Experiential Learning

I have been attending a very interesting workshop this past week on experiential learning.

This week has given me a vocabulary and clarity about things I have always believed about education. It is not about textbooks or notes on the board or even teaching. Why not? Because learning can occur without any of that. Education is about asking questions, provoking ideas, and providing experiences that create disequilibrium which can only be righted by changing our perspective. It is personal, it is purposeful, and it is earthshaking. The more we and our students can acknowledge the goggles we wear as we see the world, or take them off, or even exchange them for someone elses the more we can learn to grow from each and every experience.

This may seem unreasonable for a math teacher to be considering – after all, math is made of facts and rules, all of which are linear in our understanding. One thing cannot be learned before something else is mastered. I cannot argue that students need to understand arithmetic in order to do well in algebra, or fathom the concepts of algebra before calculus can be fully explored. Yet many math teachers expend a great deal of energy in making sure that children can perform the operations and procedures accurately and quickly, with the thought that only once they have mastered these things will they be able to think coherently about the concepts.

I attended another conference where several presenters addressed this in a way that I wholeheartedly agree with. Professor Marshall Lassak from Eastern Illinois University said,

In the process of performing manipulations, students may perceive that the manipulation itself is the important thing, but it is not. When to do the manipulation, why to do it, what the before and after forms mean, and what they are used for are far more important than manipulation skills. [...] the concept is more important than the process.

Manipulations are not unimportant, nor is knowing the skills that make up mathematical processes. But too many classes focus solely on the mastery of the skills. Even when teachers say they value the conceptual understanding, one has only to look at their tests to see that the way they assess it is by watching students perform the skills – and then extrapolate a students understanding based on the ability (or inability) to accurately perform the skills quickly. I am guilty of this as well. Students need to do math, not study math. That is how they will learn it.

At this second conference another presenter suggested a new way to look at the problem solving process. Traditionally the paradigm has been that a problem presents itself in math, the mathematician creates a model which leads them to some insight about the problem, and then through hard work – and much ‘by hand’ computation – they arrive at a solution. The difficulty with this paradigm is that our students (and probably many of us) are not the type of people who have these insights on a regular basis. Thus we are left with teaching them procedures that have already been hammered out by our betters. This presenter, in advocating the use of technology, suggested a new paradigm. One in which we still begin with a problem followed by a model. But instead of waiting for the insight that rarely, if ever, comes we utilize the technology to do the brute force computations that lead to a solution. But we do not stop there. We look at the information, the data, and the patterns to find amongst them the insight that exists. What is funny is that this is how I have always learned math. It is probably why most of the traditional math classes frustrated me… and teaching them did as well. As the first presenter suggested, and my week learning about experiential learning reinforced, this person too seems to agree that the way for students to learn math is to do math.

I leave you with one other quote, which I am not citing at the moment. This was what one, great, teacher I know says to his students at the beginning of the year.

One of your [the students] fundamental obligations is to tell me when my teaching is getting in the way of your learning.

Cheers!

A Thank You for Calculus BC

This year our seniors wrote thank you cards to various teachers. I got several of them, but the following, printed here in its entirety (minus identifying names) was from a student who has a lot of potential, but really spread himself thin this year. His mom came up to me several times in the last month to apologize for his apparent lack of effort. Well, read on and see:

“Dear Mr. _________,

I just wanted to let you know how much I appreciated this year as your math student. I can think of no other <school name> class that challenged me the same way that Calculus BC has, and I learned more about problem solving and creative thinking than I ever have before. I know that our class wasn’t always the most diligent or focused, but I hope that you know that it was far more a result of our being seniors than anything in the way the course was taught. You know from my journals that I would have tweaked (and, in fact, you did tweak) some aspects of the course, but overall I want to thank you for forcing me to step outside of my comfort zone and really struggle. You’ve certainly helped me to start to develop valuable skills for college and beyond.

Thanks,”

What made this nice to hear is that I had already received all the cards I thought I would get, the grades were turned in, and the seniors had graduated, but this turned up in my box today. If one of my goals was to change my students attitudes about math and how they approach it, then I think this is some evidence that I made progress toward that goal. As for did they learn the material? I will wait with baited breath until the results of the AP exams show up.

As I share other pieces of my editing process this summer I will also put out specific pieces of advice I asked this years class to write for next years.

Buddha is a god?

I recently signed up to Facebook, and I have been exploring all the various little goodies that one can add to their pages. I am pretty fond of the Never-Ending Movie Quiz, and I am looking at my slowly growing list of friends to see what they have on their pages. The vast number of applications that can be added is daunting, so I usually just breeze past them.

However, on one person’s page I saw the “What God are You?” application. Apparently you take some type of quiz and then it tells you which god you are most like. What struck me was that they had Buddha listed as one of the gods you can be most like. What bothers me are the number of people who actually believe that Buddhists worship Buddha as a god. As a recovering Roman Catholic most of my family still is, and my wife’s family is protestant ranging from fairly open minded to the creationist “the bible is the literal truth” variety. The thing that some of them do not understand (we have not really had this conversation with the creationists) is that when we became Buddhist we did not just “switch allegiance” to a new god.

Are there some sects of Buddhism that revere Buddha as a god? Perhaps. Several weeks ago I read an article that suggested as much. But all the Buddhists I know, and the Sangha I am part of, would never say such a thing. I can accept that there are people that just do not know any better, although that is a pretty big part of a major religion to mix-up, but I get really frustrated when, after explaining to people that Buddha is not a god, they just assume he must be like the Buddhist version of Jesus Christ. Ah well.

Of course, I found out after my wife visited the fundamentalist branch of her family that they thought that Catholics worshiped Mary as a god(dess). Funny.

J.K. Rowling Speaks at Harvard Commencement : Harvard Magazine

The Fringe Benefits of Failure, and the Importance of Imagination

This was the title of J.K. Rowling’s speech for the Harvard Commencement. I do not really want to say anything about the apparent controversy of her speaking there (talk about elitist!) or the imagination part of her speech. Rather it was her talking about failure that struck me. As I have said in previous posts my school seems to do everything it can to protect our students from failure, and then when we ask them to take risks it is no wonder that they do not want to unless there is a safety net beneath them.

At any rate, excellent speech. I recommend everyone either view the video or read the text of the speech.

Elitist or not?

I have been reading a number of posts lately in the AP-Calc listserv through MathForum that have been talking about the screening process that various schools have for their AP Calculus AB courses. I know that the CollegeBoard recommends making it available to as many students as possible, and many people say that the only calculus that should be offered, at any level, is a college-level (and hence, AP) calculus course, as well as those that say passing a precalculus class – any precalculus class – should qualify a student to at least try AP Calculus AB.

Of course, many others weighed in about the idea that just passing one class does not necessarily indicate readiness for the next course (why not?), or that students unprepared (in whose judgment, and why?) will only bring the class down, as well as many other reasons.

Our school has probably one of the more restrictive placement systems of any school I know. In many cases to continue onto the next course only a B or B- will suffice (except in certain instances such as older students in a class where we allow a C, but they only get to go into the lower tracks). Of course, we typically have 30 or more students get into AP Calculus AB. With class sizes around 80 this typically means that roughly 40% of our students make it into at least AP Calc AB. More than half of the remaining students take something we call Calculus Fundamentals, which is a non-AP level calculus course. To get into AP Calculus BC, the course that I teach, a student must complete AB first. We have such a short year (because we are an independent school we do not have to have the mandatory 180 days) that we found very few students could successfully complete the full BC course in one year without severe academic bloodshed. As a result I have never had more than 9 students in BC and as few as 4. I feel that more could handle it.

We make a big deal, and we hold as a carrot to parents and students, that students can make it into the honors classes at any point – these are the only ones that lead to AP – but the reality is that few do because we don’t want to take a chance that a student might earn less than a B in a class. It has gotten so that a C at our school is practically considered failing and there are teachers and departments who only very rarely assign these grades to students. Sitting in our roll calls, where we look at each class of students and review their grades, it is clear that C’s pretty much only show up in math and science with any regularity, and that infrequent at best. And to be honest, if B or B- is the grade required to continue in the same ‘track’, then I guess we really do consider a C failing – because it fails to let a student into the next course.

I worry that we cushion the students way too much, never letting them take risks because we have decided that we know what is best for them. Sure, we may have the benefit of experience, but most of what I have learned came from those times when I screwed up. Why don’t we allow our students to do the same thing?

I read something recently from a teacher whose opinion I value which basically split students into two groups. The first being those only taking math courses to get an A and because their parents or college counselors told them they need to, and the second being those students willing to work at it because they will probably go into some math reliant major in college. And then this teacher suggested that we should be designing most of our courses for the latter group, and let the former group fend for themselves.

Then I realized that until recently I would probably have wholeheartedly agreed with him.

And now I do not. So where do I go from here?