Conference Material

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Or they can try as two different groups to come to some sort of working consensus where the stakeholders agree. Half the time, they separate out. Half the time, they work together. You can never predict how it's going to work. And so these are some guidelines for them, what they're going to have to think about, when they're going to have to build their model. I give them a computer program where it's a spatially explicit model from Bioquest where they make an ecosystem, where they break the Yellowstone ecosystem down into three areas; inside the park, one area outside the park that's part of the ecosystem, where cattle are. They then can add vegetation, wildlife, livestock, and wolves.

They first have to make it stable before the wolves. Then they can add the wolves and see the consequences. It's not that easy to do. They have to go to the literature and find out the critical parameters of the life history that shape the demography and the population dynamics. I don't give it to them. They have to go discover it. They can put it into the model. And then they can run their model. And here's an example of one of the runs. That you can see in the four areas. Three inside the park and one outside the park. That when you add wolves, the wolves at this density level stay in just the one area where you add them.

They don't destabilize the environment. The grass keeps growing, the wildlife are there. They haven't spread into the cattle areas. Notice in the cattle areas though, there's no wildlife. That the cattle out compete the wildlife and the cycles are more severe. So they learn a lot about the dynamics. Now if they're going to get into the economic impact of these wolves on the cattle, how would you do that? When you're starting to look at the fluctuation in the livestock, where do you make that decision on the loss of one cow? At the trough or at the top? Will it really have an impact on the stability of the population?

These are the issues they start to learn to wrestle with when they start to look at the dynamics of the problem. And so it becomes a very rich learning experience.

Well, as I say, that project burned itself out. So I had to create another project. And I created a project, also using the computer, which is about conserving endangered species. It's entitled, "Where Have All the Species Gone? On Bringing Them Back by Managing Population Dynamics". And so again, they have to deal with the natural dynamics of populations and then the anthropogenic impact on those populations. And this is all the students get. They get a rationale and they get a description of the type of analyses they should think about. Very simple. Not a lot of meat here, certainly not cookbook.

And what they get in between here is a hard core dose of mathematics. That's the (...inaudible microphone blip) model. It's a population projection matrix. It's linear algebra. And all it is is a way of predicting the size of the population in ceratin age classes, infants, juveniles, and adults, as a function of the number of infants, juveniles, adults today times the dynamical processes. Which are all functions of density. Okay? Mathematics. Very straightforward. And in here are terms feeding back on age of first reproduction, on fecundity, and on survival.

They have to go to the literature. I give them six life tables. They can choose the species they want. Red caugated(sp?) woodpeckers, manatees. Dolphins in New Zealand that are going extinct. They get facts. They get on the Web and they get all the data. And they find these critical parameters. They have to make some estimates and assumptions because the density dependence hasn't always been worked out.

And they plug it in, they get their graphs, and they have to come in and give a half hour presentation to the class on what was happening, what the human effect was to exacerbate the problem, and, most importantly, what biologically should be done, but what practically should be done. So it ties it to the real world. What can be done given the politics and the economics of the situation. So these are two exercises that we use just to let students explore and do that if then sort of process.

How should you engage and empower your students? How can you take the same processes that I've been talking about? Okay. This is where the note taking should start. That was all introduction about how I think as a scientist and how I think as a teacher. I've tried to do with from generalities with specifics from the examples that I use. Tried to model the process of both the scientist and the science teacher. Now let's step back and decompose what I've been telling you and put it in your context. A context that I do understand. One, I work with teachers. I'm on the executive committee of E=MC2.

Two, I train a lot of teachers, believe it or not. That a lot of the students in my class, by the time they're done, want to go into teaching. Okay? They (...inaudible microphone blip) teaching in the high schools or elementary school teaching. Okay? Training program board at Princeton as well. I've also been on the school board. So I do know something about banging your head against the wall when it comes to the taxpayer and systemic change. And so I've tried to take all of that as the filter through the ideas that I've just talked about.

We all understand the problem of burning textbooks and we all understand the benefits of moving into inquiry based science as provided by the kits. But we have to caution ourselves. Because before all knowledge was between two hard backed covers, okay? Students didn't go to the library. Students' experiences were limited by the cause and effect relationships. Now the students' experiences are shaped by what's in the box. Now in that box, there's a lot more creativity. There's a lot more compelling issues to be studied. But are they always the same issues the students would find compelling? Or are they the issues that we, as adults, somehow think should be the compelling issues?

So I want to come back to that in a moment. And this is where partnerships with scientists can expand the potential for the kits. Move away from the quick and dirty demonstrations of processes. I could demonstrate to my students the dynamics on the computer as I quickly move through it. You change this, you do this, you change this, you do this. That would not teach them anything. Because I would have already fine tuned those that are stable or those that are not stable to make pedagogical points. They have to explore it on their own. And so I have to move away to let them accentuate the minds on experience.

Now comes the toughest one. The one I still wrestle with every time. And that's this trade off between focusing a student's perception, taking away some of the freedom to choose a compelling problem, and liberating the problem posing phase. If I am going to help, in a pedagogical sense, a student to learn how to ask questions and to parse those questions into testable parts with predictions, I have to be in command of the subject matter. I have to be in charge of what's going on in that classroom.

So if I let every student pick every problem that they want, I'd be thinking on my feet all the time. I'd be exhausted intellectually. And so there's this balance between the compelling problem the student actually really wants to know and the one that I can shape into a way where I can bring my disciplined training and thinking and share and model that for the students. That's a tradeoff that is not easy to solve. And it's hopefully something that will come up in your discussions in the next few days. Because as you get curriculum from manufacturers and from the school districts themselves, a lot of thinking goes into shaping and focusing so that the resources are there to do the hands on part.

But we don't want to escape from the minds on part which is the fun part, which is that model building, that problem posing phase with the predictive elements. So that's a tough one.

Meld facts and process. Don't separate them. It's not about content or process. The two go together. If there's anything I've illustrated to you, it's that wealth of past experience that becomes the basis for using the process. How many of you as teachers know what happens in the year before? Not in general because it's in the curriculum guide that they work on rocks and you work on plants, but how many of you know what the students have mastered? That integration becomes critical for melding process with fact. Because those are the facts they've mastered. They've been tested on it. They may not recall them all, but they're there in their deep recesses.

And so when you do that, then you can bring the new processes on their past experience and you don't reinvent the wheel. Abandon the drudgery to enrichment model. And to a large extent, the science (...inaudible microphone blip). Has it been done in mathematics? What do you think? Have we gotten away from you must master these manipulative techniques to actually solving problems and thinking in a quan-- (...inaudible microphone blip) Make sense? I don't necessarily think so. I think we're moving there, but I don't think we're there yet. I think the science may be ahead of the mathematics in that regard.

One of the reasons I can say that is that, for many years, I would go into the high school and teach calculus. The AB calculus class. The reason I did that is it wasn't until I was a sophomore in college that I realized that calculus was useful. Okay? I mean, why should (...inaudible microphone blip) I mean, think about it. It's about rates of change, it's about dynamical systems. How many of us ever really wrestle with dynamical systems in our lives? And we don't do it that quantitatively. But we learn the tools and we learn techniques. And I found that I got tired of looking at ballistic trajectories on bombs or trying to minimize the amount of cement that went into a swimming pool.

Got tired of doing those sorts of problems. Or minimizing the chain link fence around someone's ball field. I couldn't understand why I wanted to do that. It wasn't compelling. So I didn't. I just learned it, I got my grades, I was done. It wasn't until I got to biology in college that I said, "My God, I can't (...inaudible microphone blip) problems unless I know this mathematics." So I go (...inaudible microphone blip) into the senior class after I've learned (...inaudible microphone blip) Certain processes. And I asked them, along with the AP, the biology class, to come together and to explore how to solve problems.

Now I don't tell them to go into (...inaudible microphone blip) They're not dumb enough to realize that I wouldn't be in a math class if it wasn't to do that. So they know that they've got to solve the problems using calculus. Okay, we're graduating to a microphone here. I set them biological problems. And we go through basic evolution. You know, let the kids from AP biology share their thoughts on how natural selection shapes adaptation. And they, dee-dee-dee-dee, get it all right. And then the calculus kids say, yeah, these are techniques we know. Okay, great.

And I say how are we going to put these together to solve a problem on what size territory should a bird have when the environment is of poor quality? As opposed to when the environment's of high quality? Okay? You know, that was one question they were talking about. They were learning breeding in birds. And I said, "Okay, how would you solve that problem?" They look at me, you know, as I just said something in Greek. And I then walk them through. Well, what would be the benefit of having a territory? What would be the costs of having a territory? And they write them all down. In English. Great. We're moving. At least we got our list of possible explanations.

They're starting to pose the problem. Then what do we do? I say, "Well, how will we put these together?" And they go I haven't a clue. So I ask them to come to the board and draw some graphs. And that's easy. They come up and they say, "Well, the benefits are going to look like this and the costs are going to go like this." I go, "Great. Let's put some equations to those." And we make everything a function of their x axis. And it's all in algebra. I say, "Take the derivatives. Solve it." And they go, "But there's no numbers there." I go, "That's right, they're letters. Use primes. Go ahead, take the derivatives."

And they do that and they eventually get this funny looking expression that ultimately has something like the derivative of the function over the function. They've just derived a major formula in evolutionary biology called the marginal value theorem. They don't know it's that, but it's there. And they go, "Well, we can't do anything with that. It's two letters." It's function A primed over function A. I go, "You've got a lot of information there. What if A gets big? What happens?" Oh. Well, it depends on the shape. Because the derivative, first derivative. Is it positive? Is it negative?

They start to say, oh, my God, we don't know enough. Okay? And they look at their graph and the shape of the line that they put in. Which is the assumptions. That shape is an assumption. And eventually they get it. They know the sign, if it's positive, negative. And it's now still Greek. Then I say, "Well, why don't we write some equations for these generalized relationships?" And then they solve it. And then they know what they've done. They've taken it from the abstract to the concrete and then back to the abstract. And so they're able to reason from biological and mathematical first principles through to using their calculus in a meaningful way.

So it can be very powerful not to just learn techniques first and then apply them to the real world. Do the two iteratively. Do it with a problem that comes from their own volition, guide them through the explanatory power of Cripps' mathematics to solve that problem. And by doing it, you're moving into real world experiences. They're the compelling problems that fascinate them. That they didn't understand from their textbook. Because if any of you teach AP biology, you know it's pretty regimented and it's not of the science that we're talking about here.

I also suggest do things that students can touch, see, feel, and use their senses for. So ecology and behavior when they're young. Don't do cell biology and molecular biology because that's abstract. They can't see it. It's a bunch of letters connected by lines on paper. They can do that when they're older, when they believe a few things based on their past experience. Okay?

Lastly, in the classroom alone, alter your testing. Think about the example with the wolves in Yellowstone. What was the test there? What was the assessment? It was their ability to convince me that they understood the dynamics to convince me as a manager to follow their policy. It wasn't a written test. I didn't ask them specific questions. I could evaluate what they were doing. I knew the end; what I wanted before I started. I structured the investigation to get to the pattern of result that I wanted in terms of assessment. So that becomes important. Match your goals to questioning.

And I think that's a critical area when it gets to assessment. Because that keeps the process alive. It stops stagnating. Because each year as you build upon the knowledge base that you have as teachers, your ability to test will change. Even if you have the same students that have to go through the process the same way.

And lastly, make your scientists and parents partners in the process. Because they can provide the support from down under, the parents, and the support from top down, the scientists, that help keep this process creative. Now there's many prescriptions to do this. And here's one that I've culled. I'm going to talk about the prescriptions for the teachers first and then the prescriptions for the scientists. And my prescriptions are worse than your prescriptions. I have a lot of don'ts in mine. I don't have any don'ts for you guys.

The key is to identify key ideas to explore. Okay? Think about what that means. Find those compelling problems that will engage your students. Then search for activities with potential and focus those activities into explorations. Now many of you will do that in your own classrooms. Some of you will not. They'll be provided by the curriculum that you're using or the kit that you have. They will do those three steps. And I'll argue that we have to be careful not to lose point number one. That's where partnerships with scientists become important. Because that's the bookend number one that provides the tying and aligning of this process to the content area.

Then stand back and let your students explore. There should be a help here. Then comes the coaching style of teaching, the Socratic method. Help translate and transform student observations into testable questions. Those if then predictions. And then when they test them, help students persuade others that they have the right answer. And then lastly, help students extract generalizations or general rules from their observations. This is the other bookend that's tied to content that most of us in the classroom do not do.

We stop when the students have convinced you and themselves that they have mastered the material before them. When do we give the opportunity for reflection? I heard Susan talking about you wanted last year more reflective time. Do we give our kids time to reflect? Do we give our kids time to take the experiment in the laboratory or the classroom and pull it out to the real world? Do they make comparisons? Do they do those discomfirmatory tests that I was talking about? Oftentimes not. Because we don't have time in the curriculum for that. They've mastered it, they've demonstrated the process, they've done well on a test. So be it. Let's move on to whole language or let's move on to the next science unit.

But I would argue these two bookends are as important as the dynamical processes which, for the most part, teachers have mastered today. The ability to stand back and to coach and let kids explore, as scientists, you all do. I'm convinced of that. Because the teachers I've seen doing it, do it very well. But the question then becomes can we do these? Can we enrich the project by identifying the real key ideas? And can we draw upon that knowledge base for the comparisons for the generalizations? That's where increased content becomes critically important.