Darren-

You are 100% right on here! I tell students that our brains are meaning-making, pattern seeking machines. Patterns fuel our sense-making capabilities. We literally and physically make the brain work harder when we present "content" is isolated, de-contexualized ways.

When we break the brains rules, we and make learning much harder than it needs to be. We need to help our students become masterful pattern detectors, creators, and connectors. It just makes sense??

Right on! I tell my students that math is just "applying what you know to a new situation." Angela mentioned the pattern-making you referenced in your post. I think this inductive reasoning should be at the core of our instruction. For example, rather than telling students that two lines are parallel if their slopes are the same, why not show them a series of lines that are parallel and allow them to figure out this pattern for themselves? Some authors call this a "discover a relationship" style of inquiry based lesson. Now if I only had time to develop better inquiry lessons for all 180 school days and the 100 concepts/skills I'm supposed to teach this year... :)

No, you're right. And what you say here is basaically the same as what I say in my most recent (and rather long) post.

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've got it right! Pattern recognition is important, but it's also important to recognize familiar concepts out of pattern. Application is important, much more important than today's educational system portrays it to be. I know kids hate it, but word problems, and the type, are so important! I suggest teaching students about the mathematicians who developed the concept, who they were, what circumstances led them to the development of that skill. This might just inspire student to engineer their own problem solving skills.

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

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I'll join the chorus. You are totally right, Darren. I have occasionally taught math at the high school level. The curricula at this level confuse math with algebra (or occasionally trig and geometry thrown in for good measure). There are so many other branches to mathematics and it is a shame that our students will never know of those areas. The underlying theme to all of these is, as you said, pattern recognition. That seems a much more exciting area of study than algebra (not to dis albegra - I enjoyed it very much as a student, but I always was one of those math nerd types).

I vigorously nod my head with Matt as he says "Now if I only had time to develop better inquiry lessons for all 180 school days and the 100 concepts/skills I'm supposed to teach this year... :)". Somehow the education system has mistaken breadth for intellectual rigour, not just in math but in all curricula. But that's a larger issue worthy of its own blog post.

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.