable level of efficiency in a given position, they hire someone who doesn't know much about history or geography. Even more reason to trust that they are doing their very best to give me the lowest prices.
I wonder what would happen if I asked the staff at Whole Foods whether they accept ID from Puerto Rico?
Update to the update: the Wal-Mart regional supervisor has directed the local store to add Puerto Rico to the list of acceptable ID's. Time between complaint and resolution: 48 hrs.
Tuesday, January 3, 2012
Puerto Rico
Today I got into a debate with the staff of WalMart over the question of whether Puerto Rico is part of the United States. They refused to sell a man cigarettes because he had a driver's license from the island, and they only accept US ID cards. I couldn't convince them of their folly, and eventually left in exasperation. afterward I wrote a letter to corporate asking them to add a picture of a Puerto Rico license to the accepted documents list so their staff would understand. But maybe they have a secret reason for denying cigarettes to these customers.
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