Explore Dynamic Geometry in the New Year!

A GeoGebra design such as this would have a student confront the notions of circle, radius, vector, rotation, translation, and angular speed.

Why can't students learn the vocabulary first and all the fancy math details later?

Dynamic geometry can be an entrance into mathematics.

For more, see www.davemath.com

One circle and one pentagon....

Dynamic geometry can be the door of entry to higher mathematics if it can be the "hook" that captures a students imagination.
This was made in free GeoGebra. Click on tyhe image to get the actual geoGebra file behind it.
For more on dynamic geometry, visit www.davemath.com

Cause and effect

Since we spend a great deal of our lives claiming to causes of the effects we see, I would think it would not be too big a deal to describe the cause of the following path.

If you wish, you can click here to see a small web page about it.

Good luck!



Hub cap math

Blog Error

Starting this morning, whenever I click the "design" link at the top of my main blog page, I get this error.
ERROR: Possible problem with your *.gwt.xml module file.
The compile time user.agent value (ie9) does not match the runtime
user.agent value (ie10). Expect more errors.

No matter what I have tried, the error persists. Does anyone have a clue about this?

Golden Arches: Parabolas or Not?

Click on the leftmost toolbar icon, then drag the 5 points to either the inside or outside edge of one of the arches. GeoGebra will show you what it gets. (The other tools are, in order,  for moving everything, zoom out, zoom in, and selecting a single item.) EXPERIMENT!!!
Note: You mat have to click in the space below to activate the graphic.

Omega Unlimited: The Golden Arches ® Exposed!

This is a link to an older post by someone else. Tomorrow I will post a page that will let you test this theory for yourself. Omega Unlimited: The Golden Arches ® Exposed!

Amazing Midpoints

In the sketch below, drag "pointP" and "pointW" to see different graphs. At any time, to see how the graph is created, right click the "a= " slider at the top and choose "animation on". To stop, right click the "a= " and deselect animation. To start over, click the little symbol at the top right corner of the sketch box. Note: you might have to click in the frame below to activate the sketch. For more, visit www.davemath.com

My 1st GeoGebra "implant" into Blogger

Drag the point labeled "a" in the segment at the top of the sketch. I need to improve the sizes somehow so that an entire GeoGebra page will fit in the space provided in Blogger. Stay tuned. You may need to click inside the space below to start the graphic.

Parabolas are tough????

I submit the above two animated gifs (made in GeoGebra) for the sake of those who think that ellipses and parabolas are too complicated to understand.

The 1st gif, yielding an ellipse, only requires that the red segment and the blue segment always have the same combined length.

The 2nd gif, the parabola, only requires that the two dotted lines are always equal in length.

That's it. The rest comes from there.

So Sine and Cosine are related? (or...have you ever seen the rain?)

Dynamic geometry can help show relationships that printed textbooks can only hint at.  If all you saw in this was that the getting from the cosine graph to the sine graph all you had to do was "shove" it to the right a bit, you'd be well on your way.

For the more utilitarian, here is a more well known device.

For more, see www.davemath.com

Do you Know or Remember?

Inside every triangle is imprisoned a very special circle. Did you know that? It is called the inscribed circle, and it is one of the many connections between circles and triangles.

 

A shipping question relates to this: if you were shipping a million sections of fragile pipe, would you consider a triangular-shaped box? If so, what type of triangle would be best?

 

When I was in school the closest I got to diagrams like above were those frozen in time and printed in books. Students can know learn about such things in a much more dynamic environment. Make sure they can and do!!!!

For more examples, go to www.davemath.com

 

 

 

 

Dynamic Geometry Opens Doors!

The sky is the limit as to what can be designed using the basics of GeoGebra.
Click here to see the actual file behind this graphic.
It is all basic Geometry (with just a smattering of trig so I could get flashing lights to work when the file was converted to an animated gif.

Software such as GeoGebra just be used as a math motivator as well as a math tool.

Dynamic Geometry is the Way to a Child's Brain!!

The above was created in GeoGebra some basic geometry such as midpoints, chords, rotation, with a dose of basic trig. Doable by ant middle-schooler in a short bit of time. (Exported as animated gif, then shrunk using Fireworks to reduce file size and make it fit here.)

Amazing

FOX just interrupted a moment of silence in honor of Nelson Mandela prior to the Eagles-Lions game for a Cialis ad!!!!!!

Another new story about math in schools

"American 15-Year-Olds Lag, Mainly in Math, on International Standardized Tests" goes the title of the article in the NY Times of this past week. (see here)

Who knows what might be possible if the teachers and students had the opportunity and the teachers had the knowledge base and the resources.

A simple item such as below, a great motivating skill for a number of students, is not considered often. Since it is not testable, it is probably falling further outside the arena.

GeoGebra allows the user to manipulate images.
Here a simple 2-blade propeller was made to rotate using GeoGebra.
A simple task.

Check out http://www.angelfire.com/ny3/golfgolf/amtnys/AMTNYS2014/Plan2014.htm for other examples.

Math IS fun!

Isn't it about time that people started to appreciate the fun of Mathematics? Maybe if the public would encourage (maybe even allow(?)) teachers to use technology creatively, we could get past this sickening stumbling block of math-hatred. Check here for more samples!!

 

 

 

Note: Did you ever see this when first learning trig?

 

Mathematical motivation is KEY!

It's a little known fact that students with a PC and a free piece of software called GeoGebra could easily create images such as the one above.
It is my feeling that student creation of such images and designs can be a springboard toward student interest in mathematics in general.

Can't we just get it right?

" Sue and John have marbles in the ratio of 2:3. Together, they have 35 marbles. How many marbles do each one have ?"

The above statement was used in a PowerPoint slide by representatives on the New York State Education Department in a presentation at the annual conference of the Association of Mathematics Teachers of New York State in Buffalo earlier this month.

It makes me cringe.

Many would look at it as just a careless mistake. That bothers me. I find a hard time accepting errors that could have and should have been caught. At any rate, does that imply that there are errors that are "careful"?

I would expect that NYSED would hold itself to a higher standard than to leave items like this uncorrected. I would also point out that the statement is in a 6th grade example, whereas subject-verb agreement is a 3rd grade item in the Common Core (see this)
 

---

9. Will the Regents have low scores like the 3-8 State Tests did? Or will there be a change in the cut score to account for low scores until we fill in the gaps created by the rushed implementation of the Common Core?

A. For the Algebra I (Common Core) Regents Exam in June, a meeting will be held following the exam to set the performance standards. A large, representative committee of teachers is convened to review the items and performance level descriptions, and make decisions about what students must be able to do to be classified into each level of performance. The committee of teachers (known as panelists) recommend ‘cut-scores’ to the Commissioner. The Commissioner sets the final cut scores.

10. Will the passing rate for the Algebra Regents change drastically?

A. For the Algebra I (Common Core) Regents Exam in June, a meeting will be held following the exam to set the performance standards. A large, representative committee of teachers is convened to review the items and performance level descriptions, and make decisions about what students must be able to do to be classified into each level of performance. The committee of teachers (known as panelists) recommend ‘cut-scores’ to the Commissioner. The Commissioner sets the final cut scores.

 

The above paragraph is a written answer to a question submitted to NYSED prior to the AMTNYS conference. The red is mine. I bring emphasis to the fact that the performance standards are set after a test rather than before. This helps perpetuate the impression among in-service teachers that they are dealing with a moving and vague target. I would also point out the identical response to two different questions. It reminds me of the "name, rank, serial number" reply soldiers gave: acknowledge a question but contribute little in the form of an information.

Smart Scheduling?

Before I pass judgement on the June 2014 New York State Regents Examination schedule, I would like to hear from the source why two key exams in mathemtics were scheduled for the same time.
All I need to know is who set the schedule, and why it was set  this way.
I cannot find this information anywhere on the NYSED web site.

FYI

NYTimes: The Not-So-Hidden Cause Behind the A.D.H.D. Epidemic http://nyti.ms/1gfEIcr

AMTNYS 2013

November 6 to 9 this fall the Association of Mathematics Teachers of New York State (AMTNYS) will be having their fall conference at the Adams Mark Hotel in Buffalo.

As the AMTNYS County Chair for Columbia and Greene Counties I strongly urge anyone with any connection to math education to attend.

The organization is not limited to teachers, and is not limited to NY residents. The only requirement is that you join AMTNYS.

Schools: a reflection of the community

I just finished reading a column in the Albany Times Union  written by Questar III superintendent James Baldwin about the results from the round of testing that took place this spring in our schools.  This article just added confirmation to the feeling I have been gradually acquiring for decades that schools are a mirror of the community. Testing is measuring the community, and by that I do not mean just the school community. I mean the while darn thing.

 

There is a whole slew of people around who seem to think that schools drive the community: you can tell those the way that talk about "the work force of tomorrow"  and other such stock phrases.  Those people are the ones constantly campaigning for some new school reform or another with the goal in mind of "making America number 1 again!"

 

These people are putting the cart before the horse. These people would see global warming and try to fight it by redesigning a thermometer.

 

In order for our schools to get better and stronger, our whole country must get better and stronger. We have to build better buildings for our schools. We have to make better movies and TV shows for the youth of America.  We have to feed the whole country better if we expect children to eat better. We have to have better lives with the kids before we can expect them to live better without us. We need to take pride in educating our youth outside of school before we can expect them to take pride in educating themselves inside of school.

Change all the tests and curricula you want. Nothing will work unless and until the community makes a solid effort to change itself.

I hate to be the bearer of bad tidings, but if you want to get a sense of any local area in this country, visit its school(s).

Spelling?

I have just been browsing through (see it here)

PARCC MODEL CONTENT FRAMEWORKS

ENGLISH LANGUAGE ARTS/LITERACY

GRADES 3–11



A big title for what might be important, as a whole slew of states are officially in on it.

Exercising my normal skepticism, I checked it for some of what I think should be the normal curricular phrases. One search has me wondering. I used the Acrobat Reader's find tool and asked to find uses of the words spell " and "spelling".  The word "spell" did not appear in the 98 page document, and the word "spelling" appeared once, in the sentence "In all student writing, the use of specific facts and descriptive details is emphasized, as is correct spelling and punctuation. "

Now that sentence appears in the 3rd grade section of the document. At no other grade level do these words appear.

Having seen careless spelling gradually expand into our media (perhaps largely in thanks to the not-always-correct spell checker), this scares me.

Evidently the laxity in spelling exists outside of the US. According to an article by Laura Clark in the Daily Mail out of England (see it here), a school has the following guideline in its policies: "Teaching staff are not to highlight any more than three incorrect spellings on any piece of work. This is in order that the children’s self-confidence is not damaged."

The placing of self-confidence (and its companion self-esteem) above the development of skills and knowledge that lead to self-confidence and self-esteem is another of the well-intentioned but misguided trends in education.  Feeling good about oneself has been taking precedence as a means rather than an end.

The Long Division of the Literati

Just as I feel that long division has been tossed out incorrectly, I also feel that the teaching of grammar (what's that) has gone by the wayside, especially its art of diagramming sentences.

An article from the New York Times by Kitty Burns Florey (see here) from June 2012, referring to diagramming sentences, says "First of all, diagramming is not for everyone." That hits our problem nail on the head, especially when compared with the term "Common Core": what we teach in our schools now is supposed to be for everyone.

In my experience as both a student and a teacher, what our public schools have failed to acknowledge is that there are some items that are beneficial to some, not for all, and that it is okay to teach some things in the classroom that not all students can master.

We put all New York students through Algebra 1 (whatever it is called now), force them through it all, calling them failures when there are parts they cannot handle. Those "failures" become negative marks on both the students' and teachers' records, and generally on the students' home lives as well  as we pass the buck to such things as single parent families and poverty.

What New York should be looking at are two key issues: are students' ready to learn what they are being taught? Does everybody need to master what they are taught? If we were all to recognize that some skills that need to be known and can be learned by some, but not necessarily by all, then it would be okay for a student to not succeed at that skill. Can you imagine students' getting things wrong in school and being told "that's okay."

We do it in PE quite masterfully. We take all through basketball, football, baseball, soccer, etc., identify those who want to do well and can do well, and proceed with them. We do not condemn the others and call them failures.

We have gone through a patch in the last 25 years where knowledge and skills obtainable by few have slowly been weaned out of our curriculum. This has been a downward trend. The Common Core will not change this trend.

I recommend reading "Common Core Standards Will Impose an Unproven ‘One Size Fits All’ Curriculum on North Carolina" (get it here).  This article says "Common Core Standards have been sold as a tool to raise academic standards and improve education for all students across America. However, an untested assumption underlies CCS: all students should learn the same things and have the same education."

The Common Core standards website says "The standards promote equity by ensuring all students, no matter where they live, are well prepared with the skills and knowledge necessary to collaborate and compete with their peers in the United States and abroad." Take note that equity can be obtained by bringing down the top as ell as raising the bottom.  Also take note that it says "no matter where they live". Really? ANYwhere?

More oddball division

Check out University students learn a new long division algorithm .

I am still trying to decipher what this new method really is. There is just one example given, and I am trying to decipher the recipe from it.

This paper does seem odd that it asks college students if this recipe for division is simpler or more difficult than a method they had probably learned in grade school, or at least tried to learn. ( I do believe that school mathematics often "moves on" before many skills are truly developed in the students' minds. If long division was taught initially to college students, their more highly developed brains might catch on more easily than do the brains of pre-teens.)

Another long division befuddlement

In "The case Against Long Division" (see it here), Tony Ralston gives this example of an alternate method for division:
Then, at the first step, multiply by 10, record the integer portion of 10A/B as a, subtract a from both sides to get .bcdefg… on the right and then repeat this idea to get b, c, … . When B has one, or even two, digits, the subtraction can be done mentally. Thus, for example, to compute the decimal equivalent of 3/7 we would set 3/7 = .abcdefg… and proceed as follows:

30/7 = 4.bcdefg… so 2/7 = 30/7 – 4 = .bcdefg…
20/7 = 2.cdefg… 6/7 = 20/7 – 2 = .cdefg…
60/7 = 8.defg… 4/7 = 60/7 – 8 = .defg…
40/7 = 5.efg… 5/7 = 40/7 – 5 = .efg…
50/7 = 7.fg… 1/7 = 50/7 – 7 = .fg…
10/7 = 1.g… 3/7 = 10/7 – 1 = .g…
30/7 = 4. …

and now it is clear that the sequence 428571 just repeats since once one digit of the quotient is repeated, subsequent digits must also repeat. Note that in practice students should be expected to write down just the successive quotient digits with all other calculation done mentally.

I, for one, don't picture this as easier than long division, but it doesn't really matter, as the steps taken are EXACTLY the same as in long division, just written out differently. In this case I think the presentation creates a greater chance for careless error, but that is just my opinion.

Mr. Ralston, further down in his writings, claims that the long division algorithm is no better than the almost defunct pencil-and-paper square root algorithm. From my perspective, the reason the square root algorithm was marginal in school mathematics is because 1) it is time consuming, with a number of false starts for weak arithmetic students; 2) rarely tested, probably because of its time consuming nature; and 3) rarely needed since perfect squares and radical notation "ruled".

What I find scary about this long division issue is that the reformers are claiming that long division itself is irrelevant. By this logic, no one will learn it, helping to firm up the calculator dependency that is taking over. Skills will mean less and stock in batteries will mean more.

PS: I should add that Tony Ralston is (was) a computer science professor at SUNY Buffalo. His algorithm as exampled above is easily programmable for a single digit divisor, provided it is recognized that the integral part of the quotient is identified first. Otherwise, his algorithm needs many more examples in order to be understood.

More on Long Division (and short division, too)

There has been an article out for a few years titled "Long division is so last century" (find it here), In this article she quotes Lawrence Spector of Manhattan Community College as follows: "Spector says that long division "belongs in the history of mathematics" because short division is so much simpler and faster.  I have to say I absolutely agree with him!"

 

This is absolutely scary! Not because short division is incorrect or improper or not effective, but because it is merely a short way of writing long division, which in itself was shortened by leaving out some trailing zeroes.

 

I am of the strong belief that long division can be understood better as it relates more directly to the commonly known distributive law of multiplication over division (more on that in a future blog entry.)

 

The web site Ask.com, in response to the question "What is the Difference Between Short and Long Division?" replies by stating: "The difference between long division and short division is that Long division is a typical procedure suitable for dividing simple or complex multi-digit numbers. It breaks down a division problem into a series of easier steps, whereas short division requires neither complex technology nor mental gymnastics; it is only suitable if the divisor is small - typically less than 10."

 

I believe this reply to be extremely misleading in three ways: it fails to recognize that the "mental gymnastics" in short and long division is exactly the same, it claims that long division "breaks" a problem down, when that is exactly what short division is also doing, and it confuses "small" divisors with single digits divisors (after all, .674327 is much smaller than 1)

 

Wikipedia is a bit more precise, when it says "Short division is an abbreviated form of long division."

 

Needless to say, long division is greatly misrepresented, misunderstood, belittled, and unappreciated. Could it be that no one got rich teaching long division but a lot of people got rich selling calculators?

 

Click here for a comparison of a long division example and the information it contains.

Long Division (continued)

In yesterday's blog entry, I left out mention of an excellent article entitled The Role of Long Division in the K-12 Curriculum written by David Klein of California State Northridge and R. James Milgram of Stanford.  You can find there article here.

Within this article they stress California educational politics, but their arguments transcend the region. I do have to confess that readers with a lack of comfort with numbers will probably not finish this article (but I pointed out yesterday that familiarity with our number system along with addition, multiplication, and subtraction, is a prerequisite to handling long division).

Please remember that long division is an algorithm that historically became the standard process for doing division. It also is an excellent tool in discussing the differences between terminating, repeating, and non-terminating non-repeating decimals, which help distinguish between rational and irrational numbers, which is muddled up by hand-held calculators.

For the record, students with hand held calculators have big trouble handling our national debt (in dollars and cents it's 15 digits long). Ask a student to divide the debt equally among the 50 states and see what they do.

More to come...

Long Division as a Symptom

Laurie H. Rodgers, in an article entitled "In defense of long division: Pro-reform professor capably shows why reform math doesn't work" has given a strong defense of long division, and has done so while helping to show that the math reforms (coalescing nationwide the in Common Core State Standards) are not necessarily for the better. (Her article can be found here.)

Speaking of long division, I will add to Ms. Rodgers' argument by using a quote from her article: "Many don’t even know their basic multiplication facts.."

Long division, when it is taught, is generally the last of the four basic operations. As an algorithm, its use and fluidity depends greatly on the user's knowledge of basic multiplication facts. It also relies heavily on subtraction. We all know that multiplication and subtraction are largely taught in their relation to addition. Thus virtually all one's basic number knowledge comes into play in constructing long division. Weaknesses in addition, subtraction, and multiplication will all help in crashing the edifice known as long division.

When confronting long division, anyone with weak mental arithmetic skills in the other operations will generally "mess up."

As a high school teacher I made a point when analyzing a student's work of trying to detect the specific error(s) that caused students' solutions to be in error. I wish I had kept good solid data, because the fact was that the vast majority of errors (my sense is 80% or more) were caused by faulty addition, subtraction, and or multiplication. (Leaving out the errors based solely on conceptual weakness). And this was in the calculator age in the New York Regents program.

Speaking of calculators, I found students "dial wrong numbers", get erroneous calculation results, and have no clue that they were off. Even adding 23 and 2, for example, a student might hit the multiplication button, and then write down 23 + 2= 46, and let it stand.

Needless to say, I hold the position that our educational systems are having students race through arithmetic so we can get them to "higher thinking skills" where all the alleged importance is. I believe this if more time, energy, effort, and resources were spent teaching children all about numbers, the higher order items along with a love of math would follow in due course.

I suspect the English Language standards are missing the mark as well, since I have searched through them and I have not yet found a reference to "spelling".  The first grade has a standard that says "Write opinion pieces in which they introduce the topic or name the book they are writing about, state an opinion, supply a reason for the opinion, and provide some sense of closure."  In essence, first graders will be writing book reviews. Without being able to spell?

Math Wars

There was an excellent article in the New York Times a on June 16 called The Faulty Logic of the ‘Math Wars’.  It is a must-read for anybody who even cares about what happens in public education in this country.

Having been a 32-year math teacher myself, I can attest to the fact that there have been massive changes in the way mathematics is intended to be taught. I say "intended" because there has always been a hope, on my part, that teachers can handle the new wave and still manage to sneak in some real and true math instruction.  Unfortunately, as time passes, more and more of our teachers will have been products of the new era, almost to the point that pencil-and-paper calculations become anachronistic, and algorithmic thinking becomes a rarity.

Failing to teach students a good dose of algorithmic skills, especially in arithmetic, forces them to fall into one, or possibly both, of two pits: that of calculator dependency and that of failure to achieve progress because they have to constantly reinvent everything that was discovered before.

Alfred North Whitehead once said "Civilization advances by extending the number of important operations which we can perform without thinking of them."  I could change that and add that basic human life operates the same way. Do you believe that a good driver thinks his/her way through a road trip? Or has the process been, in essence, "built in" to the brain?  Now imagine how that process was embedded, and you have little picture of how people acquire competences.

Thinking and reasoning are massively important in life, but people should not need to rely on them continuously and constantly. There should be a significant portion of our life's activities that we can do competently and successfully without having to think and re-think every action.

Musicians who are successful have taken complete unknowns and turned them into second nature habits. Especially those who read music. The brain power used to develop the skills that a classical pianist exhibits is the same brain power used developing a world-class mathematician. Leibniz once said "Music is the pleasure the human soul experiences from counting without being aware that it is counting.”  A case could be made that the deaf Beethoven wrote his works using mathematical thinking.

I truly wish our educational system would look at how people like Beethoven, Whitehead, and Leibniz succeeded, and adjust our schools accordingly. Instead they listen to the business world, if anybody.
Much more on this to come....

On China

I have been reading Henry Kissinger's book On China. It is a slow read, but extremely well written, and massively informative.  And extremely scary.

By scary I mean two things. First: I consider myself rather well educated, you know, Cornell undergraduate, Colgate graduate school, etc. But I have to say that 90% of what I now know about China and the US-China relationship I know from this book. Maybe younger people are better off. My school years were ending as the Sino-American interface was just beginning to thaw, back in the 1970s.

The second scary thing is this: I read this book via Kindle (which is not scary at all), which limited me to 2 weeks and no chance of renewing. This 600+ page book, being read for maybe an hour a day tops, has now taken me 6 weeks. I have now had to "return" the book 3 times, and reborrow it 3 times.  With our library system you cannot borrow a book you already have out, so you have to hand it in and go to the end of the queue. But...

Three times now there has been no queue. And our library system only has one ecopy to lend out. This system includes all of Columbia, Dutchess, Green, Ulster, and Putnam counties in New York State.  In that whole system not one other person expressed an interest in the ebook. No waiting list at all.

Then again, at Amazon this book has, as of today, 186 reviews, while Harry Potter #7 has over 4000.

I know there are people out there who will say that this is totally understandable, categorizing Kissenger as just another conservative who doesn't know anything anyway.

I guess I'll just have to read a lot more about China to get a broader feeling. Even I know that relying on just one source for anything is not healthy.

the power of twitter

Yesterday a hoax tweet about a White House bomb caused a stock fall. It only lasted minutes, but it did happen.  That stock drop was caused by people reacting to a text message.

Maybe it is a good thing that our government is not quite so quick on the trigger. If this message had been about a nuke hitting a US city, an equally momentary response could have been horrific.

I propose that the stock market fix the problem that it has. It cannot be this susceptible to emotional responses.

NRA, where are you?

Today's Albany Times Union has an article headlined "U.N. OKs treaty to regulate global arms" . In the article the Secretary General of the UN is quoted.  “This is a victory for the world’s people,” U.N. Secretary-General Ban Ki-moon said. “The Arms Trade Treaty will make it more difficult for deadly weapons to be diverted into the illicit market. ... It will be a powerful new tool in our efforts to prevent grave human rights abuses or violations of international humanitarian law.”

I assume that the NRA will rise up against this. If the 2nd amendment is so correct in their interpretation, I cannot see them in favor of arms control of any type.

Come on NRA! Don't disappoint me!

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?

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?

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?

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.

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.

What's wrong with the sequester?

With all the budget stories that have been out in the news media recently, I have tried to focus on the issue as simply as possible: income vs outgo.  The US government has, in simplest terms, been letting the outgo vastly exceed its income, borrowing what it needs to balance its budget. In essence that achieves the goal of postponing actual payment of its costs, and adding interest on top of it.

Just imagine the typical homeowner deciding that they can only pay a portion of their weekly grocery bill, and borrow to pay the balance.  The next week he would have a debt, so that the new weekly costs would include the new groceries, interest on what was borrowed, and hopefully at least a tad of a payment on the debt that had been incurred.  Should income that week have risen, that option might be survivable. Otherwise, the new homeowner will have to borrow more to pay the shortfall, increasing the debt, and thereby be in need of more money to cover larger debt costs, plus the next week's grocery bill. 

When would this process end? Only when one of three things occurred: the homeowner finds a way to earn more money, enough to pay off the debt and cover the grocery costs. An alternative is to cut the grocery costs back to a level that can be covered in light of limited income and the need to pay off the debt. A third alternative involves a combination of increased income along with a cut in grocery spending.

That's it. No other options, at least of the legal sort.

In essence, that's what federal sequestration would do: cut the government's costs while hopefully adding some extra income. It has to happen. If it doesn't, the US follows the homeowner into financial death.

Now imagine the homeowner making a conscious decision that at a certain date, if he cannot figure out what to cut or how to get money, he will be forced to cut 10% of everything and do something to get a raise or a new job or a second job. No questions. Has to.

That's where we are at. Our homeowner (Congress) could not come up with a plan to meet its needs, so now a sequester plan will kick in: cut spending, increase revenues.

Seeing how our Congress comes to its decisions (or, should I say, how it doesn't come to its decisions), at least sequester will be a step getting back on sounder financial ground.

As a last word, I will add that I, as a homeowner, spend a lot of time dealing with how I will spend the money I do have coming in. I don't spend much time at all dealing with how to spend money I haven't got. (Car loans and mortgages are exceptions, of course. Their payments are, however, a solid part of my "grocery bill.")

Employer Contribution rates High????



There has been a bit of noise lately in the New York media about the upcoming contribution rates for the public employees' retirement plans. Since I was once a teacher, I will limit my comments initially to the New York State Teachers, Retirement Association (NYSTRS).

 

The upcoming Employer Contribution Rate (ECR)  is estimated at 16.25% of payroll.  That means that that percentage of salary of employees who are members of NYSTRS will be paid to the system.  There is a lot of groaning about that, perhaps rightfully so. There are stories as to how this cost will force districts to cut staff, cut program, etc.  That may be so, but...

 

Until 1988-89 school year, the ECR was constantly higher than that 16.25%, frequently much higher.  I will not comment on any complaining during those years: there was some, so be it.

 

I will comment, however, on the lack of complaining during the 1990s, as the ECR fell dramatically to an all-time low of .36%.  A slight bump in 1994, but the big trend was down.  School districts were steadily liable for retirement costs that were declining. Now, what did they do with that money they were saving? They spent it!!

 

During the 1990s some of the best teacher contracts were approved and enforced, best from the teachers' point of you. For that I say to the public, thank you. I benefited from those years. I can't say I was ever paid what I should have been. My union negotiated not a minimum salary for its members, but a maximum salary as well. (For that matter, find me a teacher who gets paid for overtime in a manner like many many other public employees)

 

Districts were not allowed to take part of the savings and bank it, saving it for the rainy day that we are in now.  Collecting and saving money would be considered, in simplest terms, a misuse of taxpayer funds. Not wanting to create the situation whereby they are given less money one year because they didn't need what they got the year before, districts made sure they spent what they had. They prided themselves on "no tax increase this year". They found ways to spend dollars that were available.  (A number of times we were told in our district, towards the end of a school year, that we could purchase items because "money was available". Those occasions were always at the end of the year, never the beginning, when we would then have an opportunity to use what we purchased.)

 

Enough grumbling. The fact is that salaries went up at little or no cost to school districts. The costs of those days are being paid now.

 

Take note that this is occurring in a state that has had several "retirement incentives" geared towards getting the more highly paid experienced worker off the payroll and onto the retirement dole, the idea being that employers could bail out of paying those higher salaries. That might seem fine, but this is the same state the turned around and bumped the minimum age for retirement with full benefits by 8 years (Tier 5 brought it from 55 to 57, and Tier 6 brought it to 62). What's the message now? Hey old timers we need you to stick around longer?

 

Part of the problems that have arisen in public education funding in New York is that school budgets are done one year at a time and basically preclude any long range planning. No business could succeed under those parameters.

 

That concept does lead to strange circumstances. For example, in my district teachers had to fill out requisitions for the following year as early as January. Of course, you would not know the courses you would teach until June, or even later at times.

 

Enough for now.

 

As a final word I will say that the education of our youth will never be excellent as long as we run it as a caveman's business. Click the graphic to access the entire NYSTRS briefing booklet (the source.)

The people??????

Amendment II

A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.

Amendment X

The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

Do you note a common phrase in these two Constitutional amendments? In fact, the words "the people" are used nine times in the Constitution of the United States.  Is the meaning of these words constant throughout the document? Does the meaning vary?

The phrase is used regarding selection of Congressmen (Article I, Section 2) and Senators (17th Amendment). Take note that even in these two instances, the meaning of the phrase has evolved.

The 17 the Amendment states "The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each State shall have the qualifications requisite for electors of the most numerous branch of the State legislatures."

Note that in that amendment, the definition of "the people" who do the electing is left to the states. That is the same phrasing as in Article I, Section 2. In fact, the 15th Amendment changed the meaning by, expanding the voting public, as did the 19th Amendment, which granted women the right to vote.  (On a sober note, the 19th Amendment lies between the 18th and 21st, which began and ended prohibition. )

The 26th Amendment set the minimum voting age at 18, which again changed the meaning of "the people". 

When it comes to voting, one would be hard pressed to prove via the Constitution that voting for President is even required. (Choosing electors is required, but the manner is unspecified).

Speaking of voting, those living in Washington D.C. do not have representatives in either house of Congress. Do they have to pay income taxes? Is that not taxation without representation, one of the items this country was created to avoid?  Are D.C. residents really included in "the people"?

Indeed, there are inconsistencies within our Constitution. It is not a complete description of the way we need to live, but at least it has a built-in process to correct itself.

Remember: those who originally approved  2nd Amendment did not allow women or people of color to vote, left it to the states to decide any age limits, and actually created a document in which slavery was okay.  Did the authors of the original Constitution make mistakes? You bet they did.

Now what about that Second Amendment?

Thank you Mr. Metallo

The article below is from the Albany Times Union of February 8, 2013. Rather than trying to improve on his work or using out-of-context snippets, I decided its best if everyone could just read it. Then reread it. And read it again. This short article includes the majority of the snags that have stymied public schools over the last 25 years.
Everything below here is from the paper.

Denying kids junk food isn’t the answer

  By John Metallo

     I feel a lot better after reading the newspaper this morning. The federal government has banned the sale of junk food in schools across this great nation.

   Finally, our schools will have something to do that is productive instead of worrying about things like teaching and learning, curriculum development, accurate budgeting, state testing, teacher evaluation, bus safety, sex education, character schools, bullying prevention, safety and security, psychological counseling, social work, health screening, etc., etc., etc.

   Monitoring the sale of junk food in schools?

   Give me a break, please.

   What a waste of time and energy, from the Capitol building in Washington to the corridors of every school in the nation. This is simply something that is not the headache of the schools.

   I worked in public schools for over four decades. And guess what? I really don’t care what your kids eat. I don’t care
  how much they weigh. I don’t care if they develop lousy eating habits that will negatively affect them in future life.

   How about the parents of these little cherubs of ours doing something instead of expecting the schools to raise their children for them? Schools would be a lot better off if we ask them to do what they are supposed to do — educate kids.

   Schools are not meant to raise kids, nor will they ever be able to do that.

   Let’s look at a few facts. Kids are not getting fat because of what they eat in school. They consume most of the food they consume at home or some place other than school.

   Based upon a 180-day school year and three meals per day, a student consumes 180 lunches in school during a calendar year. That would move up to 360 meals per year if the student eats breakfast in school each day.

   By the way, why are kids being fed breakfast in school? Shouldn’t that meal be consumed at home in the presence of a caring and loving family? Given lunches only, a student would consume
  180 meals in school and 730 other meals at home or elsewhere. That diminishes to 550 meals consumed outside the school for those who have breakfast and lunch in school.

   If schools serve healthy lunches, that does not necessarily mean that kids will eat them. Some kids bring their own junk food with them from home. Some buy it from the corner store on the way to school. Some trade for it with their friends. If there is junk out there, kids will find it.

   It is what kids do. Much more junk food is brought into a school each day than is sold in any vending machine in a school.

   By the way, vending machines are in schools to help support things like sports and school activities, which are cut when budgets are slashed. If we properly fund those programs, we don’t even need the vending machines. But that is another article.

   We cannot expect our schools to solve all of the ills of society. How about we start taking responsibility for ourselves
  and our families?

  

   Education begins at home. Teaching kids how to eat properly should begin long before the school years begin. Kids who eat healthy at home tend to eat healthy elsewhere.

   Face it. Junk food is not going away. As a matter of fact, I like my chips on occasion. Trying to eliminate it from school or anywhere else for that matter is an abject waste of time and money.

   Let’s concentrate on teaching youngsters to make good decisions when it comes to food. That teaching begins at home.

   Are you with me, Mom and Dad?

   • John Metallo lives in Slingerlands. He is the retired principal of Albany High School and a former adjunct instructor at the University at Albany and SUNY Plattsburgh.

Is caffeine OK?

To continue my notes from yesterday, I will have to acknowledge that now we have Vijay Singh's bout with IGF-1 to contend with.

I still have trouble pretending that this is an issue. Every professional, you would think, does all they can to improve his/her performance in their own field. Naturally there are limitations, normally involving actions that actually bring harm to others.  In the case of PEDs, the only harm, if any, is to the user himself.  Taking PED's does not guarantee anything for the user other than quicker healing and perhaps greater strength, but do not negatively affect other people.

An argument could be made that the users of PEDs are not the strongest or fastest or best. If they were, they would not need these supplements. So maybe the use of PEDs is really an attempt to level the playing field.

Whatever the case may be, let's stop pretending that we need purity in athletics.  If we extended our PED logic to the rest of our lives, we would be banning caffeine, sleeping pills, vitamin waters, even quality food, for those things merely allow us to do what we otherwise would not be able to do.

By the way: it has nothing to do with PEDs, but Pete Rose does belong in the hall of fame. He was the best, and nobody ever tried to throw a game that they had bet they would win.  Did his bets perhaps make him try harder than the others? Perhaps. Does that mean its bad to exert greater effort?

Enhancing Performance?

Front page of sports section of Albany Times Union of January 30, 2012 had two headline articles. The headlines were "AHL players OK with PED testing" and

"A-Rod denies new PED claims".  Beneath one of them was a pointer to an inside article: "Ray Lewis denies PED use".

 

How nuts are we? Viagra and Cialis ads all over the place, Beyonce faking it at the inauguration, and we cry "let's be pure" to the sports world?

The CBS station in Miami says today "As for the New York Yankees, there were reports late Tuesday the team was looking into whether it could void A-Rod’s contract with the team"(see here.) Perhaps that is pointing the finger where it should point: massive payments to top athletes. Would baseball players go after these drugs to improve performance at a $50,000 a year job? I'll bet Rodriguez's contract had enough incentives in it to push anyone to do whatever they could to improve performance. I don't know if he did or didn't. That doesn't matter to me. What does matter is how the American public is quietly gulled into paying massive salaries to entertainers (of all types).

That stuff falling from the sky...

The image below was snipped from the National Weather Service web page this evening.
Read carefully!
If they do not know, who does?

The red item here appeared as a letter to the editor in the Albany Times Union today.

Math of gun law flawed hypothesis

If a gun control law saves even a single life, it’s worth it. As a mathematician, I call this logical fallacy reasoning by incomplete hypothesis.

My favorite example of this form of unreason is the following.

Adolf Hitler’s persecution of the Jews drove many of them out of Europe who otherwise would have remained in Germany to be killed in Allied bombing raids. We can therefore conclude that Hitler’s persecution saved Jewish lives and was therefore worth it. Note that I’ve left the Holocaust out of the hypothesis.

ROYCE SOANES Waterford

Regarding  this letter, I, also as a mathematician, pose two questions.

First, if there was a fallacy by incomplete hypothesis, then what was missing? I'll accept the Hitler item as a definition by example, but nowhere in the letter is any missing information regarding gun laws.

Second, compare the opening conditional statement in the letter to its converse: "If it's worth it, then a gun control law saves at least a single life". This does force us to acknowedge that to the person making the statement, being "worth it" is not the same as "saving a single life." Should that be the case, what is it that makes a gun control law "worth it"?

What's your anti today?

Over the past few days it has begun to hit me that our news media has a "thing" about what I will call the anti-approach to news. The papers, web, and TV have all hit me with anti-tax, anti-spend, anti-gun, anti-abortion, anti-fracking, anti-gay marriage, anti-smoking, anti-big business, anti-immigration, anti-Muslim, anti-this, anti-that, and, yes, even anti-Bush.

 

Does the American public have a rather negative view of things these days? Could we as a society perhaps begin to view our roles in life as promoting what we would like to see rather than merely continually chastising that with which we disapprove? This really hit me when I read items about people who are against flu shots. I mean, anti-flu shot? I guess in the ideal world everyone who wanted a flu shot would get one. (What really flabbergasts me is those who have said they would get one if insurance paid for it, otherwise they would not.)

 

The antis are really taking over, so it seems. The anti- that really bothers me is the anti-intellectualism that is taking over our society. Paul Stoller wrote a good article entitled "My Struggles With Anti-Intellectualism" that appeared in The Huffington Post on May 8, 2012 (click here). In it, he writes "There is a deep tradition of anti-intellectualism in American cultural and political life. It has a long history, spreading its messages into every nook and cranny of American social and political life. We are the "can-do" nation that values "common-sense" solutions to our problems. We are suspicious of "egg heads," dreamers and "pointy-headed" intellectuals ..."

 

Written by Susan Jacoby, "The Dumbing of America" appeared in The Washington Post on February 17, 2008. In it, she states "Americans are in serious intellectual trouble -- in danger of losing our hard-won cultural capital to a virulent mixture of anti-intellectualism, anti-rationalism and low expectations. "   This article is a must-read, and it actually reminded me of a book by Richard Hofstadter called Anti-intellectualism in American Life. I have not read it, but I will.

Counting my own years in school, I spent 50 years in classrooms. I will be one of the first to point out that there has been a creeping ever-stronger anti-intellectual tide washing over our schools. I attribute part of it to a law of unintended consequences: if you want more horses to be able to jump the fence, lower the fence. (It's easier than raising then ground, and doesn't require stronger horses.)

In a culture of anti-intellectualism, it is much easier to adopt a position against something that is, than in favor of something that isn't.  Anti-intellectualism is most likely the source of the "anti"-culture.

Robert F. Kennedy is quoted as saying "Few are willing to brave the disapproval of their peers, the censure of their colleagues, the wrath of their society. Moral courage is a rarer commodity than bravery in battle or great intelligence. Yet it is the one essential, vital quality for those who seek to change a world that yields most painfully to change."  He was onto something, but what he could have added is that moral courage together with great intelligence is stronger than any weapon of battle. Gandhi knew. 

Does this work?

Hyper Smash

This video was created in Geometer's Sketchpad, and shows the trace of the midpoint of the segment connecting two points moving in opposite directions on concentric circles.

Kudos to The Bay State, I Hope..

Massachusetts is going to consider, as part of a bill, requiring gun owners to have liability insurance on their guns. I mentioned this in this blog back on December 24, and now I can mention it again with an least a smidgen of hope.

I do find it unfortunate that Massachusetts will not vote on this as a stand-alone bill. The website masslive.com has an AP story this morning that says "The initiative is included in a gun control measure which would also change standards for gun licenses and outlaw large capacity magazines."  Unfortunately this will prevent the issue from being judged on its own merits, (Which seems to be getting rarer and rarer in our political world of "buy-my-vote")

A different AP story in the Albany Times Union states "Craig Baenziger, who works at a gun- and ammunition-seller in North Attleboro, Mass., called Northeast Trading Co., said requiring liability insurance for guns makes little sense because it targets people who buy the weapons legally instead of going after criminals who illegally possess them."  This is at least one more instance of gun advocates hiding behind criminals. I was brought up to believe that the criminal element does try to hide itself among the law-abiding people, but using the criminal element as a shield and an excuse for inaction is in itself just plain despicable. I suppose Mr. Baenziger is also against speed traps because they mostly just force good drivers to slow down and they miss most of the speeders anyway.

Remember: automobile insurance pays for the costs of damage and injury caused by owners and operators of vehicles. Gun insurance could and should do the same. What's fair is fair.

Wish I'd written that...

This letter to the editor appeared in the Albany Times Union of 1/14/2013. I'd wish I'd written it...

Are guns a lure or a deterrent?  

Gun owners are complaining that publication of their home addresses puts them in danger from criminals in search of guns. But gun groups have been saying for years that a home known to have a gun is safest, because criminals will steer clear of it. Come on, folks, you can’t have it both ways.

ALAN LEVENSOHN Delmar

The 2nd Amendment? The 29th Amendment?

Those who hide behind the Second Amendment ought to recognize that earlier in the US Constitution sat Section 2 of Article 4, which was subsequently changed via the 13th Amendment.

 

To use as a rationale for an interpretation of the Constitution the intent of the Founding Fathers must acknowledge the society within which they lived. Semiautomatic weapons with large ammo clips did not exist. Slavery did. Not only did they not use any powers of prophesy as regards weapons, they accepted the ownership of one human being by another.

 

In many respects they were quite good, but the Founding fathers were not omniscient, and they recognized that there words might need to be corrected. They built in the process for amending their work.

 

Perhaps the time has come to correct their words regarding the "right to bear arms."

 

Anyone ready to help the campaign? Ready for the 29th Amendment?

Kids with guns??!!???!!

Daniel Akst, in the Albany Times Union of January 4, 2013, has an op-ed column entitled "Our kids deserve a fair shot" (see here) in which he claims to see the need for requiring school students to carry guns and learn all about them.  I would like to think this article was meant for a satire magazine, but there is nothing in the article nor near it to indicate that it is anything but serious.

He writes " Just as the answer to offensive speech is more speech, the answer to a firearm is another firearm — or a couple dozen, which is about how many should be in the average classroom."
Can you imagine? He claims that this would also stop bullying, because "even the smallest, most socially awkward child can put a bullet between the eyes of his or her tormentor if properly armed."

Is this guy nuts or what?  Shootings of 20 or more in a school over a year would be the norm unless the entire school's resources went into everything from flak jackets to bullet proof vests to armored lockers (after all, a student who left his gun at home should not find it too easy to steal someone else's).

Akst's article was in the LA Times on Dec.28, one week ago, yet I do not recall hearing or reading anything about it in the intervening week. That surprises me, as news of a truly wacky nature generally travels pretty fast. Maybe the New Year's festivities got in the way..

I did a bit more exploring regarding Mr. Akst, and I discovered that back on July 29, 2012, in Newsday (a Long Island publication) he wrote " Children get hold of a weapon. It's awful." Alas! A bit of evidence for the satire theory!! You can find that article here. In the article he was commenting that guns are a leading tool of suicides and that eliminating guns will help alleviate the suicide rate.

A bit more exploring uncovered Mr. Akst's authorship of a book entitled Temptation: Finding Self-Control in an Age of Excess. I have not read this book, but in a bit of googling I did uncover the comment "Suicide prevention is probably one of the best arguments for gun control - and a good example of how people sometimes need to be protected from themselves."

By then I was almost convinced that the article I read this morning was indeed a farce. I am convinced that the author intended it as such, but I am not convinced that all readers will take it as such. There are people who will use this article as evidence of another NRA supporter and feel good as a result. I have heard people say, in all seriousness, that if people in schools were armed there would not be as many school shootings. I find it scary, but I find it more scary for a newspaper to feed them "evidence".

A major problem in communication is when creators of a message presume to know how recipients of the message will receive it.  After 30+ years in a classroom, I know that what is intended to be heard and what is actually understood by the listener can be worlds apart. That, right there, is the number one reason why constant student-to-teacher feedback is needed and why lecture classes succeed only sporadically.

I would respect the Albany Times Union much more had they had a side-bar comment giving some information about Mr. Akst and alluding to the tongue-in-cheek nature of his column. Immediately below Mr. Akst's column is another column entitled "Hydrofracking panel will put children first". Is that one for real or just for "fun"?