In my last post, I expressed incredulity at Pat Rogan’s statement that by limiting education degrees to no more than 30 hours of pedagogy courses, the state of Indiana would be “put[ting] educators without essential teaching skills into classrooms”. I brought up the example of one-room schoolhouse teachers and homeschooling parents as examples of people who teach successfully without anywhere near that amount of coursework. Another example I realized this morning was my own profession of college teaching. Most college professors have never had a pedagogy course in their lives, and yet many of those are among the best classroom educators our society has to offer. They certainly have “essential teaching skills”.

Of course there are also many professors whose teaching is atrocious. But there are also high school teachers with 30+ hours of pedagogy courses whose teaching is equally atrocious, and it’s highly questionable whether they have “essential teaching skills” despite surviving all that coursework.

What exactly are “essential teaching skills”? How do these differ from one teaching situation to the next — the preschool classroom, elementary schools, public high schools, private high schools, college classrooms, homeschoolers’ living rooms? Is there a single set of “essential teaching skills” that is common to all teachers, regardless of their context? And what role does education coursework play in conveying those skills?

Indiana Superintendent of Public Instruction Tony Bennett is announcing today a plan to overhaul the state’s system for teacher licensure. The announcement is here, and there are three PDF’s linked at the bottom of that page that go into more depth. [Update: There's now a 7-minute video of the press conference at this site as well.] And here’s an Indianapolis Star article (written prior to the announcement, so it’s a bit short on detail) that gives a thumbnail overview and some reactions from local education people. Those reactions seem pretty heated, and when you read the details of the program, you can begin to understand why.

The first point listed in the plan, and the one that seems to have the most impact, is that requirements for content knowledge for pre-service teachers are going to be ratcheted up several notches. Secondary education teachers will now be required to earn a baccalaureate degree in a content area — not in education — and earn a minor in education. Elementary education majors may do this as well, or earn a baccalaureate degree in education with a minor in a content area. Those aspiring to change careers into teaching do not have to get any formal coursework in education at all, but rather be certified by the American Board for Certification of Teacher Excellence, which involves holding a baccalaureate degree (in any subject) and then passing a teaching exam and attending teaching workshops. In addition, the very definition of a “major” or “minor” in education will be changed: A degree in education can have no more than 30 hours in pedagogy, and a minor can have no more than 15. This new system, along with the other changes in the plan I didn’t just mention, would be put into place on July 1, 2010.

In a word: Whoa. These are some  monumental changes to the way things are currently done here in Indiana. And that’s not all. There is more to the proposal than what I mentioned above — in particular, a big change is that the PRAXIS I exam will no longer be required, and there’s an end to a portfolio and mentoring requirement that I know firsthand new teachers hate — but let me give some quick thoughts about the changes I did mention, which affect directly those of us involved with training new teachers.

• The change in degree requirement for secondary education teachers is huge. Consider the Mathematics Teaching major at my college. As it is, students in this program take 24 hours of “professional secondary education” courses (including methods courses, instructional strategies, etc.) along with several semesters of field experiences plus internships in education. Most, if not all, of the courses in the “professional” category would probably be considered “pedagogy” courses under this new system (although there’s no clear definition of that term). Under the new system, all of that would be replaced with an education minor that consists of no more than 15 hours of pedagogy courses. This is effectively cutting the “professional secondary education” courseload for these students by half or more.
• Like I said, this is huge — for both students and faculty. Students wanting to be secondary educators are now going to have much more flexible schedules and greater choice. And if, previously, a secondary education content major was heavy on the education and light on the content area, the whole world will be changing for students in that major. For us, our Math Education degree is just one course away (an independent research project) from a Mathematics degree; students in Math Education usually just double-major, so this change is not going to affect us much. But it could completely change the landscape for other programs where the math (or science or whatever) education major is something like half of the associated content area major plus a bunch of education courses.
• It’s also huge for faculty. Now you can see why some of the teacher education people might be very concerned. Their pedagogy courses are going to be depopulated. And the effect will be far worse at larger schools, where education professors tend to specialize more and you might have some profs whose entire course load, year in and year out, consists of pedagogy courses — courses that are now being taken by only a fraction of the former number of people. If universities were like industry, we’d simply lay off, reassign, or let go the profs whose services are no longer needed. But what if such a prof has tenure? Things get complicated.
• I think the change to allow ABCTE certification is brilliant. I have talked to dozens of students and their parents who are thinking about going into teaching, but they aren’t sure, so they want to major in a content area and then “go back and get a teaching certificate later” if they felt the call. It is painful to have to explain that, in Indiana, you can’t just “go back and get a teaching certificate”. Well, now you can. (Assuming this all passes.) This is an excellent way especially to get more teachers in math and science. I know a lot of scientists and engineers who have wanted to get out of industry and into teaching, but the amount of coursework required was a real hindrance. Now there is an alternate route.
• The speed of the timeline is shocking. If this goes through, this upcoming academic year will be the last one in which the traditional secondary education majors exist. That’s got to be a major jolt to the system for many colleges, especially the education departments. But it becomes less shocking when you read the “Licensing Summary” PDF and see that the Obama administration’s Race to the Top program, and the tons of money contained therein for states with “streamlined” licensing processes, is listed there. They want to get this deal done quickly so they can be better positioned for money from this program.

A telling statement from Pat Rogan, executive associate dean of the education school at IUPUI, was in the Indy Star article: that the proposed limits of 30 credit hours of pedagogy in education courses for education majors and 15 for education minors would “put educators without essential teaching skills into classrooms”. Seriously? It takes more than 30 credit hours — the equivalent of an entire academic year of coursework if the student took nothing but pedagogy courses, 15 hours a week solid for nine months — to convey “essential teaching skills”? Somebody needs to send a memo back to the great one-room schoolhouse teachers from 100 years ago, and all the successful homeschooling parents today, letting them know that they are badly lacking in their preparation and need more pedagogy courses. Perhaps the gist of this entire plan is to say to education schools: Align your conception of what constitutes “essential teaching skills” with reality, and then redesign your programs to match.

I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and  as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.

When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:

• I thought that competency in calculus consisted in the ability to think through difficult mechanical calculations. For example, calculating using multiplication by the conjugate was an essential component of learning limits.
• There were certain kinds of problems which I felt were inseparable from a proper understanding of calculus itself: related rates, trigonometric integrals, and a few others.
• I thought nothing of calculus that didn’t involve algebra. I’m not saying I held a low opinion of numerical or graphical calculus problems or concepts; I’m saying I didn’t even have them on my radar screen. I spent no time on them, because I didn’t know they were there.
• Mechanical mastery was the main, and in some cases the sole, criterion for student learning.

Since then, I’ve replaced those criteria/priorities with these:

• I care a lot less about mechanical fluency in algebra and trig, and I care a lot more about whether a student can read a problem for comprehension and then get an optimal solution for it in a reasonable amount of time and using a reasonable method.
• I don’t think twice about jettisoning any of the following topics from a calculus course if they impede the students’ attainment of the previous bullet point: epsilon-delta proofs of limits*, algebraic limits that involve sophisticated algebra tricks that students saw five times three years ago, formal definitions of continuity, related rates problems, calculation of integrals using limits of Riemann sums, and so on. I always want to include these, and I do it if I can afford to do so from the standpoint of managing class time and maximizing student learning. But if they get in the way, out they go.
• I care very much about whether students can do calculus on functions of all shapes and sizes — not only formulas but also tables of data and graphs — and whether students can convert one kind of function to the other, and whether students can judge the relative pros and cons of doing calculus on one kind of function versus another. The vast majority of functions real people encounter are not formulas — they are mostly evenly split between tables and graphs — and it makes no sense to spend 90% of our time in calculus working with formulas if they are so rarely the only option.
• I don’t get bent out of shape if a student struggles with u-substitution and the like; but it drives me up the wall if a student gets the units of a derivative wrong, or doesn’t grasp that a derivative is a rate of change, or doesn’t realize that the primary purpose of calculus is to quantify what we mean by “rate of change”. I guess that means my priorities for student learning are much more about the big picture and the main ideas than they are the minute, party-trick algebra/trig calculations.

Perhaps the story would have been different if I’d remained tasked with teaching calculus to an all-engineer audience. But here, my classes are usually 50% business majors, about 25% biology or chemistry majors, and 15% undecided with only a fraction of the remaining 10% being declared majors in mathematics (which includes students in our 3:2 engineering program). But that’s the story as it is, and I’m sticking to it.

* Technically I never have to omit these, because we don’t do them in our intro Calculus class here.

The Australians are also facing critical shortages of students choosing to study science, math, engineering, and technology (SMET) fields:

“It is no exaggeration to say that the relative decline in the science, technology, engineering and mathematics capability and literacy of South Australian school students is a very serious situation that requires decisive remedial action by the government,” said Engineers Australia state president Bill Filmer.

“There is an urgent need for reprioritisation in schools, staffing and curricula to overcome this problem to enable South Australia to be more competitive in the knowledge-based economy.”

The report also identified the lack of training in science given to primary school teachers as a key issue and questioned their commitment to teaching science. [Emphasis added]

As to that last sentence above, insofar as I can understand the teacher education curriculum in Australia from a little bit of Googling, the curriculum for primary teachers does seem awfully lightweight on the math and science end. The curriculum at the University of South Australia has students take a three-course sequence in “Studies in Science, Mathematics, and Society and Environment Education”, and the course descriptions go like this:

[For the first course] This course engages students with constructivist perspectives of student learning; social constructivist pedagogies including interactive approaches to teaching; thinking and working mathematically scientifically, environmentally and socially from socially inclusive and critical perspectives; planning for learning in mathematics, science and society and environment; key concepts embedded in sorting and classifying, pattern, number, living things, interdependence and ecologically sustainable components of Years 3 to 9 curriculum.

[Second course] This course engages students with constructivist perspectives of student learning and focuses on interactive approaches to teaching and student questions; thinking and working mathematically, scientifically, socially and environmentally from socially inclusive and critical perspectives; planning for learning in Mathematics, Science and SOSE; student centred inquiry; equity (fair trade) governance (political, social and economic systems); key concepts embedded in spatial sense and geometric reasoning, energy systems, matter and fair tests, personal footprints, democratic participation and poverty as aspects of the Years 3 – 9 curriculum.

[Third course] This course engages students with constructivist perspectives of student learning and focuses on interactive approaches to teaching and student questions; thinking and working mathematically, scientifically, socially and environmentally from socially inclusive and critical perspectives; planning for learning in Mathematics, Science and SOSE; student centred inquiry; the SOSE value of social justice and equity through refugees and Indigenous Australians; key concepts embedded in measurement, earth systems (soils and weather), plant and animal relationships as aspects of the Years 3 – 9 curriculum.

Like I said, it seems light on the actual science and math content, but the students will certain get lots of social justice issues and a bias towards constructivism as the religion pedagogy of choice. Perhaps I don’t understand Australian culture as I should, but if I were a student being taught by someone thoroughly drilled in this kind of thing, I probably wouldn’t like math or science either. And if I were a teacher who wanted to teach math and science because, well, I really liked math and science, I would be a little put off by the back seat that the actual disciplines take to all this constructivism and social justice stuff.

It would be interesting to take the countries who are having these kinds of problems in one column, and the countries that are eating our lunch in SMET fields in the other column, and compare how science and math teachers are trained in each column.

Yesterday on Twitter, I posted a “tweet” venting asking why it is that a student who keeps making the same mistakes over and over again on assignments, and who receives feedback clearly telling him/her about this mistake and even telling her/him that this is the nth time they’ve made the mistake, won’t come to office hours to get some additional feedback or at least ask some clarifying questions about the mistake they are making. I don’t usually expect replies on Twitter, but I got this one:

Two points in response to this.

• Professors aren’t scary. Well, OK, some are — but most are normal human beings who really want to help students. And besides, if the student never comes to office hours, on what basis do they say professors are “scary”? How would they know?
• Even if professors are scary, so what? You need the help; you take initiative to get it. This is the way life works, and if students are not learning this in college, exactly how are they getting ready for life? If, in the future, this hypothetical student gets a job and lands an important role in a difficult project and needs to get help from her/his supervisor, what happens if s/he feels like the supervisor is “scary”? Does s/he suck it up and ask for help or guidance (possibly to find out that the supervisor isn’t so scary after all)? Or does s/he keep running around not getting his/her project done, hiding behind the canard of “that supervisor sure is scary“?

To answer my responder’s question, yes, there is a tutor s/he can visit: THE PROFESSOR. And students’ tuition has pre-paid for a semester’s worth of on-demand, unlimited one-on-one tutoring with that prof. The rumor is that s/he even understands the material. Will the student take advantage? The invitation is out there; hopefully common sense will overcome juvenile fear.

This is the second installment of Monday GTD Moment, where I take a post to blog about Getting Things Done and how it applies in an academic setting. Here’s the first post. If you’re unfamiliar with GTD, here’s a good overview, and make sure to read David Allen’s book that started it all.

It’s week 5 of the semester for us, which is crunch time for students — and professors. This is the time of the semester when everybody has tests and papers all due, usually on the same day, which means there’s lots of grading. I don’t like grading, but it has to be done. And if I treat grading lightly or let it pile up, I will make mistakes when I grade and students won’t get the feedback they need to improve in a timely way. As an academic type, grading is one of the most important, difficult, and time-consuming features of my job and therefore requires careful management. But it doesn’t fit neatly within the GTD framework which I use semi-religiously to handle all my other productivity issues. So here are a few lessons I’ve learned that have helped me use GTD to be a better grader.

1. Grading is its own separate context. This is not canonical GTD doctrine. In David Allen’s book, contexts refer to physical locations or tools that you have on hand, such as @home, @phone, and @errand. Contexts are about where you are or what you have available to work with. But I’ve found that, unlike in the business world, in the academic world there are not only contexts for locations and tools but also activities where I have to be in a certain “place” in a mental sense to get them done. Grading is one of those contexts. I can grade basically anywhere and I don’t always need particular tools to do it, but I do need to be in a particular state of mind to grade. And while I am grading (and even for a few minutes after I am done grading) I can’t easily switch gears — much the same as if I were running an errand or talking on the phone. Also, and importantly, I’ve found that if I try to shoehorn my grading into a typical place/tool context like @macbook or @office, grading tasks get put into the same place as non-grading tasks, and guess what kind of task would usually win my attention? Having an @grading context helps me to maintain my focus regardless of location or tool needed. When I have some time and space to grade, I focus in only on the tasks under @grading without interference from stuff that is not grading (unless I choose to look at multiple contexts at once).

2. Most grading “tasks” are really projects. One of the main problems I had early in my career with grading was that when confronted with a grading task, like “Grade Calculus test 1″, it took so long to get it done that it became demoralizing even to think about it, and so I’d shelve it and procrastinate. What I learned through GTD is the crucial definition of the project: A project is a thing that needs to get done but which requires more than one physical action to complete it. That’s all. Nothing more. Projects are not always highly complex long-term constructions — making a peanut butter sandwich is technically a project. Upon adopting this wide-open criterion for projects, I realized that the reason it’s easy to get demoralized by the sheer quantity of grading is that we think of grading a test as a task, when in fact it’s a whole lot of tasks all related to the same thing that needs to get done. As David Allen says, you cannot “do” a project. You can only “do” a physical action. So where a prof can get discouraged by looking at the one “task” of grading a calculus test because it takes hours and hours of work, by seeing one’s progress through the various tasks of the project (Grade problem 1, grade problem 2, etc.) you can feel, rightly, like there’s stuff getting done and you can see your progress. This is just an extension of what we math profs always tell students: When you have a difficult problem in front of you, break it down into manageable parts that you can do, and then do the first one.

3. Grading tasks have to be broken down into granular tasks — but not too granular. There’s a question related to what I just wrote in (2): What exactly is a single, physical action when it comes to grading? It’s possible to be too granular in breaking a project down into tasks. For example, if I took my calculus test and broke it way down into tasks — Grade Alice’s problem 1(a), Grade Bob’s problem 1(a), and so on throughout all my students’ work on problem 1(a) in all my sections of calculus and then throughout the entire test — then the list of tasks is so detailed that it takes longer to enter it than it does to actually do the tasks. But we do need to break projects down into their constituent tasks, so some happy medium needs to be found. For me, the middle ground is to consider the grading of one section’s worth of papers on one problem — or perhaps one page of an exam — as a “task”. So “Grade Calculus Test 1″ would go in as a project, and the tasks in this project might be “Grade Section A page 1″, “Grade Section B page 1″, “Grade Section A page 2″, and so on. And of course these are all filed under the @grading context (unless there is a good reason not to, for instance if grading requires the use of a computer).

Here’s what it looks like right now in my system for one particular item, the Quarter-Term Exam for my calculus classes:

The A and B refer to the sections I teach. As you can see, I still have a way to go. But because I am conceiving of grading this exam as a project rather than as a single monolithic task, I can at least feel good about the fact that I got page 1 graded for both sections, and I have a reasonable quantified sense of how much longer I have to go. So I don’t get discouraged. (Or if I do, it’s because of what I see when I grade, not because of the grading itself!)

Have a productive week, and don’t let the stacks of papers get you down!

[Photo by wudzy]

Why is the concept of the difference quotient so hard for beginning calculus students to handle? The idea is not as hard as some other concepts at this level that students have fewer problems with. You start with a function f and a point a. You are asked to write, and then simplify completely, the fraction

or

This involves four clearly-defined steps. (1) Compute all the function values in the numerator. (2) Perform the subtraction between the two objects in the numerator and simplify. (3) Factor the result out completely, and (4) see if you can find a common factor to cancel. And there’s a step (5): Since you know that every time you’ve done or seen a problem like this, there’s a factor/cancel step at the end, you know you screwed up if there isn’t one.

But somehow, the fact that this is a totally algorithmic, almost automatic process that is the same procedure every single time — and even the slight variations among instances only consist in algebra tricks — doesn’t stop students from suffering a complete brain-freeze at the sight of them. They convince themselves they don’t know how or where to start (despite worked-out examples or even difference quotient exercises that they themselves have worked out before). They plug in to f. They use instead of . And so on. There’s a massive wall of intimidation that these exercises lay down, and even those who make it over that wall end up talking themselves into doing all kinds of stuff that is wrong bordering on bizarre. These exercises get inside their heads somehow. And these are young men and women smart and capable enough of getting into college, mind you — not dummies.

The cure for math intimidation is a disciplined heuristic for solving problems and a faith in your algorithms for more mechanical exercises. But with difference quotients somehow the heuristics and algorithms run fleeing like Tokyo residents before Godzilla. It’s not just difference quotients, either — there are lots of algebra components that throw calculus students for an absolute loop, and I cannot figure out why. Any ideas?

Interesting, well-produced, and potentially controversial video here:

“Good teaching is good teaching. And teachers don’t have to adjust their teaching to individual students’ learning styles.”

The Fort Wayne Journal-Gazette has this article today by Karen Francisco which is an excellent, if troubling, survey of the problem of rising textbook costs and the things people are doing to offset those costs. I was interviewed by Ms. Francisco last week for this article, and I am happy to say that unlike in my previous newspaper interview experience, she got my comments exactly right (and asked if my name and position could appear in the interview). Here’s what I had to say, although you should read the whole thing:

Robert Talbert, an associate professor of mathematics and computing science at Indiana’s Franklin College, is one of several hundred U.S. college faculty members who have signed on to PIRG’s online pledge to help control textbook costs. He’s passionate about the issue.

“Many of my students are either first-generation college students, students from middle- to lower-income families, or both. They are struggling to afford college as it is – often having to work off campus, which then affects their class performance – and it really pains me to see textbook companies charge more and more for a less and less useful product,” he said in an e-mail.

Talbert said he’s bothered not just by the cost, but by the quality of the books, which he said are often “poorly written, chaotically organized and full of so many irrelevant graphical elements and sidebars” that the information students need is difficult to find. If he can avoid it, Talbert doesn’t require a textbook or directs his students to an inexpensive one.

“In my abstract algebra course last fall, I used no textbook but rather homemade course notes and a handful of helpful Web sites,” he wrote.

Of course this is all old news to Casting Out Nines readers!

Later, after discussing Rice University’s adoption of an open source textbook for their introductory statistics class, she went on to quote me about the potential for open source textbooks:

“Imagine having a calculus textbook, the contributors to which are some of the best calculus professors in practice today, and which includes not only text material but also links to Web sites, embedded video, interactive applets for visual/kinesthetic learners, and user-contributed problem sets – for free,” he wrote.

“There’s a stigma against such things now, just as there is a continuing stigma against Wikipedia in academia (because academicians have a hard time accepting the legitimacy of something that is not peer-reviewed), but I think once students start learning and getting engaged with material through these things, that stigma will go away quickly,” he wrote.

Actually I think the stigma isn’t so much against Wikipedia itself as it is the notion of putting Wikipedia on the same level of authority as, say, a peer-reviewed monograph or a published encyclopedia. But a lot of academic types let their stigma start there and pretty soon the entire concept of an open-source informational source is stigmatized. That’s just throwing the baby out with the bathwater.

Again, go read the whole article, especially for the stories from students about what they are made to buy at a high price that can be had elsewhere for next to nothing, comparatively. It’s shocking.

The video post from the other day about handling ungraded homework assignments went so well that I thought I’d let you all have another crack and designing my courses for me! This time, I have a question about really bad mistakes that can be made in a proof.

One correction to the video — the rubric I am developing for proof grading gives scores of 0, 2, 4, 6, 8, or 10. A “0″ is a proof that simply isn’t handed in at all. And any proof that shows serious effort and a modicum of correctness will get at least a 4. I am reserving the grade of “2″ for proofs that commit any of the “fatal errors” I describe (and solicit) in the video.