"The first things i learned about mathematics was that there is a hell of a difference between, on the one hand, listening to math being talked about by someone else and thinking that you are understanding, and, on the other, thinking about math and understanding it yourself and talking about it to someone else." Sarah Flannery
Reuben Hirsch from his book What Is Mathematics Really? "Mathematics is learned by computing, by solving problems, and by conversing, more than by reading and listening."
It is fair to say that mathematicians enjoy doing mathematics. We enjoy solving puzzles, working on ideas, making connections, and all that. It is the so called "heart of the matter," and one of our goals as instructors is to make this happen for all our students, at whatever level they are at.
Thursday, February 23, 2012
Wednesday, February 15, 2012
IBL Instructor Perspectives: Professor Dana Ernst
A Q&A session with Professor Dana Ernst, Department of Mathematics, Plymouth State University. Dana is an award-winning, highly energetic math professor, who has recently learned to teach using IBL. In this blog post, we ask him a handful of questions related to his IBL teaching.
1. How long have you been teaching and what was your teaching style before you started using IBL?
I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1998. My classes have always been interactive, but initially they were predominately lecture-based. Probably like most teachers, I modeled my teaching style after my favorite teachers. I was aware of the Moore method and IBL, but I had never experienced this paradigm as a student. By most metrics, my approach in the classroom seemed to be working. My teaching evaluations have been consistently high and I have received several teaching awards. However, prior to implementing IBL, I was suspicious that I could get so much more out of my teaching. More precisely, I was suspicious that I could provide my students with the opportunity to get so much more out of my classes.
2. How did you learn about IBL and when did you begin using it in your classes?
It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to consider using IBL. Carol's workshop was about implementing a modified-Moore method approach in an undergraduate real analysis course. I do not remember the details of the workshop, but by the end, I was inspired to give IBL a shot. In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course. The course I chose is called Logic, Proof, and Axiomatic Systems, which is meant to be the introduction to proof course at Plymouth State University. Perhaps surprisingly (since it was my first go), the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student's learning and character development. I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me. I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well. However, the IBL version was a vast improvement. I have since had students from all three variations in upper-level proof-based courses and the students from the IBL version are much more independent and, in general, better proof-writers.
3. What was one of your best IBL experiences?
There are so many! Here is one event that illustrates why I am hooked on IBL. During the Fall 2011 semester, I was chosen for jury duty, which required me to miss six days of classes. For all but two of these days, I was able to find a faculty member to cover my classes. For the two days that I did not have faculty coverage, I convinced a graduate student in education to cover my IBL abstract algebra course. This student had taken my IBL introduction to proof course, so he had some IBL experience, but he had never had an abstract algebra course. On the days the graduate student covered for me, the class ran as usual and the students were highly productive. They didn't need me! The students were so proud of what they achieved while I was gone, they sent me pictures of the work that was presented on the board. The graduate student that covered for me indicated that all he had to do was jot down who came to the board to present.
4. What advice do you have for new IBL instructors?
In order for IBL to be successful, the students have to buy into it. To pull this off, instructor need to do some marketing at the beginning of semester. The right amount of marketing varies from class to class and semester to semester. For classes filled with students with prior IBL experience, I don't have to convince them of the benefits of IBL. These students are generally ready to dive in and get started. For classes consisting of students that are new to IBL, it is important to explicitly spell out the format, expectations, and goals of the course. One approach I take on the first day is to ask the students what skills a college student, specifically a math major, should have upon graduating and how best to acquire these skills. Through some Socratic questioning, I am able to get them to tell me that we should be doing something like IBL. I'm not trying to trick them, but rather give them some ownership in the philosophy behind the structure of the course.
Even if the students are sold on IBL, you still have to be willing to adapt, overcome, and improvise. Issues will come up that you couldn't have predicted. Building a community of trust will make any challenges a lot easier to deal with. I believe that the two most important qualities of an IBL instructor, heck any instructor, are patience and being "Mr./Mrs. Friggin' Positive." Lastly, I would like to echo something Ed Parker has said. Sit back, shut up, and "see what they can do".
[Added by Stan Yoshinobu] Summary:
1. How long have you been teaching and what was your teaching style before you started using IBL?
I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1998. My classes have always been interactive, but initially they were predominately lecture-based. Probably like most teachers, I modeled my teaching style after my favorite teachers. I was aware of the Moore method and IBL, but I had never experienced this paradigm as a student. By most metrics, my approach in the classroom seemed to be working. My teaching evaluations have been consistently high and I have received several teaching awards. However, prior to implementing IBL, I was suspicious that I could get so much more out of my teaching. More precisely, I was suspicious that I could provide my students with the opportunity to get so much more out of my classes.
2. How did you learn about IBL and when did you begin using it in your classes?
It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to consider using IBL. Carol's workshop was about implementing a modified-Moore method approach in an undergraduate real analysis course. I do not remember the details of the workshop, but by the end, I was inspired to give IBL a shot. In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course. The course I chose is called Logic, Proof, and Axiomatic Systems, which is meant to be the introduction to proof course at Plymouth State University. Perhaps surprisingly (since it was my first go), the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student's learning and character development. I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me. I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well. However, the IBL version was a vast improvement. I have since had students from all three variations in upper-level proof-based courses and the students from the IBL version are much more independent and, in general, better proof-writers.
3. What was one of your best IBL experiences?
There are so many! Here is one event that illustrates why I am hooked on IBL. During the Fall 2011 semester, I was chosen for jury duty, which required me to miss six days of classes. For all but two of these days, I was able to find a faculty member to cover my classes. For the two days that I did not have faculty coverage, I convinced a graduate student in education to cover my IBL abstract algebra course. This student had taken my IBL introduction to proof course, so he had some IBL experience, but he had never had an abstract algebra course. On the days the graduate student covered for me, the class ran as usual and the students were highly productive. They didn't need me! The students were so proud of what they achieved while I was gone, they sent me pictures of the work that was presented on the board. The graduate student that covered for me indicated that all he had to do was jot down who came to the board to present.
4. What advice do you have for new IBL instructors?
In order for IBL to be successful, the students have to buy into it. To pull this off, instructor need to do some marketing at the beginning of semester. The right amount of marketing varies from class to class and semester to semester. For classes filled with students with prior IBL experience, I don't have to convince them of the benefits of IBL. These students are generally ready to dive in and get started. For classes consisting of students that are new to IBL, it is important to explicitly spell out the format, expectations, and goals of the course. One approach I take on the first day is to ask the students what skills a college student, specifically a math major, should have upon graduating and how best to acquire these skills. Through some Socratic questioning, I am able to get them to tell me that we should be doing something like IBL. I'm not trying to trick them, but rather give them some ownership in the philosophy behind the structure of the course.
Even if the students are sold on IBL, you still have to be willing to adapt, overcome, and improvise. Issues will come up that you couldn't have predicted. Building a community of trust will make any challenges a lot easier to deal with. I believe that the two most important qualities of an IBL instructor, heck any instructor, are patience and being "Mr./Mrs. Friggin' Positive." Lastly, I would like to echo something Ed Parker has said. Sit back, shut up, and "see what they can do".
[Added by Stan Yoshinobu] Summary:
- Go to a workshop if you can.
- We can all improve our teaching!
- We are not just teaching facts, we are developing appropriate habits of mind, such as independence
- Marketing to students (i.e. getting them on board with an IBL class) at the beginning of term is one of the keys to success.
- Be flexible! Be patient! Be super positive!!
Thanks Dana!
Thursday, February 9, 2012
Architecture and Education Part 2
In part 1 I described an ideal situation where the physical space of a building can encourage learning. Now I turn to a pragmatic issue -- how to make your classroom physically and mentally a good learning space. Small groups are useful in a wide variety of situations. Although some IBL instructors forbid collaboration of the group work kind, many instructors use group work highly successfully. Both style have merit, and what I highlight here are ways to setup a classroom, should you choose to use group work (which I recommend as a good way to get started using IBL methods).
Physically most of our college classrooms are setup with the factory model. Students are sitting in rows.
Non-physical caveat: The buildings, setting up groups, desks, these are all physical and can assist with developing a healthy, productive learning environment. By themselves they are not sufficient of course. Instructor skill, leadership, facilitation, and coaching are major drivers and can overcome almost any physical boundaries.
Ideally we would have small tables in one part of the room for small group work, and chairs in another part of the room for whole-group discussions and presentations. I hope that future buildings are designed for class discussions, collaboration in small groups, as well as some private space for those times when students need to think by themselves. This is doable and low cost.
In a room like this, the desks can be moved. In this case you can easily ask students to move their desks into small groups of size two to four. One suggestion, if possible, is to arrange it so that it is easy for you to walk through the classroom. Creating a boulevard in the middle allows you to walk down the boulevard and get to all the groups (e.g. ask the first two rows, and last two rows to move together) .
If your room has fixed desks, all is not lost. You have to ask students to work with neighboring students and get them to turn their bodies at face on another. It's not ideal, but it works.
Don't be shy about moving people. This is part of IBL instruction -- move students to where they will be successful! It is better to be mildly intrusive than to allow other factors to inhibit learning.
Non-physical caveat: The buildings, setting up groups, desks, these are all physical and can assist with developing a healthy, productive learning environment. By themselves they are not sufficient of course. Instructor skill, leadership, facilitation, and coaching are major drivers and can overcome almost any physical boundaries.
Ideally we would have small tables in one part of the room for small group work, and chairs in another part of the room for whole-group discussions and presentations. I hope that future buildings are designed for class discussions, collaboration in small groups, as well as some private space for those times when students need to think by themselves. This is doable and low cost.
Wednesday, February 1, 2012
Teaching Tips: Chaos
Another title for this post is "Chaos" vs Chaos. True chaos is not appropriate in a classroom for obvious reason. As an instructor transitions from lecture to IBL methods one has to recalibrate what and orderly classroom looks and sounds like.
Talking -- there will be more talking, especially if one uses collaborative groups. If the tasks are implemented appropriately, then there will be noise. This is a good thing. Listen to the discussions about math and get insights into how students think.
Side note: If students are talking about other stuff, redirect. Ask them them what they tried, if they can work on the next problem, give them another task,...
Non-textbook route -- students who are learning math should not be expected to produce clean proofs like a seasoned, professional mathematician. It's hard for instructors to remember the challenges we faced when we were learning ideas for the first time. To instructors the work looks chaotic, disorganized, messy. But that is a fact of life. Learning is messy and nonlinear. That's a good sign. Students are trying and working and building and exploring. If there's nothing on the page, that's when you need to step in and provide guidance and mentoring. When there's stuff happening, keep an eye on them, but don't mess it up by intervening.
For pure Modified Moore Method classes, such interactions in classrooms as described above may not happen. That's fine. You know it is happening outside of class, but it is important to know and be conscience of the fact that students are struggling. If they come in for help, then it is our job as instructors to listen compassionately and understand that the students asking for help are stuck and are developing. We can shape the path for them without giving away answers, and we can be supportive and point out all of the positive steps they have taken.
The purpose of class discussions, presentations, homework, portfolios, etc. is to put all this chaos into a final form that makes sense. Our goal is to produce mathematics, and the way mathematics is produced is by proving or justifying why statements are true. Class discussions, presentations, homework, portfolios are some ways to channel the chaos into a product.
Short story: Embrace good chaos. Channel good chaos into a final product of some sort. Don't expect students to take your path to a proof or solution (i.e. what you think is chaos might just be another way). Help and support those who need it without diminishing the intellectual value of the task.
Note: If the homework collected is not high quality, use homework templates to encourage your students to improve their process.
Talking -- there will be more talking, especially if one uses collaborative groups. If the tasks are implemented appropriately, then there will be noise. This is a good thing. Listen to the discussions about math and get insights into how students think.
Side note: If students are talking about other stuff, redirect. Ask them them what they tried, if they can work on the next problem, give them another task,...
Non-textbook route -- students who are learning math should not be expected to produce clean proofs like a seasoned, professional mathematician. It's hard for instructors to remember the challenges we faced when we were learning ideas for the first time. To instructors the work looks chaotic, disorganized, messy. But that is a fact of life. Learning is messy and nonlinear. That's a good sign. Students are trying and working and building and exploring. If there's nothing on the page, that's when you need to step in and provide guidance and mentoring. When there's stuff happening, keep an eye on them, but don't mess it up by intervening.
For pure Modified Moore Method classes, such interactions in classrooms as described above may not happen. That's fine. You know it is happening outside of class, but it is important to know and be conscience of the fact that students are struggling. If they come in for help, then it is our job as instructors to listen compassionately and understand that the students asking for help are stuck and are developing. We can shape the path for them without giving away answers, and we can be supportive and point out all of the positive steps they have taken.
The purpose of class discussions, presentations, homework, portfolios, etc. is to put all this chaos into a final form that makes sense. Our goal is to produce mathematics, and the way mathematics is produced is by proving or justifying why statements are true. Class discussions, presentations, homework, portfolios are some ways to channel the chaos into a product.
Short story: Embrace good chaos. Channel good chaos into a final product of some sort. Don't expect students to take your path to a proof or solution (i.e. what you think is chaos might just be another way). Help and support those who need it without diminishing the intellectual value of the task.
Note: If the homework collected is not high quality, use homework templates to encourage your students to improve their process.
Tuesday, January 24, 2012
Physicists Seek To Lose The Lecture As Teaching Tool
Note: Thank you Matt Jones, CSU Dominguez Hills for the heads up.
Emily Hanford of NPR has a nice article about physicists, who are lecturing less and getting better results. Be sure to also click on the link "Listen to the Story"for the radio broadcast.
http://www.npr.org/2012/01/01/144550920/physicists-seek-to-lose-the-lecture-as-teaching-tool
Physicists have collected data providing evidence that learning gains are better when students are deeply engaged in the subject and allowed to collaborate with peers. Sandra Laursen, University of Colorado Boulder, also has data collected at the college level from mathematics courses. Laursen's results are in alignment with this article. Laursen's group has a number of findings. Among the long list of findings, there is statistically significant evidence suggesting the more an instructor talks the less their students learn. (Link to Colorado Study).
As Ed Parker says, "See What They Can Do."
Emily Hanford of NPR has a nice article about physicists, who are lecturing less and getting better results. Be sure to also click on the link "Listen to the Story"for the radio broadcast.
http://www.npr.org/2012/01/01/144550920/physicists-seek-to-lose-the-lecture-as-teaching-tool
Physicists have collected data providing evidence that learning gains are better when students are deeply engaged in the subject and allowed to collaborate with peers. Sandra Laursen, University of Colorado Boulder, also has data collected at the college level from mathematics courses. Laursen's results are in alignment with this article. Laursen's group has a number of findings. Among the long list of findings, there is statistically significant evidence suggesting the more an instructor talks the less their students learn. (Link to Colorado Study).
As Ed Parker says, "See What They Can Do."
Sunday, January 22, 2012
A Peek into the (Near) Future
Apple Inc. introduced iTunes 10.5.3, which includes interactive textbooks. This is almost surely the wave of the future, no matter what teachers and instructors might think of technology. Just as the CCSS represents a great opportunity, so does advancing technology. The Math community should experiment and investigate the options.
For those who say, "Run what brung ya," see Kodak. Kodak, once a giant, the inventor of the digital camera sensor is now headed for bankruptcy. The point isn't just about our individual preferences as instructors. We also have to consider what our students need to compete in the 21st century.
For those who say, "Run what brung ya," see Kodak. Kodak, once a giant, the inventor of the digital camera sensor is now headed for bankruptcy. The point isn't just about our individual preferences as instructors. We also have to consider what our students need to compete in the 21st century.
Do we have to have answers today? No.
Should we explore and study this? Absolutely yes!
Are/will international competitors be exploring technology and education? Absolutely yes.
Thursday, January 19, 2012
Teaching a Course for the Second Time (or Third...)
Winter term has started here at Cal Poly. In fact we are in week 3. This winter quarter I am teaching two sections of Math 330 Algebraic Thinking with Technology for future elementary school teachers. The course title isn't the point of this post, though I may talk about this course in the future. What I want to get to is that I taught this course last term for the first time. This quarter I am at an advantage as an IBL instructor, since I have laid out the course materials once. For the second time around I can now review my notes, look for problems that did well and try to fine tune problems to get students into the learning zone, where the problems are not obvious and not out of reach.
A hidden "side" advantage of IBL is that we become much more efficient as we repeatedly teach a course. So prep time declines with each iteration until it reaches a steady state. Moreover, we gain valuable insights into the learning issues. This is part of Math Knowledge for Teaching that helps instructors help students.
Example 1: Let's say students really struggled with Problem N last term. They tried it, but no one got it and things stalled big time. Some extra time and special cases were needed to get students going. This time around I can put into the problem sequence an exploration, where students investigate some given special cases and are asked to build several more examples. Then using what they learned from this experience, they hopefully are better prepared for Problem N. I don't want to make Problem N too easy or give away too much, but want to get the core ideas out into play through investigating related examples.
Example 2: Problem K was way easy for nearly all of the students. This time I'll leave Problem K as is, but add in a few more challenging problems after Problem K to ramp up the level. (Note: In math finding really hard problems isn't the issue. The issue is finding appropriate challenges for the developmental level of your students. This changes with each class, so one is always getting things "in the ballpark" and adapting to each situations.)
It's Jazz. You have Rhythm Changes in mind, but it's not the same every time.
A hidden "side" advantage of IBL is that we become much more efficient as we repeatedly teach a course. So prep time declines with each iteration until it reaches a steady state. Moreover, we gain valuable insights into the learning issues. This is part of Math Knowledge for Teaching that helps instructors help students.
Example 1: Let's say students really struggled with Problem N last term. They tried it, but no one got it and things stalled big time. Some extra time and special cases were needed to get students going. This time around I can put into the problem sequence an exploration, where students investigate some given special cases and are asked to build several more examples. Then using what they learned from this experience, they hopefully are better prepared for Problem N. I don't want to make Problem N too easy or give away too much, but want to get the core ideas out into play through investigating related examples.
Example 2: Problem K was way easy for nearly all of the students. This time I'll leave Problem K as is, but add in a few more challenging problems after Problem K to ramp up the level. (Note: In math finding really hard problems isn't the issue. The issue is finding appropriate challenges for the developmental level of your students. This changes with each class, so one is always getting things "in the ballpark" and adapting to each situations.)
It's Jazz. You have Rhythm Changes in mind, but it's not the same every time.
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