Wednesday, March 27, 2013

Good question? Bad question?

The question above is from the January 2013 New York State Regents exam in Integrated Algebra.

It is an example of a well-intentioned, but poorly designed, question.

Simply put, the student who has absolutely no idea, and randomly guesses, has 25% chance of selecting the correct answer.

The student who knows exactly how to graph the quadratic, who can plot its parabolic shape with mastery, and can state the coordinates of its vertex, state the axis of symmetry, identify its intercepts, and so on, but happens to forget the "number names" of the quadrants, has a 25% of selecting the correct answer.

The student who knows everything that the second student knows, but remembers that quadrant I is to the northeast, but forgets whether they are numbered in a clockwise or counterclockwise manner, will select choice (3), the correct answer, since the vertex lies in the southwest quadrant.

The student who knows everything that the second student knows, but believes that quadrant I is to the northwest,and remembers that they are numbered in a clockwise manner, will select choice (4), the an incorrect answer, and have a 0% chance of getting this question answered correctly.

My question is this: What is question 14 actually measuring?

Tuesday, March 26, 2013

How do you add?

A number of papers today have an Associated Press story about the recognition that the development of math skills starts early in life, before first grade. The version from the web page of station WKRG out of Mobile, Alabama,  (click here) says "Scientists say that what children know about numbers as they begin first grade seems to play a big role in how well they do everyday calculations later on."

They just figured that out? How many billions of dollars have spent deducing that fact?

Counting my own days as a student I have spent 50 years in classrooms, and it hit me in the head early on that those who started strong tended to finish strong, and those who started weak tended to finish weak. There were crossovers, strong starters who weakened and weak starters who got stronger, but they were exceptions. The weak starters becoming strong finishers were the rarest. I am not speaking only about mathematics, either.

As long as they are addressing reasons for poor student performance in mathematics, perhaps they should look at some other oddities in the learning and teaching of the subject.

The biggest oddity is the standard method for adding multi-digit numbers using paper and pencil. We read left-to-right, we write left-to-right, yet most people learn to add (and subtract and multiply) right-to-left. Operating the natural way, left-to-right, is even considered by some of those "in the know" as a TRICK!  For an example, check out this site: http://mathtricks.org/addition-tricks/addition-tricks-addition-from-left-to-right/.

The web site FoxyMath even says "Right to Left is so important when solving column addition equations that FoxyMath gives it its own page." (original page here). Right-to-left is so embedded in our culture, and so awkward for many, that it becomes one of the big stumbling blocks in math education, as well as an cause of the early demand for calculators. (Note: on most calculators, we enter numbers left-to-right). Right-to-left arithmetic is one of our major cultural flaws, not as big as the Roman Empire and its number system, but big enough.

Now there is a socio-political reason for operating left-to-right. That has to do with the tendency, when performing a calculation with many steps, to start out correctly before any errors creep in. With left-to-right arithmetic, the portion of an answer most likely correct is to the left, the part of a number that is more meaningful in general. If you question that, go to http://www.brillig.com/debt_clock/ and tell me what is more significant: the digits after the dollar sign or the digits after the decimal point.

Worth noting is that the skill (art?) of estimating is geared around the leftmost digits in any number. To be a good estimator you must have some basic knowledge that the left hand portion of a number is the more important portion.

Our country has become so addicted to right-to-left arithmetic that other methods are largely ignored, if not treated with suspicion. To get a taste of some options, check out this page, from Rockwood School District, in Eureka, Mo. It only shows one example of each method, but it might be eye opening.

This country spends hours and weeks and months and years teaching youngsters how to do arithmetic operating from right to left. Just about the time we have them totally befuddled and a far ways down the path of math phobia, we introduce division, operating from, drum roll here, LEFT TO RIGHT! Paper-and-pencil long division then becomes the least-learned, and perhaps least-taught, of all the basic operations.

Back to the beginning, today's news story. What are the chances that a parent with a weak base in basic arithmetic will be a positive role model for his/her child in this regard?  Won't the behavior the child picks up on have more to do with picking up a calculator, or, worse yet, ignoring arithmetic completely?

Wednesday, March 20, 2013

A well regulated militia...

One of my neighbors owns a gun. I don't know who it would be, I don't know what type of gun, but,based on statistics and probabilities, someone around here has to be a gun owner.
What I also don't know, and can't figure out, is how this unknown neighbor is involved with a well regulated militia in this neighborhood. After all, that is the sole reason given in the US Constitution for the right to keep and bear arms.  The concept of crime prevention is not even mentioned in the Constitution, militia is. Yet gun defenders have begun to jump on defensive gun use as a bit of evidence to support their gun rights.
The web site www.saf.com, where the Second Amendment Foundation hangs its hat, states "Firearms are used defensively roughly 2.5 million times per year, more than four times as many as criminal uses. This amounts to 2,575 lives protected for every life lost to a gun ." The Cato Institute, on their web site, states "The estimates of defensive gun use range between the tens of thousands to as high as two million each year."
I find the first statement extremely alarming: alleged victims pull out a gun 4 times for every single time an alleged criminal pulls out a gun. Either the other 3 times the criminal is unarmed, or armed with an alternate weapon. This tells me that the victims are overarmed. We don't allow our police to shoot unarmed people, yet it's okay for individual citizens to do it?
The second quote points out that the writers of the first quote are making up statistics. Perhaps their 4-to-1 gun use ratio is totally made up as well.
What I do know is that the gun-toting public has a number who are falling victim to a logical trap known as the fallacy of the converse.  In a nutshell, they identify a conclusion they want (gun ownership) and then search out information to support it. In that situation, absent good solid evidence, it is very easy to fall into the fiction trap. After all, their conclusion HAS to be true, hasn't it?

Tuesday, March 19, 2013

Today is the Day

At long last, the NCAA men's basketball tournament is here.

The strangest aspect of this annual event is not that it claims to crown a "national champion" of collegiate basketball. That has always been debatable. What is not debatable is that the NCAA tournament gathers 68 of the top teams and assures that 67 of them will end their year with a loss. The last memory that the majority of seniors on the teams will have is the sting of defeat.

We claim to watch the tournament as a validation of our picks in the ubiquitous bracket sheets we fill out with reckless abandon.  I believe our real gut-based interest in watching the tournament is to see how success-targeted young men deal with an ultimate defeat. After all, dealing with failure in a positive manner is life's biggest lesson.  None of us want to be failures in life, but the realists know that failure along the way is inevitable. Dealing with failure is a skill all need.

What bothers me about our world of team sports is the manner in which colleges take a losing season and use it as a basis for firing a coach or manager. It's as if they are placing blame for losing on the coach. They need to recognize that the leading cause of losing is winning. Losers are required for winners to exist. Sure, nobody wants to lose more than they win, but it happens.

A bigger test for me were I a college "decider" is whether or not the coach was successful in helping his players learn from losing. After all, even in this tournament, all but one will have a guaranteed loss.

Wednesday, March 6, 2013

How do people justify their ownership of guns?

Rather than attempt to answer this question here, I will instead refer you to a place where it has been answered excellently. Michael Boylan has done so in The Opinionator at the New York Times. His complete article can be found here.