Teaching High Points
I've decided to start featuring key moments of success or joy or learning in my teaching here. Basically, this is my teaching "shameless plug" area, but hopefully others can learn from it.
Never Give Up!
This month's teaching high point is a plug for the book The 5 Elements of Thinking. I explored using it with my First-Year Seminar class this fall, and while we weren't all sure it was the best book of its nature, I found the life lessons of the book came back again and again to help students see where they could improve and change.
The elements are:
- Understand Deeply (Grounding Your Thinking)
- Fail to Succeed (Ignite Insights through Mistakes)
- Be Your Own Socrates (Creating Good Questions)
- Look Back, Look Forward (seeing the flow of ideas)
- Transform Yourself (Engaging Change)
These all are such an important part of making meaning and constructing knowledge (aka learning) in mathematics, and provided for great critical eyes in learning how to write about math, about ficiton, and about mathematical fiction. The book is stock full of exercises and examples. I hope to hear more about how my students have internalized its lessons over this academic year.
Loose Curves Integral Choose Your Own Adventures
On the first day of my condensed summer Calculus II: Integration class, I showed up with a stack of different-width rectangles (each in a different color) and a graph of a sinusoidal curve whose vertical lines matched widths of my rectangles. I told the students that their task was to consider the graph I had printed to be their "fence" and the rectangles were "fenceposts". I challenged them to use the fenceposts to get rough estimates on the amount of wood needed, since the lumber yard could only "squarely" cut fenceposts. They just had to make the closest fit to the curve they could.
The resulting 15 minutes involved a lot of play and honestly, very few people making it all the way to computing any areas, but what happened was natural, practical. They lined up those rectangles, some of them different widths at different places to best match the curve, others sometimes above and sometimes below, one student even selecting to lay his fence posts horizontally (technically against my instructions, but just as good!) and recorded how they worked on it.
You can guess how many students built what looked like a right-hand Riemann sum estimate of the area under that curve. However, at least one did something like a Midpoint Rule. They traced their various rectangles on to paper and worked with them. They learned that area estimated rectangles. We connected the height of the fence at any point to the value of a function, and talked about the "traditional" models. Introduced a definite integral, allowed for partitions that weren't even or sample points within each subinterval that were more random, just like their work. I didn't wrap it up as nicely as I had wanted to, but at the end of that first day, the concrete idea of signed area as limits of Riemann sums that they could make hands-on like that, was at the forefront. I put it on a piece of chart paper and tirelessly brought it to bear again and again all term. (Don't worry, we then visited antidifferentiation on the second day, which was better for everyone since they got a day to brush up on derivatives first).
About a week later, I gave students another draw-your-own integral curve adventure when exploring area between curves. We had done several basic examples and I gave them several copies of two different pairs of graphs. They were instructed to draw at least one vertical, and up to two other lines on the page, some of which should intersect their curves, and then to shade in or label two different regions that their lines had created. They made second copies of each, traded among each other, and were assigned to go home with three graphs: two from classmates and one from themselves. The task was to come up with a symbolic/integral description of the area of the region, assuming the curves just were named f and g, and identifying how they would find the bounds for any of their integrals (what steps it might take, knowing f and g and the x-coordinate of their vertical line). The next day they came in and compared, and we tried some of the ones where there was disagreement. In this way we could get the algebra out of the way long enough to talk about what we were visualizing, thus separating the two, equally challenging skills for area between curves. We briefly revisited some of the easier regions they had drawn when we studied solids of revolution. The most fascinating part was always seeing how students might draw what they perceived to be an easy problem, that turned out to be much harder to set up, or vice versa.
The summer class was small and went fast. One can't get the same expertise in antidifferentiation over a month that one can with the greater amount of time for repetition in a full term. However, the one thing students did well was connect the visuals to the integrals. Integral as area, and what that meant for volume, had become ingrained in them. It was a high point for me, and one I want to refine and explore again, the next time I get to teach integration.
How do you know when you're at the top of a mountain?
After attending the Legacy of R.L. Moore Conference in Denver in June 2014, which celebrates and shares the methods of inquiry-based learning, I got to thinking about what it might look like to blend inquiry with discourse and develop an entire Calculus I course around answering (seemingly) vague open-ended questions that brought the concepts of calculus together in a neat and tidy package. I had hoped a list as short as three or four questions or problems might cut it, and I won't say this was my most successful teaching term in calculus, but I did find two moments when questions I drew from really worked.
The questions were generated based on the following qualifications:
- They had to involve calculus ideas
- They had to involve real world vocabulary
- They had to be able to bring multiple levels of depth and calculus concepts together.
The first question was my opener on the first day.
What does it mean to say you traveled at a speed of 50 miles per hour?
Students give varied answers. "Your rate of change is 50". "Rate equals distance over time". The ones who have already had calculus introduce answers that include vocabulary terms like "instantaneous" and "average rate of change". Even "derivative". Pretty soon to make sense of what everyone has shared, between verbatim quotes of what they have said and my verbal and written paraphrases of their ideas, we have graphs of functions (possibly even a question about what a function is and the importance of knowing what you have "input" versus "output"), and maybe even a tangent line.
The idea of rates of change "approaching" the instantaneous speed shows up. If you come back to this one near the end of term, it can also give you a conversation about the "common sense" nature of the Mean Value Theorem.
Then came the one I like the best, because of all it pulls together. I have used it now in Business Calculus as well:
How do you know when you're at the top of a mountain?
(and follow-up): Does that tell you you're at the tallest place on earth?
This question is designed to get students thinking about the visual and physical experience of being at the highest point of something. A mountain has one distinct feature when you are at the top, which is the very one that finds you a local maximum: everything around you has lower altitude. But along with that answer, especially since students have been learning calculus, you gets lots of other answers. "You can't go any higher". "If you keep walking in the same direction, you will start heading down." "The rate of change of your altitude is zero" (is that the same as saying the top of the mountain is flat? Let's assume it's not flat, but jagged...) It's a fun day. By the end of it, we've clarified how local and global extrema are, and are not, the same thing; how we could procedurally identify locations of local extrema and their values using how the function changes direction or how the derivative changes sign; what Rolle's theorem does and does not say, and even the Extreme Value Theorem isn't far off, if you draw enough pictures.
Suffice it to say, I love this sort of stuff. I love seeing all the different voices contributing different perspectives. We challenge each other to add more and to be specific. Students are expected to explain rationales, or explain those of their classmates. Once I used this in business calculus, but with 5 minutes of group discussion and then a share-out of their group's most specific answer. We debriefed and collaged together a complete one from the work.
How do you know when you're at the top of a (teaching) mountain? When you get to fully access students constructing their own knowledge through discourse and open questions.
Self-Regulation Inspired Reflection and Study Questions
In April 2014, the Office of Teaching and Learning hosted a conference, Empowering Students to Be Self-Directed Learners. The OTL has a great summary post and a bunch of resources at http://otl.du.edu/blog/otl-conference-attendees-share-ideas-to-promote-self-directed-learning/, so I am going to restrict myself here to the story of what I'm actually doing and what inspired it.
There's a lot to be said about self-directed learning, and the the topic of self-regulated learners, which the keynote speaker, Dr. Linda Nilson, shared with us at that conference, and it was surprising to see how much of it had already made its way into my classes. In particular
What is a self-regulated learner?
My previous experiences thinking about my own study process and what I had observed to be successful for my students (thanks to surveys and conversations), brought these ideas of focusing and reflection to the forefront.
As a learning facilitator (the role as a teacher I'm most trying to identify with these days), questions are key. If I just tell people what to do, the sort of mathematical discourse and ownership of the classroom's learning that I value and embrace, will not occur. I need to channel a combination of my inner monologue, inner-Socrates, and inner-Freud, to more outwardly build in my students a sense that the questions about the questions are important, and that I ask them to encourage a certain way of thinking about the subject and the way my students are learning it. So, I began documenting for myself some of these questions were appearing in written and spoken interactions with my students. Some of what I found are listed in the attached pdf.
Student-Generated Wiki Review Pages
Student-generated review/summaries are easy to design using Wiki features in Blackboard, or as editable pages in Canvas. More importantly, assigning students to collaborate in summarizing or reviewing material from the class gets them thinking about content in a different way, and bringing more perspectives and aesthetics to review material is never a bad idea.
In my MATH 1150 Graph Theory in the Real World class, I had students contribute to a wiki-based review guide for the midterm and again for the final exam. Every student had to offer a minimum contribution of two definitions, explanations, or key ideas from the course that would be useful to review for the final. The structure mirrored a concept organizer that I had encouraged them to use in the beginning of the quarter, so for many, this just meant pulling from their notes. I put in the main headlines, left directions at the top of the page, as well as a place at the end of the page to include their questions (which I, the TA, or other students could then answer), and I left the page open to edit. When student work was incorrect or needed amending, I left comments in another color in the page indicating that an edit was needed, often with a hint. Students could add to their own posting or fellow students could correct it for credit. After the deadline for contributions, I annotated the page with any final corrections or key connections I felt were missing. The result was a comprehensive review guide that was co-constructed with my students.
In my MATH 2070 Introduction to Differential Equations class, I've used wiki page work differently, more as a running record of the class' impression of the material in the course. Every section we cover (or sometimes half-section) has a group assigned to generate a summary of that topic in a wiki page. What they do is supposed to be a clean, final annotation of what they have learned from the text and class--I post board/tablet notes separately from our class meetings, and sometimes extra worked examples, but the nice "story" of the section in digital form is left for students to create. This has been really nice in Canvas since the equation editor is easy for students to work with, and the pages allow for students to upload images of slope fields or graphs or their own hand-written work when it is relevant. The result is a digital version of everything the students have learned from the course, and it builds up a few sections at a time each week. Students can refer to these as they do their homework or study at any point, and unlike the heavy textbook, these pages are only a few smartphone swipes away.
Acknowledgements: The ideas to work on this came from a few faculty who had shared how they were using wikis: Susan Sadler (Biology), who shared her experience in the DU Teaching Online seminar, and Peter Jipsen (Chapman University), who visited the Department of Mathematics as part of our Teaching Excellence Seminar in 2012. The inspiration for the "review guide" use, and its constructive purpose in supporting student learning, came from a talk by Terry Doyle (Ferris State) I attended at the MSU-Denver Teacher-Scholar Forum in 2013.