Murphy, sickness, and calculus

So today starts an interesting day. I was already going in late because I have a doctor’s appointment. I was only missing a free period and then part (or possibly all) of a dept. chair meeting. No big deal. I would still make it in time for my Calculus BC class.

However, Murphy has a way of showing up when you least expect it and are not ready for house guests. My daughter has not been feeling well the last couple days, and she has been home from school. Fortunately she is old enough now – and I live on the campus I teach at – that leaving her at home is not a problem. This morning, though, my wife and I took her temperature after feeling her skin literally radiating heat to find her with a fever of 104!

So plans change. I’ll still get to go to my doctors appt in a little bit because I can call her doctor immediately after, but this will clearly change the rest of the day. I had three different meetings scheduled for today. Fortunately I am only leading one of them, and that nominally.

However, I will need to miss my Calculus BC class almost certainly. The good news is that I can leave work for them to do, and they are responsible enough to not just blow it off.

So here is a problem from their next set:

Verify the identity . Use this result to help you solve the logisitic differential equation , which you should recognize as separable.

(a) Show that the solution can be written in the form .

(b) Given the initial condition f(0)=0.36, express f(t) as an explicit function of t. Evaluate the limiting value of f(t) as t approaches infinity.

At this point they have already been introduced to logisitic differential equations through a couple previous problems (one of which also introduced them to the concepts behind Euler’s method). In addition they have also seen a similar partial fractions problem, although in a similar vein – verifying, not finding the fractions yet.

We’ll see how they do!

Geometry FAQ

As I have mentioned previously, I am teaching my Geometry classes in a problem based format using material originally adapted from Phillips Exeter Academy. I did a survey recently using my new best friend, Google Docs, and got some interesting feedback. Because this is the first time for my students to learn in this fashion, I decided to construct a FAQ for them based on what I thought were the big questions raised in the survey results. Interested in the raw results? Here they are!

On to the FAQ!

Geometry Honors FAQ

Why do the GeoGebra problems have so many steps?

• The GeoGebra problems are designed to take you through different ways of using the program to solve or set-up for certain types of problems. To make it accessible to everyone, I have to include all the steps I think someone might need to know about in the beginning. As the year progresses, the problems that ask you to use GeoGebra will be less explicit about what to do, instead relying on you to remember what we have done before in the problems and/or in class.

Why don’t we practice similar problems or learn the material before doing homework?

• Learning geometry is one of two objectives for this course. Learning how to attack new problems and communicate what you know is the other. You cannot learn how to do the latter through practice and lecture. They are active endeavors and, as such, you need to be actively engaged in them in order to learn.

How long should I be spending on homework?

• Maximum of 45 minutes. I am quite serious about this. This includes the abstract and 3 problems when they are do. If you follow my suggestions in the section on ‘finishing homework’ and you are regularly spending more than this – talk to me. Don’t wait. Now.
• The abstract, which is supposed to be about the previous calendar week (Mon-Fri), is not due until Thursday of the following week, giving you five days to space out your time on it. All together this should not take you more than 10-15 minutes total.
• The 3 problems should also not take more than 10-15 minutes total. You can be working on these all week long. You do not need to include original work here. What you do need to do is:
  • Recopy the problem.
  • Show the work you did (even if not complete or incorrect!)
  • Write 1-2 sentences about what you did or how you tried to solve it.

What does it mean to “finish” my math homework?

• When it comes to the regular problems (not the abstract or 3 problem assignment) you should familiarize yourself with all of the problems. You should make a serious attempt on at least three-fourths of them, and feel like you have a solution to at least half of them. This way you can come to class prepared (if you wish) to present a problem, have some specific questions on others, and at least be able to follow along on the rest.

What do I do when I have no clue how to start a problem?

• Reread the question, draw a diagram, write an equation, define difficult words, phone a friend, or actually write down specific questions you can ask the next day in class. No one will have all the answers all the time – don’t expect to. This is an unreasonable expectation.
• Demonstrating understanding of a concept often involves applying it to a new situation. Do you really want the first time this happens with a concept to be on a test?

What should I be doing in class while the problems are being posted/presented?

• While the problems are being posted on the board, and you are at your seat, you should be comparing the work on the board with what you did – particularly for those problems you had trouble with and/or did not complete. Write down specific questions to ask when the person presents. Remember to focus on what is on the board – see if you can find where your work and that on the board diverged. You may have questions about your work, but no one else can see it, so focus your questions on what is on the board.
• While the problems are being presented, pay careful attention to the person speaking and think of questions to ask based on what they say. Don’t forget to ask, “How did you even think to start the problem like that?” You want to be thinking about how to approach the questions nearly as much as how to solve them correctly.
• While the problems are presented, make notes on your own work. Make corrections! If someone says something that really makes sense to you – jot it down! Don’t rely on your memory. If Mr. Wysocki gives a mini-lecture on a topic, take notes. The pictures I take of the problems are a resource to help you, but they should not be a replacement to active listening during the presentations – and they rarely contain everything that was discussed about a problem.

How do we connect the problems we are doing with the concepts we are supposed to be learning?

• That is one reason I provide a “notes day” several days before each test – to bring what you have been doing into focus.
• Ask questions when we are listening to problems – or share your insights when you are presenting them:
  • “I noticed that this problem used the Pythagorean theorem – but in a different way – than problem 43.”
  • “I was wondering if this problem was like the one we did last week?” At which point I might ask you to elaborate, and then we can explore that.
  • You are not learning Geometry in the same ‘discrete’ way you learned other math topics – they are mixed together, spiraled, and connected on their own – the way real math is. Trust the process.

What are our goals?

• In addition to becoming better problem solvers and mathematical communicators, there are five essential questions for the geometry material we will aim to answer by the end of the year:
  • What makes a logical argument?
  • What is important about the notion of equivalence?
  • What does it mean for objects to be similar?
  • What mathematical facts are important to know?
  • When does algebra help us understand geometry, and when does geometry help us understand algebra?

Since we don’t learn the material beforehand, how are we supposed to know how to do the homework problems?

• First of all, at the beginning of the year many of the problems are based on material you have learned before – in some cases as early as third grade! That it doesn’t come quickly back to mind is something you may need to work on. As the year progresses the problems will be based on material learned through earlier problems and even when new concepts are introduced, there will be problems that define terms or give you basic information about the concept to start with.
• The real issue here is changing your paradigm about what it means to “learn math.” To give you an example – if history classes were taught strictly as names and dates or English classes focused solely on grammar and vocabulary, they would be more like the kind of math class that focuses on learning a skill, practicing it briefly, doing some homework, and then moving onto a new skill the next day. Learning math – like learning history or English – means doing math. Doing math is not repeating what your teacher showed you – it’s struggling with a problem, using the tools you have to try to solve it, and then discuss what you did.

Why do the questions have to be worded so confusingly?

• Geometry is a subject very concerned with preciseness and, as such, can lead to problems that are worded in a way to seem more complicated than you might have phrased it. This is because we want to be very careful about the assumptions we make – and in ‘everyday’ language we make a lot of assumptions.

Why do we have to learn Geometry this way?

• I don’t want you to just parrot back material that I teach you. I want you to learn how to apply knowledge; how to gather knowledge for yourself; and how to communicate your knowledge to others. This does not come about by listening to a lecture, practicing problems that you have examples for, or working in isolation. It has to be an integral part of how you learn, not an add-on after all the content is laid out.
• There are other teachers and schools throughout the country with similar philosophies – this is not new. Phillips Exeter Academy in New Hampshire has been my inspiration, and the curriculum we are using this year is originally based on their textbooks, although the Geometry curriculum was then adapted to Emma Willard School in New York.

Why won’t you just teach us the material?

• Just as we need to redefine what it means to “learn math” I have had to redefine what it means to “teach math.” In truth, the way I teach now is much harder than what I used to do because I have to pay close attention to what occurs in the classroom constantly and capitalize on moments when it is appropriate for me to step in. At the same time, if I step in too often, you will get the idea that if you don’t get it right away I will come to your ‘rescue’ and ‘save you.’ I cannot learn the material for you, and hearing the answer is never as good as figuring out for yourself. Remember Confucius’ famous saying?
  • I hear and I forget. I see and I remember. I do and I understand.

Why do we have to present the material instead of you reviewing the problems?

• The reason is twofold. First, as I mention in another answer, it is important that you are doing math to learn it. Solving the problems and then explaining it to someone else takes a level of understanding beyond watching someone else. Have you ever sat through a class, followed what the teacher is saying – even correctly using it in the class – and then go home and draw a complete blank? Explaining it to someone and answering questions about it forces you to look at it from different perspectives. Secondly, communication is becoming an even more important part of our world every day. Knowing how to articulate your ideas, have constructive disagreements, and resolve conflicts is a skill you will use more often than any math you ever learn in any class.

So that is the FAQ as it stands now. Interested in a version on Google Docs? Click Here! It is the Geometry Honors FAQ 09-10. As the year progresses I may add to it and modify it, and I intend to make a version more directed at my Calculus class.

My Classroom

I would really have liked a Harkness Table in my room, but they are too much money, and I do share my room with two other teachers. Instead, I have trapezoidal tables which I currently configure like this:

My classroom table arrangement.

This is a view from one of the front corners. I have whiteboards across two walls, a wall of windows at the back (which you can somewhat see in the picture) and then one more wall with a door to the office I share, some bulletin boards, and the projector setup.

It’s not much, but I like it.

Maple 13

I had a demonstration of Maple 13 for about one hour on Friday, October 9th. Dr. Robert Lopez led me through numerous ways of using Maple 13 to solve problems by projecting his desktop onto mine while talking me through everything on the phone. His explanations were very clear, and it was easy to follow what he was doing, but when I tried some things afterward I had problems that I would not have thought to ask while I had him on the phone. However, overall, I would rate it as a good use of time.

First of all, I like the templates menu along the left side of the Maple screen. There are actually a number of different template menus, the picture to the right showing only one of them open. As you can see from the menu that is open, it is possible to click on a number of common mathematical expressions and just ‘fill-in-the-blanks’ rather than know any complicated code. It looks like math. From integrals, both definite and indefinite, to summations to square roots to piecewise defined functions.

Another thing that is nice, and I actually think is easier to start with than Mathematica’s Documentation Center is the Maple Portal. It is a nice starting place with links to pages on how to do basic things in Maple, including links to sections specifically designed for students, teachers, and engineers.

So what does the math look like? Let’s consider for a minute the function (just to make it a little complicated!). Maple allows you to enter the equation exactly as it is supposed to look and even numbers the output.

However, here is where I had some problems when I first wanted to graph something like this. Before I show you that, it is worthwhile to show you that instead remembering all of the commands and things you might want to do to this equation, Maple 13 allows you to right click on the equation and then gives you a nice long menu of choices:

As you can see, there are many choices to choose from, and some hints as to what the problem is that I ran into. For one thing, since I wrote the equation as rather than just the expression, it treated it as an implicit equation to a certain extent and it took me a little while to manipulate the graph so that the resolution looked decent. On the left was the initial output, the right after I finally figured out how to increase the resolution.

25 x 25 Resolution

100 x 100 Resolution

Clearly the graph with 100 x 100 resolution is preferable, and before I figured out how to increase the resolution easily (and it is easy once you know how) I considered it to be a deal-breaker on using Maple regularly. However, it was fixable, and you can improve the screen resolution even more in an explicit plot. In addition, the colors, axes, graph thickness, and many other aspects are completely customizable – even after the graph is initially drawn.

How about actually doing some math? Let’s consider something simple. What if you want to show students how to solve the equation ? First of all, you can right click on the equation – before hitting enter, and choose the Manipulate Equation option, to reveal the following dialog box.

Note that the program automatically simplifies the parentheses – something I’m not as excited about, but look at the choices given in the box. It essentially allows you to show the steps, using the old “what you do to one side, you must do to the other” rule, although under the Miscellaneous Operations choices you can even do things to one side or the other. After you do this, you can click on the Return Steps button at the bottom, and your output will look like this:

There are numerous things you can do with Maple, and a lot of it has a very intuitive feel. Below are some additional examples of the math that it will do. I found it very easy to do what I wanted, without a lot of heavy syntax to struggle through. That being said, if you are familiar with Maple commands and/or you want even greater control over what you do, Maple will still accept the commands and executes them accordingly.

Additional Problems

I am interested in finding a computer algebra system that is easy to use so that teachers who want to use more technology as they plan and teach their classes can do so without a huge lead-in of time. My experience with Mathematica, although I know the new version has some templates like Maple does, suggests it is not as far along in this as Maple is, although it seems that it may be more powerful in the long run. However, for a department that utilizes little technology beyond graphing calculators, and is only at a high school where high-powered mathematics (at least professionally speaking) is not really being done, Maple may be the better software to consider.

On October 23^rd, we have a representative coming out to our school to demonstrate Mathematica for us. We’ll see what she has to say, although I am certain I will have some questions for her regarding the interface and ease of use.

Obama – Nobel Peace Prize?

First of all, I have to say that I voted for Obama, and I believed in the change he represented. Even before he was sworn in, though, I feared that it would be politics as usual once he was in office.

But the Nobel Peace Prize? If I read this article correctly, Obama has been awarded the prize based on what he has called for and the “mood” he brings to the world. Call me skeptical, but given that our country seems to be more divided than ever – wingnuts from all sides bursting onto the scene, we are still in Iraq, Guantanamo Bay is still open, and the situation seems to be worsening in Afghanistan (where my uncle is due to be deployed in about a month) this just doesn’t seem well thought out.

I want Obama to succeed. I want to see some of the “change we can believe in.” I haven’t yet. I think the Nobel Peace Prize is a bit premature, and I would not be surprised if awarding it to him did more harm than good.