I would say you're on the right track, but I tend to think one step further - we need to emphasize skills AND application. It's all fine to see some numbers and symbols written out and apply a well-practiced skill to them, but to look at an abstract problem and know how to set up the symbols, is another thing altogether.

Mastering the skills of driving, after all, is pretty meaningless if you never sit behind the wheel of a car.

Preach it, brother! I'd give anything to have my students recognize simple stuff, like seeing the similarities between solving x^2+3x+2=0 and (x-4)^4+3(x-4)^2+2=0. They can solve the first but flip out on the second. If only they could see that the second is just a more complex version of the first...

You're not at all wrong! I absolutely agree that kids need to understand how to recognize patterns and how to explain something new (that does not, as yet, have an answer in the back of the book) based on the patterns they've recognized and observed and recorded. Applying math skills, knowledge, and concepts to the patterns and coming up with new information is important for cutting-edge scientists , artists, builders, travelers, people stranded on deserted islands, and just the plain old curious.

I remember, as a kid, watching my Dad use math all the time. He was a general contractor, and throughout his entire day, he would use math to solve problems or to answer his own questions about the house, the yard, his current building project, anything. I was in elementary school, with endless math homework, and I was always a bit awed by my Dad's real-world use of math, and how he could just scratch some drawings and figures down on a scrap of paper, do some figuring and manipulating, and come up with a solution. He hadn't been assigned the problem, and there was never an answer printed anywhere for him to check.

I would be very happy if my kids could graduate knowing enough about pattern-recognition and how to apply math skills and concepts to be able to think outside the textbook when called upon to do so. Take care,

Alexa Harrington
EducatedNation.com

Or is there too much emphasis on content and skills and not enough on conceptual understanding?

Is problem solving a skill that can be taught? If a student doesn't have a conceptual understanding of the math - be it calculus, algebra, or arithmetic - how can she hope to approach a truly novel or unfamiliar problem? Sure, past experience will guide her some of the way but to truly proceed with confidence she will need a solid conceptual understanding of the mathematics she hopes to use.

Pattern recognition will help a student discover a rule. But don't we also want him to understand why that rule works? Does understanding why fall into content or skill or something else?

Sadly, where I teach, we use a curriculum that focuses not at all on basic facts and standard algorithms. Students spend all of their time on projects and group discovery - none on mastery of facts they are learning about. The result is that students around here are incapable of simple multiplication without a calculator...forget about division. Many students haven't even learned how to do addition and subtraction with regrouping. They estimate answers quite well, but if a math problem wants an accurate answer, they have no clue how to find the answer. Or they make basic errors in calculations.

I think the problem is that our teachers always go overboard in one direction or the other - when clearly you need both a mastery of math facts and algorithms as well as the conceptual understanding necessary to apply that knowledge, but you CAN NOT do one without the other.