The Big Questions

Seeking Help Finding a Needle
A long while ago, I read a great article written by a college history professor. The article was about the professor’s frustrations with the mindset about history that his students had at the beginning of the semester. In particular, he talked about how students would enter his course thinking that the point of history class was to memorize a bunch of related names, dates, places, and battles. But as an academic historian, the professor saw the teaching of history as the re-telling a long narrative about human events, what we’ve accomplished, what our failures were, and how we can try our best to avoid huge tragedies like those we’ve seen in the past.

The professor admitted that throughout the semester, he would remind his students:

“The point of what we’re studying is not that the Battle of Hastings was in 1066AD. We want to focus on the big picture, we want to answer the big questions, we want to tell and reflect on the big story.”

At the end of the semester, the professor added a new question on his final exam: “Tell me something you’ve learned from our class that will stick with you.” –and, of course, the number one most popular response was, “I learned that the Battle of Hastings was in 1066AD.

This is my re-telling of the article. I cannot remember where I read it. I cannot remember who wrote it. I have lost so many of the details. Do you know of this article, professor, or story? I would really appreciate if anyone could point me to where this was published, or by whom.

My Big Picture, Big Questions, Big Story
The reason the above story stuck with me is that I am trying to focus my attention on what I want my students to learn about mathematics, apart from any particular topic or course that I might be teaching. What are the important things I want them to know? What do I want them to know about the discipline of mathematics? What do I want them to know about what it means to think like a mathematician?

Despite feeling like I have a ton of course content to cover (and feeling like I’m always behind schedule), I’m forcing myself to create time in class to address these big ideas. While I absolutely want my students to master the process of integration by parts, in ten years, I really don’t want them to remember our course as “the place I learned integration by parts.”

Instead, I hope my students will remember our course as “the place I got excited about mathematical ideas” or “the place that I learned to be mathematically curious” or “the place I learned to think like a mathematician.”

I don’t know if I’m successful at this goal. It’s going to take a long time to find out, since I have to wait at least ten years. I also don’t know how this is impacting my students today & if I’m making them feel bored, or frustrated, or distracted from the stuff listed in the official Course Description.

On the purpose of examinations

I have just finished the second round of mid-semester exams in my calculus courses. As I may have mentioned before, I’m teaching two sections of “Calculus I” using a traditional grading scheme and one section of “Calculus II” using standards based grading. Both courses encountered their tests last week and had them returned with feedback this week.

While grading, I mentioned the following on Twitter:

Joe Heafner then replied: 

 

Since then, I’ve spent quite a while thinking about my purpose of giving them exams. I have the sense that students think the purpose of exams, in way exams are traditionally graded, is to compile as many points as possible. Here is a common approach:

  1. Write something down for every problem, whether or not you know how to do it, because you might get partial credit points for having at least something right.
  2. If you know how to do a problem, do it as quickly as possible; so long as you get the right answer at the end, there isn’t a need to write clearly or keep track of notation or show your chain of thought. If the answer is “7” then so long as you have “7” on your paper at the end, you’ll get full credit.
  3. Once the exam is returned, look it over. Ask questions of the form, “Why did I lose three points on this problem?”

I really dislike every step of this approach, but I don’t think I have ever communicated this well to my students and I have not been able to reward a change in behavior away from this strategy when I’ve used traditional grading. On the other hand, with standards based grading, I’ve tried to put effort into motivating my students to follow these steps instead:

  1. Keep track about what you know how to do and where you still need work. If you encounter a problem that you aren’t confident you can solve, don’t worry about it until you have more practice, seek more guidance, or have more time to study that topic.
  2. If you know how to do a problem, show all of your work. This is your chance to show off! Write neatly, explain your reasoning, and demonstrate mastery of the process. Don’t fret if the answer is “7” and you got “8”; we are interested in conceptual understanding and will happily overlook inconsequential arithmetic errors.
  3. Once the exam is returned, look it over. Ask questions of the form, “I am missing something conceptual here. Can you help clear it up for me?” or “I wasn’t sure how to attempt this problem. After I have more time to work on it on my own, can we go over it together?

I suspect that the two different grading schemes would result in very different types of solutions for process problems. One such problem on the “Calculus I” test stated, “Use the limit definition of the derivative to find the derivative of the specified function.” I was not very successful at getting my beginning calculus students to understand that I am more interested in the process they are using instead of just their final answer. Standards based grading has allowed me to have conversations during class about the reason we ask these types of problems and what constitutes a solution versus just an answer.

The purpose of giving exams in my courses is to allow the students the opportunity to communicate their level of mastery of the course material. I’m looking for their ability to demonstrate conceptual understanding and their fluency with the technical processes needed in various problem-solving situations.

I think that standards based grading has made it easier for me to explain this to my students. I don’t think my students see exams as an adversarial process where I am judging them or their abilities. It is my hope they see exams as an opportunity to show what they know, to discover what they still need to work on, and to give us both a clear picture of where we should go from here.

Postscript. One issue I need to work on in upcoming courses is motivating students toward mastery earlier. Some SBG students are a bit behind in showing mastery of the standards, and while this hasn’t been a problem for them yet, I expect they will begin struggling quite a bit very soon. Now that our course material is building on itself at a swift pace (integration by parts into improper integrals into the Integral Test for Convergence of Series) I worry that their [lack of] progress on our “integration by parts” standard will cause them difficulty keeping up with the course. My students have been great and I feel like this was my failure at putting together an accurate timeline of what they should know and when they should know it. I think maybe I focused too much on “you can always improve later” that the message “…but there’s no time like the present, so do it today!” was lost.

An Adventure in Standards Based Calculus

Today was the first day of our new semester. This spring, I’ll be teaching two sections of “Calculus I” and one section of “Calculus II.” I feel like “Calculus I” is basically on autopilot; I’ve taught the class every semester for the last couple years and so I’m very comfortable with the course content. But this will be my first time teaching “Calculus II” in many years. (I think the last time I taught it was 2006 or so, at the University of South Carolina, using an entirely different textbook.) I’ve decided that I want to try something different & I am embarking on my first attempt at Standards Based Grading (SBG) — or as someone suggested today on twitter, maybe Standards Based Learning (SBL) is more appropriate?

Why Am I Doing This?
For the last few years, I’ve noticed a few things about traditional grading (TG) that I did not like. One thing that has bothered me is that a student can go the entire semester without ever solving a problem 100% correctly, yet still do very well in the course. For example, it is entirely possible to earn a “B+” grade, by performing pretty well on everything, but never really and truly mastering a single topic or problem type. I hope that Standards Based Grading helps me motivate my students to really try to master specific sorts of problems, rather than try to bounce around, hoping they can earn enough “partial credit” points to propel them to success. Really, I want to reward a student who gets four problems absolutely correct (and skips two problems) more than a student who just writes jumbled stuff down on every page. I think SBG will allow me to do this.

Another (related) thing that has bothered me: The point of calculus class is not to earn as many points as possible, doing the least effort possible. I will admit that I have used a TG scheme for years and years; I have no idea how many college-level courses I’ve taught. And I am pretty sure that I can look at a calculus quiz question, assign it a score between 0 and 10, and accurately give a number close to what my colleagues would give for that same problem. We might all agree, “Okay, this solution is worth 7 out of 10 points for these reasons.” But I think this gives the students the idea that the reason they should study is to earn points on the quiz — after all, 9 points is better than 7 points! Instead, I think the reason they should study is to understand the material deeper than they presently do now, and I think by assigning X points out of 100 sends them the wrong message.

Something that has really bothered me recently is that when a student is struggling with the course, I am never entirely sure what to tell them. I look up their grades in my gradebook; I see that they have an average of 62%; and then I try to give them advice. But what advice should I give? The 62% in my gradebook does not tell me very much: I do not know if this student is struggling because they need more practice in trigonometry. Or maybe they were doing very well, but bombed our last test because they got some bad news the night before. Or maybe they got L’Hopital’s Rule confused with the Quotient Rule. I want to be able to tell a student exactly what they can do to improve their understanding. By tracking each student’s mastery of particular standards, if a student comes to my office for extra help, I can tell that student, “Okay, it looks like you need extra help with [insert specific topic].”

Lastly, I would like to give students more low-stakes feedback about their understanding: That is, feedback without the worry that it will negatively affect their grade in the class. I will be giving a weekly quiz, and I will grade it, offer feedback, and return it to my students; then (eventually) their score on that standard can be replaced with a newer [hopefully better!] score. I will constantly replace their previous score on a standard with their current score on a standard. This way, if they are really struggling with (say) Taylor polynomials, I can communicate this to them early, they can seek extra help and resources, and then they can be re-assessed without penalty for their original lack of understanding.

What Worries Me?
I have lots of different things worrying me about this system! For example, since this is my first time teaching Calculus II in many years, I don’t know all the “common pitfalls” that my students will encounter, so I don’t feel like I’m going to see them coming until they’re already here. Also, I am worried that students will struggle to understand this method of assessment & won’t really “get it” about how they are doing in the course — or won’t take the opportunity to re-assess when they need it. Lastly, despite reading online that “before a course begins, start by making a list of what you want them to master (a.k.a, the standards)” I was unable to do this. I have the first half (or so), but I don’t know how good they are. Am I being too vague? Am I being too specific? Do I have too many? Too few? How difficult will they be to assess?

Some Resources
In my own course planning, here are links to resources I found helpful:

Wish me luck!

 

Digital Plan for Digital Action

It turns out that several people had some great suggestions about my wish for digital exam grading. I’ve decided to attempt it for my next Calculus exam, scheduled for Tuesday, March 26th. Here’s an outline of the plan:

  1. Photocopy exams single-sided and unstapled. Place a copy of each exam into an empty file folder.
  2. Subject unsuspecting Calculus students to grueling exam on these topics: Related Rates; Linear Approximation; Mean Value Theorem; Derivatives and Graphs.
  3. Alphabetize exams as they are turned in according to course roster. For absent students, place blank exam where theirs should be.
  4. Use department copy machine to scan all ~350 pages to a single PDF file and send it to me via e-mail.
  5. Thank my husband profusely for writing pdftk bash script that will take the single PDF file and break it apart, at every ~9th page, and rename the files according to last name (keeping alphabetical order in place). If this works, I should end up with 36 PDF files where each student has a file called “Owens-Calculus-Exam3.pdf” or something similar.
  6. Create Dropbox folders for the ungraded exam PDFs and the graded exam PDFs. Use GoodNotes to grade the exams on my iPad. Export the finished product back to Dropbox.
  7. Disseminate graded exams and grades to students.

It’s likely my first attempt at this will take longer than nondigital grading. One of the things I will have to do as I go is come up with “Correction JPGs” for those errors that happen most frequently and store them somewhere on Dropbox. I think these should be easy to add to each exam using the “import JPG” feature of GoodNotes. Usually I estimate that grading will take no longer than 10 minutes per exam. For my 36 calculus students, this means regular grading should take me about six hours. Hopefully this digital grading effort won’t take too much longer than this.

For Step 7, I also need to find out about FERPA. Provided I have a “sign for consent” on my exam header page, is that enough for it to be okay for me to e-mail each student her graded exam? Alternatively, is there a way using our Desire2Learn-Dropbox (on our Learning Management System) to return the exams to the students in some easy way?

Wish me luck!

Want Some Free Red Pens?

I’m about 75% through this round of midterm exam grading. Overall, I’m down to around 100 students in total, over three classes. I’ll give four midterm exams and a final exam at the end of the semester. This requires a lot of red ink.

A while ago, I read an inspiring article in the MAA FOCUS called “Abandon the Red Pen!” written by Maria H. Andersen. The article was about digital grading. Since I read it, digital grading has been a dream of mine. Ideally, here’s what I’d like to do with the pile of exams currently sitting on my dining room table:

  1. Students take exams in class, on paper, like usual.
  2. After students turn in exams, magic happens. I end up having a PDF file of each individual exam paper, titled something like “StudentLastName-Calculus-Exam2.pdf”
  3. I dump all of the PDF files into a Dropbox folder and then I do all of the exam grading on my iPad.
  4. Once I’m done, I save each file as “StudentLastName-Calculus-Exam2Graded.pdf” and then more magic happens, and each student gains access to their graded exam — perhaps over e-mail, or through the file server in our Learning Management System, or some other solution.

Overlooking the requisite magic requirements, let me explain why I’d prefer this to offline grading:

  • I wouldn’t have to carry 100 exams home, keep them away from my toddler, make sure I don’t lose any to black hole of my desk, try to avoid spilling coffee on them, etc.
  • I would have a complete digital record of a student’s work. Occasionally a student comes to me at the end of the course and says, “I just checked the online gradebook. It says I earned grade X%, but I am certain I earned grade (X+4)%.” Sometimes they are able to produce the test paper and the gradebook indeed has an error. Sometimes they aren’t able to produce the test paper, and I can’t do anything for the student. Having a digital PDF file of every graded exam would solve this issue immediately.
  • In the unfortunate case of dishonest work, I would have a clear record. (For instance, if a student modifies their test paper after it is graded and returned, and then asks for more credit on a problem. This has happened in the past.)

But the most important reason I’d love to switch to digital exam grading is that I could give better comments in less time. On the current test, all students had to solve a similar “Optimization” problem involving having a constrained amount of fencing to build a backyard of area A. For the most part, students fell into one of three categories: (A) Response entirely correct; (B) Response entirely incorrect or missing or blank; or (C) Response partially correct, but some errors were made. In category (C), there were only about three types of errors: That is, everyone who made a mistake made one of the same three mistakes.

Digital grading would allow me to type up a full response as to what the error was, why it was not correct, and how to fix it. I would only have to type the response once. I could save it as a JPG file. Then whenever a student made that particular error, I could just “drag and drop” the response onto their test paper.

Also, eventually I’d have JPG stamps for the big “Top 100 Algebra Errors”, things like sqrt(9+16) is not the same as sqrt(9)+sqrt(16). I would never have to write anything about this mistake again because I could just drag and drop the explanation JPG!

Now, the tricky part: How do I get the magic to happen? The photocopy machine in my department is quite happy to take 8 pages, scan them to a PDF, and e-mail them to me. So, for a particular student’s exam, I could undo the staple, run it through the copy machine, and I’d be done. Unfortunately, I don’t know how to do this en masse very efficiently.

Suppose I have 100 exam papers, each 8-10 pages. How do I remove all of the staples, run each one through the copy machine individually, and rename the files? This process seems very easy, but I estimate it would take about a minute per exam. At this point, I’d rather spend 100 minutes doing the grading than 100 minutes dealing with the paper shuffle. Hence I need magical elves. Or graduate students.

Since I haven’t figured out how to do this first step, I haven’t given much thought as to how to “hand back” the graded files. I’m sure there’s probably some easy way to do this in our LMS, so maybe it wouldn’t even require magic.

Do you have any ideas about how to do the first step (i.e., scan each individual exam paper to PDF) that doesn’t require magic, graduate students, or administrative assistants? I’m happy to send you all my red pens in trade for such information.

The Life of a Professor

Here’s a quick run-down of my day today:

  • 6:07am, alarm, answer student’s e-mail, shower, dressed, breakfast
  • 7:05am, wake toddler, give hugs, change diaper, feed breakfast
  • 7:17am, run out door
  • 8:00-8:50 and 9:00-9:50, teach “Precalculus” twice, answering homework questions during passing periods
  • 10-11:45am, Office Hours. Answer the same factoring question six times for eight different students. Devour granola bar between explanations.
  • 12-12:50pm, teach “Calculus”, answering homework questions during passing periods
  • 1-1:45pm, eat lunch. Discuss upcoming “STEM Education Day” hosted by our female basketball team (the Lady Cougars). Brainstorm ideas about fun, engaging math activities to present to students from grades 4-8 (Have any ideas?)
  • 2-2:50pm, Calculus Committee Meeting
  • 3-3:35pm, answer e-mails, send e-mails, delete old e-mails, read Twitter.
  • 3:35pm-?, attempt to finish writing other blog posts.

Rational Power Functions

One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like  f(x)=x^n are called power functions, the rational power functions would be those of the form  f(x)= x^{p/q} (where  p/q \in \mathbb{Q} ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

 y = -(x-5)^{4/3}+1
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

  1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
  2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
  3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.
  4. Return to Step 1.

The students will have the entire class period to create graphs of each of the nine Functions. Since they will track down the graph most closely matching theirs, they will have a chance to check their answers. Even once they see the answer (i.e., the graph), they are still asked to do some thinking since they have to provide information about the graph’s important features.

Hopefully this process will work. The only thing that concerns me is how long it will take them. I am really terrible at gauging how long it takes students to think about things. My best guess was that it would take about 4 minutes to sketch a quick graph, and then another 3 minutes to write down its important features. So, (7 minutes)x(9 Function Cards) should be a little over an hour.

I picked the nine functions from Tom’s “Homework Handout” on this Section, available here. If the weather complies, maybe we can do this activity outside. The only thing better than math class is math class in the sunshine.

On Algebra

There have been a whole flock of article recently addressing the question, “Should we teach algebra to all high school students?” It started (I think, anyway) with Andrew Hacker’s Op-Ed post, “Is Algebra Necessary?” in a recent issue of the New York Times. His conclusion: No, algebra isn’t necessary. A careful reader ought to question, “What does Hacker mean by algebra?” and “Do we want an emeritus political science professor to make decisions about the mathematical education of the masses, given that there are so many people whose entire careers are dedicated to mathematics, the research of mathematics teaching and learning, and being mathematics educators?” But today I wasn’t planning on addressing those questions or Hacker’s article. (See Daniel Willingham’s response, “Yes, algebra is necessary” if you’re interested.)

Instead, there are several important issues that I think are worth pondering whenever anyone starts talking about the necessity of algebra.

1. What do they mean by algebra?

Out of curiosity, I asked WolframAlpha to tell me about “algebra”. It gave a lot of responses (but no definition). It did, however, provide a clear distinction that algebra is something more than equation solving. Yet in Schank’s first paragraph, he seems to conflate all of algebra with the quadratic formula:

“Whenever I meet anyone who wants to talk about education, I immediately ask them to tell me the quadratic equation. Almost no one ever can. (Even the former chairman of the College Board doesn’t know it). Yet, we all seem to believe that everyone must learn algebra.”

I’ll skip over a discussion pointing out that he means the quadratic formula but wrote “the quadratic equation” (as if there’s only one).

In any case, very few people have given the word “algebra” a good-enough definition, from my viewpoint. When I put on my research mathematician hat, I like thinking about universal algebra, which is somehow even more broad and amorphous (and beautiful) than the above definition could convey.  (I am even a published, theorem-proving universal algebraist.) I define algebra like this:

Algebra. Noun. The branch of mathematics that deals with the study of structure.

Yes, that includes studying equations. But it also includes wilder animals like  finite fields or nonfinitely axiomatizable equational theories and the varieties they generate!

2. Who is the “they” and who is the “us”?

Schank also asks,

“Are mathematicians the best thinkers you know? I know plenty of them who can’t handle their own lives very well.”

It turns out that, despite lots of evidence to the contrary, mathematicians are people. As in, real people with real lives and real feelings and real kids and real cats and real hobbies. And, sometimes, real problems. I don’t know why this is news. Surely no one would suggest that we ought not listen to music (or teach music in schools) because some musicians have had difficult lives. The people behind the subject is what makes it compelling. If only robots did math, I’d probably be less interested in math. [And more interested in robots.]

How do mathematicians think? Wow, that’s a fabulous question. Look, no mathematician I know claims that we think “the best.” That’s a “they versus us” distinction if I ever heard one. However, many of us do claim that we think differently.

To believe anything, a mathematician requires a proof.

This is different from every other academic discipline. Mathematicians have a very strict code for how we think about problems. Intuition is never enough.

3. Who taught these people mathematics?!

Moving toward his conclusion, Schank writes,

“You can live a productive and happy life without knowing anything about macroeconomics or trigonometry but you can’t function very well at all if you can’t make an accurate prediction or describe situations, or diagnose a problem, or evaluate a situation, person or object.”

(It humors me that the end of the article talks about making “sensible political choices,” but here it turns out we don’t need to understand any macroeconomics. Say what?)

I wonder who taught these people algebra! None of my students will survive my courses unless they are able to demonstrate that they can use the mathematical content knowledge from my class and apply it to real-world problems about situations that involve optimization or diagnostics or evaluations or predictions. Who are these [potentially imagined] math teachers who teach nothing but endless factoring of polynomials without any motivation?

4. What’s a better question to ask?

Schank and I agree that “[t]he ability to reason from evidence really matters in life.” He thinks algebra doesn’t help develop the skills to do this — I disagree hugely. Algebra can help develop this skill. Does every math educator teach it perfectly? No. Could all of us math educators do a better job? Yes.

My goal as an educator of mathematics is to converge asymptotically on being amazing at my job. Could teaching algebra help students understand logic, reason, and critical thinking? Absolutely. Does it always? Maybe not. But that is not a reason to say, “Don’t teach algebra.” We should say, “We need to teach algebra better.”

And, I promise, I am trying to do just that — along with thousands of my colleagues around the world.

Wordle

Earlier today, Derek Bruff (@derekbruff) tweeted a link to a Wordle done by graduate student Jessica Riviere. Jessica blogged about her Wordle, so check out this link for what she had to say. Her Wordle contained data from her teaching evaluations and what her students had commented. This was clever and fun and it inspired me to make one as well.

I used my course evaluations done by College of Charleston students during the last academic year (Fall 2011 through Summer 2012). Altogether I have data from eight courses (covering several sections of Elementary Statistics, Pre-Calculus, and Linear Algebra) for a total of 114 evaluations. To make the data collection easier, I restricted my focus just to the “Comments on Instructor” and “Comments on Teaching” prompts. This meant ignoring data from sections called comments on “Organization,” “Assignments,” “Grading,” “Learning,” and “Course.”

The most frequently used words were: and, the, to, I, is, she, a, class, was, of, her, Owens, with, Dr. Several of these were removed by Wordle since I had chosen to “Remove common English words.”  I also removed my first name and corrected some misspellings (ex: “explaiend” to “explained”). I enjoyed the following word counts: awesome, 6; funny, 5; humor, 5; and enthusiastic, 9.

Wordle: Eval Cloud

E-Seminar on “Mathematics Teaching and Learning”

In a previous post, I wrote about finding an E-Seminar from the NCTM (National Council of Teachers of Mathematics). A full list of available topics can be found here. One of them I mentioned before is called “Mathematics Teaching and Student Learning: What Does the Research Say?” Check out the description on their webpage. Today was our first day in my summer SMFT course. Since the students are all in-service math teachers I thought they would benefit from watching the seminar. I hope that they got something out of it, especially considering that it took up around 75-minutes of our limited class time. Here are the top three take-home messages I got from the re-watch:

  1. The idea that teaching is a cultural activity. In other words, we all learn how to teach in a process of cultural immersion during our school years. We get some ideas about what a classroom is “supposed” to look like, what a teacher is “supposed” to be doing during class, and what students are expected to do. Many educators are not taught effective teaching methods and it is easy to revert to teaching how we were taught instead of how we would like to teach (or, how the research says we ought to teach).
  2. The idea that effective teaching is learned. It is not an “innate talent” and it requires a lot of “hard, relentless work.” This is a freeing idea since it allows us to ask questions like, “How do I learn to become a better teacher?” and “What is effective teaching, anyway?”
  3. The idea that improving teaching is a process instead of a goal. Instead of focusing on a large (unattainable?) goal of becoming an effective teacher, instead we can aim for a concrete, step-by-step process of making tiny changes in our classrooms over a long period of time. The seminar suggests to begin “by designing a few lessons with great care” — maybe even just one or two — and after implementation, then gather evidence on the lesson’s effectiveness. A lot of important work should take place after the lesson is introduced when we can consider how to improve it next time.
With these things in mind, one of the major assignments in my SMFT course this summer is for my students to engage in this third item: They are each required to create two lessons for use in their own classrooms. Although they teach for hundreds of hours per year, by focusing a lot of energy and attention on just one or two lessons I hope that they begin to make those small changes. In the meantime, I hope to change the culture of our classroom and move away from being the “Lecturing Professor” character.