The IBL Blog focuses on promoting the use of inquiry-based learning methods in college mathematics classrooms. Learn more about IBL at The Academy of Inquiry Based Learning
The following interviews are five of Dr. Vale's students mentioned in Part 1 of the interview. Five CSUMB students, five wonderful stories!
Transformative experiences are highlighted by the IBL community. Those of us who have used IBL methods in our classes have seen students flourish in ways that changes the trajectories of their lives. It sounds like we are exaggerating, as if we are making promises of rainbows and unicorns. But there are many instances, when things go right, the students buy-in, and the entire class (including the instructor) comes together like a championship team. It doesn't always happen, it's not easy to pull off, but it's why we work so hard to try and get to that special place.
In this particular instance, Dr. Vale taught a proof-based course for math majors, where the opportunities to build Math from first principles, parallel to how research mathematicians do Math. Authentic success, the greatest confidence booster there is, did it's work. This experience left a lasting impression on these students, who proceeded to earn their undergraduate degrees and then go to on to graduate programs.
This blog post is an interview in a Q&A style. In this edition, I interview Dr. Rachel Vale, University High School in San Francisco. Her experiences teaching at both the university level and high school level, gives her interesting perspectives and insights into IBL teaching. Thank you, Rachel! (See also Part 2: Dr. Esselstein's Students)
Hi Rachel! Thank you for agreeing to be interviewed. You’ve had an interesting and inspirational trajectory in your IBL teaching. Please tell us about it.
My first attempts at IBL were as a professor at California State University, Monterey Bay where I worked for 7 years. I decided to attend an IBL conference at University of Michigan after a discussion with a friend and collaborator, François Dorais. This led me to apply for a mini-grant which I used to create an IBL Intro to Proof course with Carol Schumacher as my grant mentor. The course ran several times over two years and I found it to be a very challenging but rewarding experience.
In 2014, I accepted a position at the Bay School in San Francisco where I taught math for two years, learning the ins and outs of the independent school world and how to work with younger students. Bay’s pedagogical philosophy is student-centered and so I had many opportunities to implement IBL in my classes.
This year, I started work at another independent school, University High School in San Francisco. I am very happy in my work at UHS. I am surrounded by other faculty who are incredibly talented both as teachers and as practitioners of their discipline. UHS has a reputation for being an environment for students who appreciate a good challenge and it has been fun to work with amazing students who enjoy being pushed out of their comfort zone.
Please share with us how you use IBL methods in your classes.
It was a revelation for me to realize that I could use IBL when I found it applicable and use other pedagogies in other instances. In other words, there is no need to commit to a full IBL course. Now days, I pick and choose how and when to use IBL methods in my courses. For example, in an introduction to linear programming for Precalculus, I gave my students an open ended problem with no prior instruction about optimizing the packing of supplies for a space mission. Of course, they were not expected to discover linear programming for themselves in that one lesson but many of them created systems of linear inequalities, graphed them and then started hypothesizing about the optimal solution. We then had a week-long discussion on why the optimal solutions would happen at the corners of the fundamental region and the students came up with two different ways to explain this phenomenon. The unit wrapped up with a problem set in which the students worked collaboratively on very challenging problems that asked them to assimilate their understanding of the unit followed by a unit test that was fairly standard compared to what you might see at any other high school covering this topic.
On the other end of the spectrum, there are some topics that I have found are challenging enough and simply frustrate the students if they are asked to learn them via inquiry. This is especially true for algebraic processes with younger students such as completing the square in an Algebra 1 course. In these cases, I still rely on group work and the unit will always end with a problem set of problems that ask the students to apply and assimilate their understanding but the material is introduced using other pedagogies.
How is teaching HS similar/different compared to teaching at the college level?
One of the biggest obstacles of using IBL at the high school level is working with the parents who may or may not understand and support their child learning in a style that does not reflect their own high school experience. At independent schools, the parents are generally very supportive of the faculty but they also are very concerned with the success of their student. I have found that it is always crucial to communicate the purpose and intention of IBL with my students but, at the HS level, it is also important to get the parents on board. I try to make my grading scheme clear to both the parents and students and I make detailed notes and rubrics that help my students and their families track their progress.
At independent schools, our class sizes are much smaller than I had at CSUMB. This means that working in groups is much easier to facilitate and I can take much more detailed notes about the work each student does when they are problem solving or presenting.
The resources for finding interesting problems to give my students are quite different. While it is usually easier for me to construct challenging problems for high school level math than for college level math, my students are not as developmentally ready for things such as proofs or problems that require significant work. They also seem less adept at working with open ended problems although I do believe that this is something they can overcome (as opposed to not being developmentally ready to handle). I have found NRICH to be a wonderful resource for challenging problems as well as old Math League exams and occasionally old Math Olympiad tests. The textbooks I am asked to use are good for practice problems for my students but they don’t provide the right problems for IBL so I mostly make my own materials.
Students are very self-conscious at the high school age. This seems to be especially true for freshmen who are desperate to fit in and look smart or successful. I have found it to be even more crucial to create a safe and welcoming environment in the high school classroom in order for students to truly engage in IBL. Thankfully, my classes are smaller and the students are very well-intentioned and driven by curiosity.
I don’t emphasize presentations of solutions at the board as much at the high school level, especially with younger students. I will scribe at the board (or get one of my more restless students to scribe at the board) while the presenter describes their work. I have also found success with asking students to share back what they understood about their peer’s solution. This helps them make sense of the work for themselves rather than just having them copy the work down on their own paper.
What have been some of your biggest challenges teaching Math?
The most challenging thing is that I spend copious time and effort creating a project or problem set that is successful for one group of students but falls flat with another. Sometimes this is due to the abilities of the groups being different but it seems to mostly relate to the personalities in the class. I find that IBL at the high school level (and somewhat at the college level) really relies on finding the right hooks; the problems must be engaging and tractable. High school mathematics has many more venues for applications and so sometimes I can just tweak a problem to relate to a particular group of students. Other times, I need to completely scrap an activity that I worked so hard to create the year before.
I also found collaborative teaching to be challenging when using IBL. IBL is easiest to run if you have complete autonomy over the course and can adapt to the students’ interests and abilities. I have found that my calibrations for my students are very different than other teachers’ for their own students and it means the courses might go at different paces and focus on different activities.
IBL at the high school level gets LOUD! I have had to find ways to monitor the noise level when they are working in groups or debating a solution to a problem because their enthusiasm and energy cause the volume levels to get extremely loud. High schoolers often have less social maturity than college students and so I have found it necessary to instruct them on how to give constructive feedback to their peers.
What have been some of the successes?
I don’t know that I have been teaching at the high school level long enough to see measurable successes in terms of students going on to find success in college level math. I have had multiple students from my IBL courses at CSUMB go on to graduate school in math with great success and I have also had some students become high school teachers themselves who are using IBL in their courses. I anticipate that we will see much more IBL in high school math as the Common Core State Standards and training around them continue to roll out. Common Core has been a great touchstone for teachers to re-think their pedagogy and how it reinforces the mathematical practices and habits of mind.
I have absolutely seen an improvement in my students’ abilities to solve problems that are unfamiliar to them. I also noticed the time my students spend working on a problem before asking for assistance has increased although I haven’t recorded or measured this formally.
Many of my students describe the IBL units as the “most fun” they have had in a math class. Students like discovering the material and solving puzzles. They love being challenged when they know that the stakes are non-threatening. They find that IBL makes them feel successful in math because they aren’t just memorizing but understanding the material.
Here’s a video clip of two of your former students from CSUMB, Alfred and Diana. Something special happened in this class, where your students went were transformed learners. What happened in that class?
I don’t know how to describe what happened in that class but it really was magical for both the students and me. I had Carol Schumacher as well as you, Stan, as resources and you both were very helpful in providing me with encouragement and advice when both were needed. The stipend from the mini-grant allowed me to put much more time into preparing the course than I would have been able to spend otherwise. Still, the money wasn’t commensurate with the workload. It was a lot of work! I had a great group of students who were open-minded enough to go through this experiment with me and I had very clear outcomes that I wanted to reach. I think the success of this class was mostly due to the fact that IBL works when it is done well. Any thoughts or advice for instructors thinking about using IBL but have not tried it yet?
IBL is a lot of work but it forces you to make your teaching student-centered. You will be surprised by the things your students discover and understand. The best feeling is when a student or a group of students has a break-through and they want to celebrate it with you. As mathematicians, we are used to the elation of a breakthrough on a tough problem but, for many of our students, this is the first time they have ever experienced this. They cheer, give high-fives, post things on their refrigerators at home, and develop a more favorable opinion of math class. It is a lot of work to run an IBL course but it is so much more fun for everyone.
Because it is a lot of work, find a teacher mentor who can help you troubleshoot any issues that come up. It is best if they are at your school so that they know the school culture but even long-distance mentors are better than going at it alone.
Make the expectations clear to your students (and their parents) as well as your department chair and anyone supervising your work. Students sometimes panic that you are “not teaching” and their complaints to supervisors can be detrimental to your career if the supervisors are not aware of your pedagogy. Give the students ample positive feedback and opportunities to reflect on how far they have progressed. Happy and well-supported students will put in significantly more effort than unhappy and frustrated students.
Observe a class that is using IBL and attend conferences such as the AIBL conference. They are inspirational and motivational.
