Be careful out there!

The question given here is from the January 2016 Algebra II/Trig Regents in New York State

I have one major issue with this question.

A student needs to know nothing about circles and their equations. The point (3,-5) makes only one of the four choices true. All a student had to know was which number was x, which was y, and then substitute them in and do the arithmetic.

Yes, the student had to know something.  But they had to know nothing about circles.

Please take note that the pair (0,-2) does not satisfy any of the equations.

I can guess at the design behind the question: the student has to know the standard form of the equation of a circle, what to substitute in for its parameters, and how to "wrap up" to identify the radius and complete the solution.  As an "fill-in-the-blank" questions this would be fine.

As multiple choice, with the choices it has, it stinks.

Just a Power Breakfast...

The other morning, while at my local diner for pre-golf breakfast, I sat at a rocky table. I was talking with a man I knew at the next table, and he grabbed an unused paper placemat and started folding it. After the customary 5 or 6 folds (I don't think he got to seven folds), he handed it to me and I stuck it under the table leg and the rockiness was solved.

But, aha!, that was not the end! I, being the dastardly nasty math teacher type, asked him a simple question: If you folded just like you have been folding, but do it 100 times, how thick would it be?

The immediate response was "100 times the thickness of the paper". At that point I said "not even close, you've got to go higher". He asked how high, so rather than telling him I took out my cell phone calculator, punched in a guesstimate for the placemat thickness (I took it for about 300 to the inch) and multiplied it by 2 to the hundredth power. He agreed that exponents could be used for repeated doubling, so we calculated it out. The number is not relevant, but its magnitude (about 11 billion light-years).

He made the usual "no way!" response. Then his tablemates got into the conversation. One of them said "you must be a math teacher", and I said I had been. That's okay.

What befuddled me, however, is that they took the answer and the method for obtaining it as indicative of an esoteric nature of math, mentioning how bad they were at math. All I was doing was multiplying by two!

Then it hit me. My question was out of their comfort zone, yet they wanted to stay in their comfort zone. It was easier for them to reply based on their gut feelings, rather than thinking the problem through.

Many people get to the point where a problem like this is not really a problem. The others have to think about it, and serious thinking while eating breakfast in a diner surrounded by friends and relatives is not an easy task.

Does Whitehead's famous quote apply to individuals as well? "Civilization advances by extending the number of important operations which we can perform without thinking of them."

I wonder; is a school classroom the proper environment for serious thinking?  It is a good environment for factual learning and algorithmic learning (how to solve quadratics, write a complete sentence, learn arithmetic, learn how to spell, for example.) But deep, serious, thinking? Especially on a timed test? My best mathematical thinking is done alone, with smooth jazz or light classical playing, with the opportunity to take my dog out when the thinking is in the mind and not yet ready for putting pencil-to-paper. In college I used to go into an empty classroom in the evening and make major use of the chalkboards when working on assignments. Class time was for acquiring the specifics which, presumably, would give a helpful knowledge base for the assignments. Deep thinking in a class full while at a small lecture desk? No way.

Thinking is what we have to do when we don't know what to do. If we know what to do, there's no problem. Yes, we would like future generations to be good problem solvers, but, more than that, we need to prepare them with a factual and knowledge base so that many potential problems will not be problems. 

Stanford has a course called Introduction to Mathematical Thinking. The course description states

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

Why do they use the box analogy this way? Why don't they just say that they will be addressing mathematics as it sits in a larger box?  They put down high school math while ignoring that high school students are not professional mathematicians, and they fail to mention that "professional mathematicians" is a gross generalization. Many mathematicians I know work in very small boxes anyway. One quick web click and I find a representative of this group listed with 

Specialties: Abstract algebra, Galois theory, geometry

Interests: Commutative rings and algebra

Most professional mathematicians are equally specialized. (Just as you would not go to a dermatologist when you need a pacemaker) Stanford also overlooks the fact that all those "professional mathematicians" were once high school students. 

I would like to believe that Stanford's description is referring to mathematicians hired by business and industry to apply their special skills and talents in very specific arenas. If that is what they meant, it is not what they said.

Back to breakfast. My diner neighbors, at least one of them if not more, added that they "never saw any use in the math they were seeing in school". They missed the fact that they will never use what they do not know. Yes, I do not know the Greek language, and naturally I have never been able to use that knowledge. Would my life be different if I had learned Greek? Maybe yes, maybe no, but doors of opportunity are and were definitely closed to me. I cannot say what was on the other side of those doors since I could never go through them.

Time is of the essence

On January 22, 2015 I posted a blog about the FSA Mathematics Reference Sheets.

After thinking about the issue for over a year, it dawned on me that another part of the FSA reference sheets is just as insulting to students. That part contains the following:

Florida appears to have decided that expecting students to remember these bits of information is tantamount to cruel and unusual punishment. New York must have  thought differently, as none of the time conversions appear on its reference sheets.

It is fascinating when you compare the reference sheets from different states. Texas, for example, (see here) has no conversion facts on its reference sheets. None. At all.

What intrigues me is that each state is claiming to be aspiring to "higher standards." Are we raising standards when we no longer expect students to know that there are 60 minutes in an hour?  It's even more strange when we tell them, as fact, that 52 weeks is 1 year.  That is flat out wrong.

For clarification, 52 weeks is 364 days, and no year is 364 days in length. To obtain that fact, I had to know that 1 week is 7 days, which must be something Florida expects students to remember, as it is left out of their reference sheet.  When I was young I remember being told that there were "52 weeks IN a year" and also being told that it was not an exact match: 52 was as many as you get get without exceeding 1 year.  We were also told to use that piece of information only when estimating.

The current wave seems to look at remembering as unimportant and unnecessary. The "experts" tell us to "teach children how to think". 

Speaking of thinking...

The clip above is from the New York State reference sheets for high school math (see it here). The rectangles are mine. I used them to indicate the conversions that are expressed incorrectly.

Each of those conversions that are boxed are approximations expressed as if exact.  There is no side comment clarifying anything (such as "rounded to the nearest hundredth"). In the absence of a side comment, there are symbols commonly used that mean "approximately equal",  such as "≈".  Florida could have escaped detection by stating "52 weeks ≈ 1 year".

I do commend Ohio for proper use of "≈" on its reference sheets when stating a value for π. 

before i sign off on today's missive, I wish to encourage you to read Ben Johnson's blog post from early 2010. Read it and ask yourself if you would rather have a generation that remembered that an hour was sixty minutes, or a generation that has to look it up.

Baseball is Math!!

As I continue my experiments with GeoGebra 3D, I thought about that lace pattern. Rather than reinventing them, I borrowed from http://paulbourke.net/geometry/baseball/, and used them in this graphic. Nothing fancy. Just getting a handle on GeoGebra 3D features.

One of the features I am having issues with is getting a 3D version (the kind you have to view with special glasses) to save and embed successfully. Wish me luck!

See what one midpoint can do for you!

When you first see the image below, you will see a red ellipse. That ellipse is the result of letting two points (the blue and red X's) move at the same speed in opposite directions around their respective circles and tracing the midpoin between them. Start Animation to watch!

Change the speeds and you will change the graph. Experiment!

To modify, expand, adjust, alter, or just plane change the file, get it here.

Using GeoGebra to pose questions

I truly wish I could have had dynamic geometry such as GeoGebra to pose problems for my classes.
I suspect this problem could have fit in as a challenge to a precalculus class. It is probably also a good question for math teachers.
Well, is it?

What would they do?

Every once in a while i just try to imagine what young elementary school students might come up with if given half a chance to acclimate themselves with a program such as GeoGebra. What if...

An elliptical rose?

Every once in a while I try to create something that just acts as a"tug" to get the user moving in the direction of dynamic math. Here is a little file that just demonstrates something that a lot of teachers might never think of because without dynamic geometry it is massively ugly.

It just involves controlling the rates at which two points travel on an ellipse, and tracks the midpoint of the segment connecting them.  Many PreCalculus books include sections on polar graphs, based heavily on trig. This takes those shapes and modifies them.  A sketch based on circles can be found here.

If you would like to play around with this file, get it now!

WAVING to the crowd!!!

I was experimenting today with GeoGebra 3D, trying to see how I could simulate a "ripple tank". The Geogebra part is not too bad (so far), but getting the 3D file embedded in a web page is a nightmare. Getting browsers, devices, operating systems,. and GeoGebra to get all in sync is not nice.

At any rate, I took a rather large automated gif copy of part of a file, doctored it up to shrink it down in size a bit, and put it here.

For its parent file, you will have to wait. It is still in "production."

Enjoy!

Monkey in the middle!

Here is just another demonstration of a relationship between an ellipse and two special concentric circles. Two points will rotate around the circles at equal speeds in opposite directions. The midpoint between them will be traced.  Those rotating points will start at the top of the circles if the vertical axis is longer than the horizontal axis. Otherwise the rotating points start "due east" from the center.

Play around. 

The original file is here. Download it and experiment. GeoGebra is cool!