The proof will be in the pudding.... if the new methods really to help students grasp math concepts at a more fundamental level, it will show up down the road when those students are better all around in math then those who learned in the old-fashioned way.

I like the ideas as they explain them in the article, but sometimes I see a failure to teach the discipline involved in some of what us older folks got with rote memorization.  It's mistake to treat some things that are 'rote memorization', things that require discipline, as wrong as such.  Then it becomes too easy to say that you are freeing up creativity when it is actually just avoiding work that would be very useful later.

Creativity, grasping the fundamental nature of a process or intellectual area, being able to exercise discipline in your thinking, being open to new methods (if they add value), and being open to work - to expending effort - these are all good and all have their place.

Common core is an example of a movement that politically is a progressive trojan horse.  On the inside it is about gaining control over education.  But like all trojan horses it has things that look good on the outside.  And in this case some of those things are of real value - like having solid standards that teachers would have to live up to.

I suspect that quality teachers, no matter what method is used, will make the most difference in whether students ends up hungry to learn, putting out effort and finding it captivating.

I say all of this as one who struggled to get through my calculus courses.  I loved the concepts, but had to memorize the methods in a monkey-see monkey-do kind of way to pass the next test.  Maybe I could have used more of those creative, conceptual models of math principles in the video :-)

Luke, you are an engineer.  What is your take on all of this.

I agree that it seems very good with some potentially very bad.  I think that the overall problem is (1) assuming that everyone learns the same way counterbalanced by (2) the fallacy that every attempt is a good try.  The teacher does say that getting the right answer is the standard for success.  Nonetheless ...

It seems like a shockingly complex word problem for kindergarten: There are four spaceships. Each spaceship will hold two aliens. How many aliens can ride? ...  One student says he grouped them by four and three and one. Another by two and two and two and two. They're all correct, because they all arrive at eight."

 

Well, OK, so some prodigy notes that 8 = 20 mod 12. And 8 is the right answer.  So what?  What does "three and one" have to do with it, except as an interesting though irrelevant fact about four?  It is good that the child knows many ways to make four, but to solve any problem requires relevant processes. They can be highly creative.  We all know the many ways to find the height of a building with a barometer. (Give it to the building engineer in return for the answer...)  But "building starts with B and barometer starts with B" is not going to get you there.

On the other hand, some years ago here Ed Thompson (where is he now?) complained that his neice was learning the "box method" (apparently; he did not use that name) and he called it a weird new age progressive thing; and others piled on to condemn the evil new age progressives.  However, I pointed out that according to my copy of Capitalism and Arithmetic which reprinted The Treviso Arithmetic of the 15th century, that was indeed the traditional way to solve multiplication.  We just got more sophisticated in our understanding, as we did for calculus and algebra. The methods we all learned for multiplication and long division would have been arcane a century before.  The methods we learned are not "natural" in any sense.  

By comparison, how do you find a square root without a calculator?  Do you use the intellectually rigorous method, or do you approximate, mulitply, and adjust?

If you remember when... part of the "new math" of the 1960s was number bases.  We learned 8 and 16 and my class chose 12 as our favorite.  No parent understood why... but octal and hex proved useful later.  As for 12s... Decimal was advocated as a "new math" by Simon Stevin in the early 17th century.  As late as the mid-1800s, German states coins were denominated in halfs, fourths, and thirds, and the thaler was divided into 12ths -- and despite the rare accordance of Jefferson and Hamilton on the matter of a decimal coinage, the USA minted and still makes a quarter (not a fifth or 2/10) dollar.  My point is that it seems to be "human nature" to assume that the way you do things is the natural and superior method.  It ain't necessarily so...

(Edited by Michael E. Marotta on 2/15, 8:48pm)

Question: "Luke, what motivated you to remain childfree?"

Answer: "I have no interest in shoving a helpless child into this kind of world."

Mental

Abuse

To

Humans

(Edited by Luke Setzer on 2/18, 12:33pm)

Butthead: “Arrg. . . I hate numbers.”

Beavis: “Yeah, like there’s just too many of ’em ’n stuff.”

DD observed:

Another issue is that when Common Core was first introduced, not one teacher or administrator at my son's school was able to explain to me that these changes were being made because this method supposedly teaches "number sense." (It was actually a person over on OL who enlightened me.) Why would you adopt a methodology without knowing, or being able to explain, why you are doing so? The answer to that was never "because it teaches number sense." It was never, "because it teaches that numbers are quantities, not just symbols." It was always,"because that's how we meet the standard."

In an emergency, I feel highly tempted to let nature take its course with these fools and let them die.  What makes the lives of these mindless bureaucrats worth saving?  So far, I am not drawing any motivating answers.

In the place of our life-affirming values -- Self-Esteem, Reason, Purpose -- they have placed their opposites -- Selflessness, Mindlessness, Sheepishness -- and subjected helpless school children to them in spades.

Let them die and deserve it when nature strikes.

I have too many bitter memories of caretakers like these to feel otherwise.

Deanna, have you made plans to remove your child from this abusive environment?

(Edited by Luke Setzer on 2/19, 4:55am)

What Mike said except I think he meant "... for the state to drive a wedge between parents and children ..." instead of "... to drive a wedge between the state and parents ...."

I strongly encourage reading Separating School and State for more insight into the problem and Montessori: The Science Behind the Genius for more insight into the solution.

I have noticed a predisposition of so-called "private schools" to "follow the herd" of state schools in teaching methods and apparently Deanna's school is part of that trend.

(Edited by Luke Setzer on 2/19, 6:31am)

Thank you Luke, that's exactly what I meant to say.  I posted too early, before my coffee cooled off enough to drink.

I've never had a child of my own but my first wife had a son, about 5 yo when we got together.  I talked her into and paid for him to go to Montessori school from first through sixth grade.  I had one of the books about Montessori recommended by Ayn Rand, can't remember the name.  My first wife and I divorced in '88, and she put her son in the public school system for the rest of his K-12.  I've remained friends with him.  He's in his late thirties now, married a beautiful girl from Peru about a year ago.  He works in a pawn shop during the day and teaches karate in the evenings.  He's done a bunch of different kinds of work over the years.  Hitch in the Navy, got out, got an EMT certificate.  He's worked in a casino dealing cards, worked for Comcast as an installer, worked in a vet's office, construction, not sure what all else.  He's very personable and easy to get along with, has a nice smile.  Not hugely ambitious but I don't worry about him.  I know, too much information.  Not sure how his early Montessori experience affected his life but I think it was a positive one.

Math should be taught in the pursuit of accomplishing the needs of productive careers.  Practical.  If someone wants to learn history or art that is a different subject.

Hence math would be more integrated with finance, science (for engineering), programming, plumbing, etc anywhere math is actually used for various careers.

How math is taught should not be dictated by government (which only operates by dicating with force).

I'm ok with the grid multiplication.  I am a big fan of solving a problem multiple ways, that is how I always got 100% on my math tests, by solving all the problems two different ways in order to verify my answers.

In Surely You're Joking Mr. Feynman he tells of harrassing mathematicians by bringing them back to the real world.  "Imagine a sphere," the mathematician says.  "OK, like an orange?" Feynman replies. The mathematician agrees. "Now, take an infinitesimal slice..."  But Feynman stops him: You cannot do that with an orange because at some small slice, it ceases to be a piece of an orange and is something else.  

If you know the book, then you  know that that is just one story out of very many.  Feynman admits to being bested by a school of rabbis.  He is staying there to save money while attending a conference in NYC.  They find out that he is a physics professor.  They want him to tell them whether or not electricity is fire.  You see, the Sabbath has rules about fire. The problem is the elevator.  Can they use it?  He and the rabbis go back and forth and they decide to hire a Christian to run the elevator for them.  "Wait!" objects Feynman.  It is not moral to hire someone to do something that is wrong for you, but not wrong for him.  "Why?" they ask.  Feynman admits that he should have known better than to continue the argument, but...  

 

One possible way to group the aliens would be two and two and two and two in which case the answer is 8. Or two and two and two and one because one alien gets motion sick and doesn't want to ride in which case the answer is 7. Or two and two and two because one spaceship is in the shop getting a new warp drive in which case the answer is 6. Or one and one and one and one but there will be two trips because all the aliens want to ride alone in which case the answer is 4 for the first ride and 4 for the second ride. Or what if all the aliens want to stay home today? That's the way my kid thinks, and why this word problem is not ideal for teaching 4x2 if the answer the teacher expects is 8.

 

So, Deanna's son is just creative and insightful.  

The broader problem is that we have three different modes of learning, teaching, and enjoying mathematics.

1. Memorize the tables.  Again, at the start of the capitalist age, no one did that. They had the multiplication tables in front of them 10x10.  But we memorize today and it is handy.  

2.  Story problems.  Again, they go back 500 years... 2500 years... 3000....  because those were the motivations to create, invent, and learn arithmetic and geometry.  

3.  Math for math's sake.  Once you learn how it goes, you can go wherever you want with it.  I have several of the Martin Gardner books here.  Most of it is uninteresting to me, personaly, but I am impressed (even awed) by the range what mathematicians find interesting: folding paper, stacking coins, coloring, cutting, stealing cigarettes, ...  And then, once you learn the language, you think in it - which I do not.  Get a textbook on geometry from a university math library and you will not find a single picture, shape, or figure.  They do it all with symbols as in algebra.

Contra Deanna, it is not the purpose of a story problem to teach that 4x2=8.  If you do not know that, you cannot do the problem.   The purpose of a story problem is discover which of the infinite mathematical facts are relevant.  A train leaves Chicago...  A pail with a hole is filled from the top...

There once was an episode of Death Valley Days where a farmer lost a henhouse that was insured for fire.  The argument was over the distributive law of arithmetic.  The school marm taught him how to understand the problem and prove his case to a judge.  Story problem...

You often can exchange the people on either side of a sales counter.  People who like to shop in hardware stores could work in one. And elementary school teachers never left the 6th grade.  So what do you expect them to know about the epistemology of arithmetic?

You can blame the public system, if you want.  Even teachers in private schools graduated from state universities and hold government approval papers. That they are not prepared for the exceptional intellect is nothing new.  Ever see Mickey Rooney in Young Thomas Edison?  

Ayn Rand recommended The Tyranny of Testing by Banish Hoffmann (Crowell-Collier, 1962).  It is all still relevant today.  

Other than that, we are talking past each other.  Sally can mow the lawn in 20 minutes.  Susie can do it in 15 minutes.  How long will it take if they do it together?  If they get into an argument and stop working, there is no telling how long it will take, I agree.  The right answer is still 8 and 4/7 minutes - and it matters how you get the numbers: show your work.

I have a friend who filled in the bubbles to make patterns or draw pictures on the answer sheets.  After completing a bachelor's in urban planning and working as a drafter for a construction firm, he went to work for FedEx.  Today, he drives a city bus.  He has a heck of a nice house.  He did most of the work himself, moved windows, added doors, changed the landscape, drained the swamp ...  

(Edited by Michael E. Marotta on 3/02, 3:22pm)