Tuesday, March 31, 2015

Mathematics and Creativity go hand-in-hand

Creations such as these can be introduced to beginning Geometry students, without even needing what is in red. Demonstrating the shapes, and how they are created using chords and midpoints, can be seen as a possible motivator for the later study of trigonometry and polar coordinates.  If we have a "core" be it "common" or not, that does not allow time for such investigations, we will lose the academic race to those cultures, countries, and schools that do allow for it.

This was made in GeoGebra in 10 minutes. If I had been able to use this in the classroom....

Monday, March 30, 2015

A Butterfly Effect...

Short extension of yesterday's post that merely allows you to change the direction of one of the moving points. A simple change creates a new shape. If I was still in the classroom, I would use the creative possibilities of GeoGebra as an aid in teaching the vocabulary and basics of Geometry in lower grades, and use it as a tool in posing questions and problems in the upper grades. All along one must keep in mind that GeoGebra is just a tool, not a teacher.
While I work on this I cannot but cringe at the lack of time teachers get during the school year to create, modify, improve, and implement any of their own works. Can't have creativity in a  "right-answer" subject such as math!

Sunday, March 29, 2015

Math is simple...

I cannot forget that my primary goal is eliminating math phobia and generating "hooks" that can gab and keep interest. I use GeoGebra a lot with this in mind.
Here is a app showing how a limacon results when we trace the point exactly midway between two points rotating around a circle. One point is traveling twice as fast as the other.

Friday, March 27, 2015

What is best: weakly correct or strongly incorrect?

Here is a question from the New York Algebra 1 (Common Core) Regents Exam from January 2015. It was worth two points.
This solution is given as a model answer worth 2 points.
 Here is a model answer worth only 1 point.
 Here is a model answer worth 0 points.

Please take note that the sample answer that was incorrect is to be given more credit than an answer that was correct.

I have a serious problem with this as it seems to devalue the correctness of a response while overvaluing an explanation for an incorrect response. Euphemistically speaking, talking the talk seems to be more important that walking the walk.  Please take note that writing "fuction" where the word "function" belongs seems to be totally irrelevant.

This is not to belittle the value of being able to explain a process. Explanations are important. But the hierarchy should be:
1. Correct results well explained.
2. Correct results weakly explained.
3. Correct results not explained.
4. Incorrect results with support.
5. Incorrect results not supported.
Please take note that an incorrect result with support given is simpler to fix that an unsupported incorrect result. Just as a doctor will ask you "where do you hurt?", fixing a solution ought to begin with "where did it go wrong?" That is most easily answered when the result's alleged support is given. 

I refrain from using the concept of "explanation" in relation to an incorrect answer as such answers cannot be explained away. Hopefully they can be repaired.

Along with this dilemma is the misleading description of "function" as stated in the Common Core. Read this:
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
The words "from" and "to" in the first sentence is misleading. The idea of "with" would be an improvement, as in "A function is a pairing of elements from one set (called the domain) with elements of a second set (called the range) such that each element of the domain is paired with exactly one element of the range."

The description used by the Common Core also gives the impression that a function must have a graph. Try to graph this function: y = the first letter in the word "x", where the domain can be any list of words.  It is a function, but can you graph it?

My favorite example of a function, that I used in class plenty, is the bar code scanner at the local supermarket. To get students to latch on to the notion, I asked them what would have to happen for it NOT to be a function? They quickly agreed that a single item should result in only one price.

Getting back to question 27, is it good to penalize a student for knowing a correct answer to a simple question? I have known many students who interpret "Explain your answer" as if it said "write your answer in sentence-paragraph form."  The answer is so obvious that they do what they think they should and move on.  And they get 0 points?

To put this question in perspective, evaluate all three responses as if they were given to a student by a tutor or teacher as answers when a student posed the question. Is the 1-point answer more valuable then? Naturally the 2-point response is best, but is a supported wrong answer more valuable than an unexplained correct answer? 

Wednesday, March 25, 2015

Which way Will he go?

Today's Albany Times Union, in its short column entitled "Rodger's Quick Take", quotes WNYT's sports anchor Rodger Wyland with the following:
At some point UAlbany men’s basketball coach Will Brown might need to take the next step in his coaching career. If you have an opportunity to coach in a bigger conference for more money, you might have to consider leaving the Great Danes after 14 seasons. If it’s the right place for Brown and his family, good for Brown. He has earned it. Only Brown will know if Fordham or George Mason are the right fits. Bottom Line: When you get to the NCAA Tournament five times and you get there three straight years, you are going to have opportunities. At some point, if you want to coach at the next level, you have to take one of them.
This paragraph epitomizes a key flaw in our culture.

Please note that Will Brown looking for more lucrative employment is not part of that flaw. As an individual, he is free to do that. Maybe even obligated to his family to do that.

The flaw is that UAlbany, according to USA Today (see here) is already paid $295,000 as a base salary, with possible bonuses up to $233,000 more. For details, his contract is posted here. Can you find me any educator at UAlbany that has a comparable salary? Or comparable incentives? 

UAlbany is a university, which, to the best of my knowledge, means that its key role is to educate. Has Will Brown been one of the most successful educators at UAlbany?  Some people might claim that to be the case, but I for one would want to make sure that he is compared to educators who hand-pick students who get to attend at no cost to themselves.

In the back of my mind I suspect that this column might not have rankled me so much had it revolved around the value of Will Brown to UAlbany, how he and the university worked well together, and things would be best if he remains at UAlbany. Rodger Wyland seems to be placing Will Brown's future above that of UAlbany and its players/students.

To help put this in perspective, here is information about a fee that students pay (see here):
Intercollegiate Athletics Fee$275.50 for 12 or more credits, $206.63 for 9-11 credits, $137.75 for 6-8 credits. Funds the intercollegiate sports programs including team transportation, uniforms, etc. This fee is mandatory for undergraduate students taking 6 or more credits except those who do not have access to the campus, such as students in our Overseas Academic Programs.
The mere existence of this fee confirms that intercollegiate athletics is not self-funding, so anyone who thought that is, to put it simply, wrong. No matter how hard I looked I could not find any reference to the "Mathematics Department Fee". There are other fees, but the one the Intercollegiate Athletics Fee is by far the largest, with the exception of an insurance fee for students studying abroad and for international students.

Recently Norman Chad (aka "The Couch Slouch"), wrote a column that includes this;
Actually, why do universities even have athletic departments? That’s like an auto body shop having a produce aisle.
Now, if a college has a physical education department, that’s a different ballgame.Because, indeed, mind and body are important, but universities should prioritize those needs for the bulk of its students rather than a handful of illusionary student-athletes. What’s a greater achievement, having 15,000 undergraduates swimming three times a week or bringing in a couple of basketball recruits who can average 15 points a game?
If Maryland, say, established the nation’s most extensive intramural program, that would be athletic excellence, in my view. I would be a proud Terp.
I find Mr. Chad's comments relevant, in that UAlbany does not have a Phys Ed department. For that matter, UAlbany's website includes the following:
Physical Education Courses: Although the University no longer offers physical education courses for credit, students may apply toward their degree a maximum of six credits of physical education activity credits. These and credit for courses in coaching, recreational studies, etc., will ordinarily be designated "non-liberal arts and sciences" credits.

Evidently, UAlbany has devalued Phys. Ed. while it has gone Division I and has a basketball coach as one of its highest paid employees.

Oh, by the way, UAlbany's football coach has a base salary of $230,000. He does work with more student-athletes than the basketball coach does, yet gets paid less. Makes this morass even stranger than it already was.

Sunday, March 22, 2015

Pearson, again...

Today's Albany Times Union has two columns on the first page of the Perspectives section: "A failing grade for tests" by Michelle Adalian and an editorial entitled "The test of Common Core". Both about, essentially, the same topic.

Ms. Adalian is a 4th grade teacher in New York State. You should be aware that buried in the grade 3 through 8 tests in New York is the statement Developed and published under contract with the New York State Education Department by Pearson. 

I wrote about Pearson earlier (see here). My opinion has not changed. Pearson has a vested interest in getting states to go full bore into a "common core". As long as it is a shared curriculum, they stand to profit.

The TU editorial includes this:
The latest tests in question are for aspiring K-12 teachers, who must pass them to qualify for a teaching certificate. Teaching students started taking these tests back in September. Six months later, many are frustrated to still be waiting to find out whether they passed — because the state hasn't figured out what the passing score should be.
The reader should be aware that Pearson Education Inc. creates the teacher tests. In its own words
The New York State Teacher Certification Examinations™ (NYSTCE®) address New York Education Law and Commissioner’s Regulations, which require prospective New York State educators to pass designated tests as a requirement for receiving state certification.
The public should take the time to learn about Pearson's role, and checking out their web "sales pitch" can help. Their push on teacher licensure is here. Getting a sense of their grab into assessments in general, check this out.

Bob Schaeffer is quoted as saying (see here) "In a capitalist society, if there’s a market, somebody will figure out how to serve it. But the corporations reinforce the stupidity of the bad policies of politicians."

Ms. Adalian says in her column "I encourage the powers that be to walk in our shoes."

I would hope and expect that politicians at all levels take the time to learn a lot about the actual schools with which they are playing. When they think they know enough, go and learn more.This hope is largely due to the fact that the only change that can be made in Schaeffer's statement is the removal of "stupidity of the bad".

Pearson has too much power and it was given to them blindly.

Disclaimer: there is greatr merit to the idea of a common core curriculum in this country. But it should be managed much much much much much much much better.

Saturday, March 21, 2015

March Madness: Who are the "mad" ones?

The NCAA March Madness begins and in a flash its ending. Thirty-two games played Thursday and Friday, with only 31 left to go.  Peter Hooley of Albany has had his Warholish "15 minutes" (In my mind he deserves more).

This all brings me back to the amazing thing about March Madness: 63 of the alleged best college basketball teams in the country end their year with a loss. Even so, teams strive for the opportunity.  I have heard that there are 351 colleges that play "Division 1" basketball (see here), and that these are the teams from which the March Madness tournament pulls its teams. So 351 teams exert their energies for a chance to be the one team out of a group of sixty four that gets to end its season with a win.

It really and truly seems that a lot of time and energy is exerted chasing this remote possibility. To quote Hilary Clinton out of context, "does it really matter?".

What I would like to see is the statistic of how many professor/teacher salaries could be paid at each of these schools for the amount that it pays its basketball coach.

Side note: NCAA rules in effect for 2014-15 limited Division 1 schools to 13 scholarships for men's basketball, but 15 for women's basketball. Interesting.

Tuesday, March 17, 2015

McDonald's ambiguity

I have seen a lot over the years about the shape of the Golden Arches. Here I let you decide. Just drag the points to locations (anywhere, no special order) along the curve you are investigating, then click "show graph". Enjoy

Friday, March 13, 2015

Big business fails again...

Earlier this week this email was sent to Verizon Fios cable customers:
Dear Valued Verizon Customer, 
Verizon’s agreements to carry The Weather Channel and Weather Scan have expired, and have not been renewed.  In today’s environment, customers are increasingly accessing weather information not only from their TV but from a variety of online sources and apps.  Verizon is therefore pleased to launch the new AccuWeather Network, which will be available on FiOS® TV on channel 119/619 (HD) and on our free FiOS Mobile App starting March 10, 2015.  Verizon will also provide the FiOS TV WeatherBug “widget” application, which features hyper-local weather, on FiOS TV channel 49. WeatherBug can also be launched by pressing the “widget” button on the FiOS TV remote.
Your Verizon Team
Please take note that the official Verizon review of the Weather Channel App includes:
Whether you’re hitting the beach, planning a picnic or taking a road trip, it’s helpful to know what kind of weather to expect out there. The Weather Channel app will help you do just that. It helps that the info in it is sourced from one of the most trusted weather outlets there is. 
Thus it appears that, for the benefit of its customers, Verizon cable is dumping its most trusted weather outlet.

There are a few aspects of this that help tarnish Verizon's image. First, it appears that Verizon was not honest in dealing with its customers. The email it sent out was virtually simultaneous with the disappearance of the weather channel.  If Verizon had the opinion it quotes in its email, it could and should have communicated with its customers well before "pulling the plug".  The last minute email makes it appear that Verizon was  surprised by The Weather Channel's not giving in to Verizon's contract demands, and was trying to cover its tracks. 

I suspect the real reason for this mess is that TWC wanted more than Verizon wanted to pay. Well thanks, cable TV, for protecting us from the TWC omnivores. Now how about the same for all those sports channels that really suck out pockets dry?  Why not stand up to them?  After all, most people get the most of their scores and sports information "not only from their TV but from a variety of online sources and apps."

Admit it, Verizon. You screwed up.

Solution? Reinstate The Weather Channel. It still has more weather than the History Channel had history and the Travel Channel has travel.

Tuesday, March 10, 2015

We have to do better..

After having been away for a while, I finally had the opportunity to look at the January 2015 New York State regents Exam in Algebra I (Common Core). I looked at the first question on it, and immediately my head was scratching.
The problem is:

I know that the answer expected is choice 2, which probably should have said "hours worked per week by employee". That ambiguity made me realize a few things about this question.

First, I presumed that profit is measured in dollars (as I took this as a problem for New York State). Placing that fact together with the mathematical knowledge that no units were given in the equation, and that only like terms can be subtracted, I came to the conclusion that the 8600 was really $8600. Now the 22 could be 22 of anything, but the nearness of $22 in the problem led me to accept that most would take it to mean $22.  Basic mathematical knowledge forces the conclusion that the x must then be "unit-free". That means that the x need not be the hours worked (by the employee".

Second, and this is probably due to some basic cynicism on my part, the use of a linear function for profit is as "unreal" as you can get. Do we wish students to be pushed a bit in their minds towards the notion that employee work-time is bad for business?  Is this the best we can do in the billion-dollar industry of education in New York under the guidance of Governor Cuomo?   Who creates these tests? 

Later on appears this questions:
This problem has some style points to its credit, but a big fat goose-egg when it comes to its realism.  The acceleration due to gravity on the Moon' surface is not .8 m/sec2 but is, rounded, 1.6m/sec2.  In the big quest to make problems "real", doesn't it make sense to use real information?

All is not so gloomy, however. Here is question #2, which to me is a masterpiece when it comes to a multiple choice question testing a meaningful skill.