What Mathematicians Think...

Earlier this month Jan Nordgreen at Caymath posted about an interview of a couple of professional mathematicians talking about their work. Here's one quote.

Isadore Singer: ... when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.

Another quote:

Michael Atiyah: My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say — as a kind of joke — that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture — it may be a beautiful result but with terrible proofs — you spend your time trying to find better ones; you do not listen to the lecture. It is all about asking questions — you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.

The full interview is right here.

These are two things I find myself constantly belabouring in class when teaching problem solving:

• Take risks! Experiment, play, try something out and see where it takes you. Good math isn't knowing what to do with any problem -- good math is knowing what to do when you don't know what to do. ;-)


• Ask questions! If you don't ask questions then I can't tell whether you understand or not. I'll either go on to something new, leaving you confused in the dust, or go over and over something you already understand trying to help you but really just wasting our time.

Food for thought ...

Hare and Hounds

In this game you have to "run for it!" You can be the Hare or the Hounds. The hare has to escape; the hounds are trying to corner him.

The real question is: Are you an expert Hare or expert Hounds? ;-)

Have Fun!

What impression do you want to make?

Berrien County Intermediate School District (BCISD) is a school division in Michigan. They have an online workshop on educational blogging and other web 2.0 tools. Look at the impression you made (under the heading "Teacher Blogs").

Dean Shareski is an Educational Technologist in Saskatchewan. He made a list called The Best of the Blogosphere for New Bloggers. You're on it. More than that, he suggests that you guys are THE example of how math students can also be bloggers.

A new teacher blogger writes about you and mentions Sarah's blog in particular to demonstrate to her students how they can benefit from blogging.

On October 17, 2005, an online news article entitled Blogging 101--Web logs go to school mentions you on the second page as the only example of how blogs are being used in mathematics.

You are the number 2 hit when someone uses Google's Blogsearch to look up calculus.

You've already made an impression.

Now what are you going to do, not just to maintain it, but move it up to the next level?

Graphs Graphs Graphs

Try this little exercise/game. It deals with the definition of the derivative and matching up the related graphs of f, f' and f''.

Have fun with it!

Sunday Funday (Oops, I missed a beat...)

Here are the rules.

Here is the game.

Have fun with it!

LIMITS! LIMITS! LIMITS!

OK guys.. I've been sitting here infront of my computer, and I asked myself.. What is a limit? Today in class we were told that the definition of a limit were as follows:

lim f(x) = L

x -> a

- which means the function has a limit, L, as x approaches a.

We also looked at graphs and how you have to look at the left and right side of the function to see if the function exists.

We must remember that in order for a limit to exist we must look at both sides. If they equal, the limit exists.

I dont want to babble on because I for one am not that comfortable with teaching this stuff to you guys and discussing it because I am not sure about certain things. AND! I dont want to confuse you!

READ SARAHS BLOG! ITS AWESOME =) (high five to sarah!)
I think you'll learn more there...

tomorrow will be......
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PRINCE

Limits

Watch this to review the "intuitive definition of a limit." There is a lot of text but stick with it and pay really close attention when the ideas are illustrated on a graph.

Limit Theorems
Check out this page and review the first seven (7) limit theorems. Click on the examples next to each one to see how it gets applied.

Try the examples on this page to see how the theorems are combined to solve limit problems.

Then try your hand at this quiz on evaluating limits using graphs. You will be writing one just like it in class on Thursday so get ready for it. Each time you click that link you'll get a different quiz with similar questions. Have fun with it!

Update
Check out Sarah's blog! It's awesome. ;-)

Sunday Fun! (or Sunday Madness!)

I got this from a blog called Think Again! A great little math blog full of interesting puzzles. Lucky me, it took a while but I found my way out of the room ... can you?

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You are trapped in a room. To get out requires some thinking. Good luck!

Over two million people have tried to leave the room already. No one knows how many are still stuck.

Online Class #1: Critical Numbers

Tomorrow's class will be done entirely online in the iMac Lab, Room 58. Here we go ....

We will be learning how to use derivatives to find local minima (the plural form of minimum) and maxima (the plural form of maximum) for a given function. We'll be looking at lots of graphs to see if we can identify the relationship between:

i)  The roots of a derivative function, and
ii) where (the x-coordinate) its parent function has a maximum or minimum.

1. Watch this tutorial where we look at a cubic function.



2. Read through this tutorial. All the graphs are interactive. See what happens when you move the mouse over the graphs or pictures. Don't calculate any derivatives algebraically; use your calculators to create the graphs of derivative functions using the nDeriv command ([Math]:[8]). We'll learn the algebra next week. ;-) Pay attention to how the derivative changes where the parent function has a local max or min.



3. Look up the definition of a Critical Number (follow the link to the definition of a stationary point as well). Write it down, in your own words in your notebook.



4. Your assignment is in your textbook. You can review the introduction to this topic on pages 114 and 115. Then do all the ODD numbered exercises beginning on page 116. Also do questions #10, 16 and 22.

If you have the time, comment on this post and let me know what you thought of this lesson. Did you enjoy having the entire lesson online? What were the advantages and disadvantages of learning this way? Should we do this again? Why or why not?

Have a great day and do someone a good turn for no good reason. ;-)

Cheers,
Mr. K.

Secant Lines and Tangents

We talked some more today about how to find derivatives and exactly what the definition of a derivative is.

Play with this applet. You can use the given function or type in another one of your own choosing. You also get to pick the "step size" of h and see what happens to the slope of the secant line as it approaches the tangent line. Try changing the "step size" of h and see how this affects the estimate of the slope of the tangent line by clicking on the [+] or [-] buttons. At some point you will see that the secant slope is undefined. Can you explain why this happens?

The Difference Quotient

Watch this to see a little review of some the things we talked about in class today. Here is a movie illustrating the same thing.
This will illustrate how we can do the same thing numerically, using a table of values.

And here is an animation of what we were doing with our graphing calculators. Play with this applet too.

DRIVE SAFELY!

Today in class we went on a road trip ;) Boy! Did we learn a lot today.. We looked at a graph displaying the journey to our destination.. It looked like we took a VERY long time.. We calculated the average speed or velocity using the formula, change of distance over the change of time. With our results being 45km/hr for the whole tirp, we thought that JAYSON drove pretty darn slow. But you see there was a constant line, which meant we stopped.. Also, we thought that he was driving slow, but in fact, the more we looked into the velocity in x amount of time.. we realized that, he was driving pretty fast! Then we tried XUN! We thought she would be better! She was worse ;) We could have gotten hit into a building like Chris mentioned because on the graph we didnt have a complete stop; the car was still moving! We discussed about secants and tangents, but that will go into more detail in another class.. But anyhow, the whole point of this 'trip' of ours, in my view was to realize that depending on where you look at the graph, you can come up with one answer, and then look at the whole graph and see such a big difference. You see, calculating the average speed was determined by looking at the whole graph. By looking at only on piece of the graph, you determined the instantaneous velocity. Which is.. dun! dun! dun! The topic for tonights homework!

The next scribe will be the one and only SARAH!

The Muddiest Point (Glenney's version)

Well, well, well. It seems that my title is very self-explanatory when it came to the first test of the year. I bet that everyone was a bit nervous as soon as they entered the classroom- I know I was. To be honest, I was expecting a lot of logarithms and ready to tell myself: "A logarithm is an exponent!" But I was disappointed to see that there wasn't really any?! Or am I wrong? (Short memory span. That's why I try to write almost everything Mr. K says in detail.) I, personally, found that test to be really tough and after class I started having my doubts again. But then I read Mr. K's comment to Emjay. He basically says that it's only one test and we will have many more opportunities to possibly raise our mark from more tests in the future. So I guess I'm not going to let this bring me down so hard. So since we didn't really learn anything new, I'll keep this short.

Mr. K- Come back soon.. we need more laughs... even though we don't laugh when you tell us a joke that we don't get right away :) haha jk!
Mr.Tram- If you're reading this... you really do know your calculus. haha like me and Prince say while you describe something: "You're smart!"

Since Emjay was supposed to be the scribe today... she'll be the next one for tomorrow =)

A little something to read..

As you can probably assume, the picture to the left is an image portraying Sysiphus. He has so much on his shoulders, so much pressure to get that rock going without failing, but he knows he is set up to fail. In calculus we are being set up like Sysiphus, we have a lot on our shoulders, but we are not to fail, but to succeed.. I feel as if I have set myself up to fail.

I am doing this blog for many reasons. One being, I am still at that point where I am contemplating- to stay in or to leave. And by doing this blog, I am hoping it will help me with my decision. But to be honest, I see no progress in myself, you guys or Mr. K did not fail me, I failed you. If you have not noticed, I have not been in class for the last two days. I have already disappointed myself, but also you guys! Being in this class does have its ups and downs. Its so frustrating when its so quiet but then we get this spark of hope that we are going somewhere as a group. I guess another reason why I am doing this blog is because it may be my last. I dont know yet. It may not. You might see me tomorrow morning in class, you may not.. If you see me there, that means I am giving it another try, and if you dont see me there, I give my apologizies to you all, I didnt mean to let anyone down. We are all in that class for a reason.. believe it or not..*whisper* People think that we're that smart and capable of doing this!

I guess this can be either my TESTblog, or a final blog. So one thing that frustrated me so much was logs! I HATE LOGS! Well, when I went to get some help from Mr. K, and was shown the answer and how to do it, i was thinking to my self "THATS IT?!" I think I tend to look at things a little more difficult than it really is. I was so shocked when I found out that that is all you have to do! I FELT SMART ;)

One thing I have come accross in the 'real world' is feeling like Sysiphus. I had a talk with Mr. K, and he says he sees no effort. I see no effort myself.. SOMEONE! kick me in the butt and tell me to put some effort into this. I guess, i feel as if, every time I attempt to do the homework, I am like hey I can do this! But once I stumble accross a question where I dont know what I am doing, i give up! Unlike Sysiphus he has a rock on his shoulders, I have a lot on my shoulders.. but thats not an excuse to quit! I knew from the start that I was taking a heavy load, and I was willing to do that. I know a lot of these things. Maybe I am frustrated with my ankle and just the way this school year has started.. WHATEVER it is! I still shouldnt quit. I AM NOT A QUITTER. I do not like to quit or give up. ( i know this is long, but and im rambling on and on, but i need a place to let this out and maybe i can come to an answer!)

* A note to MR. K- if and only if i decide to go to class, do not be surprised with my mark on the test. but please be glad that i stayed. If i do stay, I will promise to try. If I do not show up, I am sorry.
*A note to you guys- I am also sorry.

Feel free to comment and voice your opinion, because really I do not know what to do! HELP!

The Muddiest Point

Test day is Wednesday. I'm away for the next two days. Use the comments of this post to share with Mr. Tram, my substitute, your personal "Muddiest Point". You can use your name or leave your comment anonymously, but, whatever you do, share your troubles here. Remember, not only can Mr. Tram help you but you can help each other too! Leave tips and advice in the comments for your classmates. And don't forget, you can form an online study group and "meet" in the chatbox of our blog! Unlike Sysiphus, you're being set up to succeed! Take advantage of every opportunity you've got!

Can You Do It? .... YES You Can!

A wealth of links and practice material to help you get ready for Wednesday's test. Pick and choose the ones you most need to review. Of course, even this extensive list doesn't cover everything we've talked about. Work hard in that review class tomorrow and ask LOTS of questions. ;-)

• Functions and Models (10 questions)



• Domain (4 questions)



• Exponent Laws (17 questions)



• Exponential Equations (8 questions)



• Graphing Trig Functions (3 questions)



• Graphing Functions Using Technology (6 questions)



• Interpreting Graphs (10 questions)



• Inverses (5 questions)



• Exponents and Logarithms (only try questions 1 - 6)



• Transformations (19 questions)



• Trigonometric Functions (15 questions)



• Trigonometric Identities (4 questions)

Do your BEST on Wednesday!

Don't Read the Textbook .... Write It!

If you write it you probably understand it a whole lot better than if you just read it.

The internet has made all textbooks out of date. By the time a book gets published the world community has learned a whole lot more and shared it on the internet. Here's your chance to do the same. ;-)

First, the modern internet textbook is written in wiki format. A wiki is basically a website that can be created as easily as creating a blog post. Watch this to see what a wiki is and how it works.

After that check this out. It's a collection of text books that have been or are in the process of being written by an international community of ordinary people like you and I. As a matter of fact, YOU can add to any of them. Go ahead and do so if you wish. ;-)

What strikes me most powerfully about this latest development on the internet is the fact that anyone can write a textbook on any subject they wish! One of the textbooks being written is called How to pass a course. One of the things I really liked about this textbook was this:

Forming an understanding of the ideas behind each lecture requires active thinking. Try to think ahead of the professor: "What is he going to say next?". If the professor asks someone else a question, answer it in your head. If you answer wrong, try to think why it was wrong.

We will be using a wiki for our story project. Each participant will post their story problem and link to the solution which will be written on another page.

Do you think we should have our very own wiki in our very own webspace or should we add a textbook to the growing list of WikiBooks? Leave your thoughts in the comments to this post.

Cheers,
Mr. K.