Tuesday, December 6, 2011

Students create the problem to solve

Math 31 Assessment
Option 1: The head of NASA has approached you asking for your assistance plotting a course for the International Space Station.  The tracking device can be found at http://spaceflight.nasa.gov/realdata/tracking/index.html .  NASA needs an equation of the path, as well as the 1st and 2nd derivative.  The director is also asking for all relevant information about the path to be explained.  Create a potential problem the satellite might encounter and provide the solution to the problem.
Option 2: Ethan Hunt, an IMF point man, is securing top secret files at the bottom a cylindrical tube.  He is being lowered by rope into the room at a constant speed.  Unfortunately, while being lowered, another man walks, at a constant rate, towards the room.  Ethan is then raised back up out of the view of the approaching man.  While suspended in air, sweat is building at a constant rate until it reaches a critical value and drops onto the floor.  Using the video (a clip can be found here http://www.youtube.com/watch?v=k-oVuQpjG3s) use calculus to help out Ethan and determine various velocities, critical values, and timing to assist him in securing the evidence.
Option 3: Taking what you’ve learned this semester you must create a problem to solve. It must be based on a real world example (or movie world) and must have at least one solution. Be sure to submit your proposed solution in a sealed envelope. Five bonus points will be awarded if you are able to stump your teacher.

No matter which option you choose you must create the question and problem in the given scenario
 Calculus Student Calculus Student Teacher Calculus Teacher Calculus Master Real Life Application:Is the problem worthwhile solving?What are the implications of the result of the problem?Who would benefit with the knowledge of the answer to your problem? Only students of this course would see the relevance of this problem. The purpose is built on strictly recalling facts. The solution is only needed to complete this assignment. Problem is created from a plausible issue with major changes. Purpose is unclear and does not go beyond the needs of the course. Students in this course would only understand the consequences of the solution to the problem. The problem is created from a plausible issue or problem with minor changes. The purpose is clear and slightly exceeds the needs of this course. Few, outside this course, would benefit from the solution to the problem. The problem is created from a real life issue or problem. The purpose is meaningful beyond the needs of the course. The solution to this problem adds to the experience of the students’ real world knowledge. Explanation of math.Are you using basic math knowledge?Are you demonstrating most of the knowledge you learned in this course? Focuses strictly on basic recall and basic knowledge of the mathematical skills. Requires few sections to apply higher level thinking to solve the problem.Math is still largely focused on recall of knowledge. All levels of understanding, from basic to higher level thinking are implemented throughout the problem. Focuses on higher level comprehension, the use of the combination of multiple skills is evident throughout the entire solution. Communication of your answer:Does your work follow sequentially throughout your solution?Are there gaps in your communication? Minimal response is given with multiple gaps in the explanation process. The use of similar explanation techniques is used throughout the solution.  Communication is limited with various gaps in the problem solving. Most of the project is easily understood, and organization is mostly logical. The use of different means of demonstration is illustrated throughout the solution.  The mathematics is clearly communicated as well as the meaning of the solution(s)

1. Dave, your first option isn't calculus, as presented I am not even sure what it is (btw I worked at NASA). The second option is simple algebra, and the third option could be anything from silly to ingenious, depending on what the student turns in.

The reason the first option runs aground is that you are trying to formulate a problem in a context (orbital mechanics) that is simply out of reach for students (and teachers) in high school. I am for injecting applicable problems but it has to be reasonable or it just becomes silly. We use contrived problems because you can't start with the truly real ones yet because the context is too complicated. The real problems will reveal themselves soon enough when you go on to take the courses that require calculus, like physics or engineering.

The second option needs some related rate thing going on, like the person lowering ethan on the rope is walking in a path that would cause the rate of descent to be variable. Calculus isn't a better way to solve algebra problems. It is a way to solve problems that can't be solved by algebra. It is also other things to, but in keeping with your theme of practical applicability, it extends our reach past algebra.

And finally, your rubrics are not mathematical. Maybe I am misunderstanding what Math 31 is. My comments are critical but you are posting a better way to teach calculus and I am not seeing anything here that resembles how we (people that use calculus later) use calculus or that captures the mathematical enligtenment we experienced in calculus that lead to us choosing a path relying heavily on mathematical (or mathematical like) reasoning.

2. Robert, I think you are missing the last sentence. I am providing students two scenarios, not two problems, and asking them to create the problem and the solution. For you to say there is no calculus used, I would say there most definitely could be.

Take scenario one: One could look at the path as a sinusoidal path, talk about slope, concavity, relative max/min, absolute max/min, points of inflection. A problem, using calculus, could be: The satellite needs to launch a drone tangent to its curve that will fly directly over a point not on the curve. Where are all the possible launching points?

The same could be done for scenario 2.

As for the rubric, I am not only asking students to create a calculus problem, but to also solve it and communicate properly the answer, and you are saying this is not calculus? You are asking for mathematical enlightenment; are you suggesting that I use traditional multiple choice and numerical response, with a psuedocontext word problem at the end to achieve this?

Robert, do you have an example of what I could use which you say “resembles how we (people that use calculus later) use calculus or that captures the mathematical enligtenment we experienced in calculus that lead to us choosing a path relying heavily on mathematical (or mathematical like) reasoning.”

3. I am beginning to believe that the different perspectives of Dave and Robert are due to their looking at school in different ways. It seems to me that Dave sees school as an opportunity to get an education, while Robert sees it as an opportunity to get a training.
Although there are overlaps, the two approaches are very different and the exchange above seems to support that.
Dave is trying to get his students to think, to question, to explore, to look for connections. Robert wants students to learn what problems there are out there "in the real world" and how to solve them.
What do you think?

4. "Robert, I think you are missing the last sentence. I am providing students two scenarios, not two problems, and asking them to create the problem and the solution. For you to say there is no calculus used, I would say there most definitely could be."

If they create actual calculus problems from these scenarios and solve them then that is of course different than what I suggested, but we have different interpretations of real world. For example, that curve you see on the map is not the trajectory of anything, it is a plot of ISS's ground position on a cylindrical projection of the globe. It would be "real world" to me to unwind all that and produce the function that returns a map point {x,y} from ISS's orbital elements (which define an ellipse). None of that is calculus yet, but if you added something of the sort where the student is to find the lowest rate of change (velocity) in that curve, maybe to maximize viewing of the ISS from the ground, then it would involve calculus. Your example of finding a tangent to that sinusoidal plot would be calculus, but would have nothing to do with the context of ISS. Hitting a target from the ISS would be an orbital transfer problem and the path of the projectile would be another orbit with a slightly different sinusoidal ground path, not a tangent. I am not trying to be picky here, but what you have done is taken a standard problem like finding the tangent to a curve, put it in front of a global map and called it "real world". It isn't a real problem.

But, I am more interested in real math at this point, not a real orbital mechanics problem, which would obviously have to come after they have learned the math, and then after they have learned orbital mechanics. And your example of finding the tangent to a sinusoidal curve counts as that, but only if the following criteria are met...

1. The curve has to be mathematically defined. This can't just be a student making a poster where they draw a tangent to the curve and label it dy/dx. Since you did not mathematically define the curve then the student will have to do so and since they do not know orbital mechanics and the trig is probably beyond them to do the projection then they will have to invent a sin(x) plot that approximates that curve. I am fine with that. I would be more fine if they gave it a bit more personality and flexibility by making it a*sin(b*x) or something, but as long as they arrive at a mathematically defined curve.

2. Next, they must mathematically define the problem. Given a target at {x,y} find the tangent from that sine curve that intersects it.

3. And finally, they must provide a mathematical solution.

And I would judge all of that based on the mathematics they employed, not their creative writing or graphics skills.

5. "You are asking for mathematical enlightenment; are you suggesting that I use traditional multiple choice and numerical response, with a psuedocontext word problem at the end to achieve this?"

Pseudo context? What exactly was your ISS example? But to your point, no, I do not think that the standard fare of questions is enough to own calculus. But they are an absolutely necessary component, like the addition and times tables are absolutely necessary components of pre-algebra. But they are not the tipping point of owning calculus. Taking another course, either simultaneously or soon thereafter, that uses calculus, like physics for example, is the tipping point to ownership.

As far as multiple choice goes, if you don't like that format then don't use it. Personally, after reviewing over a million results, I don't see any significant summative assessment issue there. There is a formative issue there and in an ideal world, they are not the best format mid term. But I guess teachers don't have that kind of time. But even in that ideal world, I have no qualms about them being used at the end of term, none whatsoever. The only caveat being that students should be allowed to take a course again, and that should fulfill their graduation requirements. In other words, even if a student had to take algebra 1 twice and algebra 2 twice, that should fulfill their 4 years of math. Naturally, this is going to affect what they pursue after high school, but as long as they were trying, that is all we can ask of them.

You are trying to teach an advanced subject with advanced applications. If the student doesn't bring with them an interest and personal purpose for what it is you are teaching, they will not get it. I realize you are trying to reach them, and I would be less critical if you were trying to do this as a precursor to learning calculus rather than during. I don't know what the philosophy of Math 31 is, maybe it is a precursor to AP calculus or college calculus, in which case my comments do not apply.

6. "I am beginning to believe that the different perspectives of Dave and Robert are due to their looking at school in different ways. It seems to me that Dave sees school as an opportunity to get an education, while Robert sees it as an opportunity to get a training."

I think it is both education and training, and a huge part of your transformation from a student into a productive member of society. I can't think of any larger purpose for school, but I can think of other purposes. But in this particular case, I think if you teach a course titled calculus it should actually be calculus. If you want to relabel it "Calculus Appreciation" like they do with "Music Appreciation" then I am more than fine with that.

"Dave is trying to get his students to think, to question, to explore, to look for connections. Robert wants students to learn what problems there are out there "in the real world" and how to solve them.
What do you think?"

You cannot use or apply calculus in the real world, or in any world for that matter, unless you are already VERY MUCH prone to exploring and looking for connections. I literally assume that the student is already prone to exploring and looking for connections when they step into a class as advanced as calculus. That is the difference between Dave and myself. That is all that is going on here.

That defines all recent reform efforts. Forget the requisite interest. Forget the requisite development. Just put the damn students smack in the middle of these advanced courses anyways. I do not think there is any gain in that, in fact I think there is harm in that. This fad started in the early to mid 90's and I see signs that it has finally run its course. The results have been nothing short of awful and a whole class of students straddled with tuition debt they have no chance of repaying has been created. There are real consequences to all of this.

I realize that Dave is trying to make the experience more meaningful, but at the same time, a 1 year calculus class still has to add up to a 1 year calculus class. I mean the students should make roughly the same number of connections in that year as students in other 1 year calculus classes have. Unless of course, you label the class differently. But when teachers do this (label the class differently), they get criticized for "tracking". And Dave is so programmed by all this that even though what he is doing is tracking, if you call it that then he gets bent out of shape.

7. "You cannot use or apply calculus in the real world, or in any world for that matter, unless you are already VERY MUCH prone to exploring and looking for connections. I literally assume that the student is already prone to exploring and looking for connections when they step into a class as advanced as calculus."
That can be said of any subject in school. Unfortunately many of the students I get in college are not like that, or at least they do not show such inclination, so I can imagine what happens in high school.
So, what do you do with a student who has been allowed to take calculus without being prone to it: kick them out of class or tell them they are not worthy to be there?

"That defines all recent reform efforts. Forget the requisite interest. Forget the requisite development. Just put the damn students smack in the middle of these advanced courses anyways. I do not think there is any gain in that, in fact I think there is harm in that."
Here I totally agree! There is a fundamental problem with the way students are channeled and pushed through school and I too wish it was not so. I have always been a proponent of having mostly "appreciation courses" in school, supported by deeper courses that students would choose because they WANT to, in preparation for life, both professionally and personally.

"This fad started in the early to mid 90's and I see signs that it has finally run its course."
In math this started in the 60's, as a part of the space race and, unfortunately, I don't see it on the way out, except for a few "rebels" like Dave (and me?) who are trying to expose the problems and propose a new way.

But asking that the current approach be maintained by continuing with standard tests, standard course topics and activities and strict selection hoops will not change the system a bit, nor encourage it to do so.