However, for many students such as myself, it is important not to lose sight of the big picture. I constantly need to be making connections with what I previously learned and how it ties to the endless possibilities of mathematical language.

The idea behind math as a “top down” thinking process is correct in many ways. But you can’t teach it that way. Start with the basics, let kids learn what mathematicians are saying, how they are saying it, and what it means. Then, once they have grasped what the language means, they can begin to understand the more interesting and complex aspects.

I advocate keeping kids interested not through “Real world!” Applications that are completely irrelevant and recognized as bull by kids AND instructors – but through the many interesting things within the subject. I’ve found that most of my instructors, while computing very well and knowing what they were doing had trouble reaching the majority of the class because they themselves were only teaching as a way to make a living, and weren’t truly interested in the subject. After having explored mathematics myself a bit, I don’t understand how someone can lecture to a class without getting excited about what’s being taught or the possibilities.

That’s just me…someone who used to do poorly in math because of a lack of computational skills. Learning about the bigger web and how all the rules played into the bigger picture, what I call true understanding – was the deciding factor.

Forgive the haphazard organization of my reply, it’s late.

]]>Yup. Fractals, not proofs. Pie-chart understanding, not understanding that comes from application of fundamental identities and theorems.

Geometry is an interesting topic, and one can argue that not all students need the formality of proofs, but that’s not the point in a education world that gushes about the beauty of mathematics and about thinking like a real mathematician. And, one can see the same mismatch of conceptual and formal understanding going on with algebra. Going from algebra I to algebra II requires a lot of mathematical understanding, not conceptual understanding.

The problem is that few educators want to delve into the details of what understanding really means for math, and this goes well beyond whether proofs should be taught or not. Too many educators see math as some sort of top-down thinking process, when in reality, it’s based on a rigorous understanding of a large set of basic definitions, axioms, identities, and proofs. One can play at math with guess and check, but that’s not what a trained mathematician or engineer does.

My favorite examples have to do with basic identities like:

a/1 = a

Do students really understand what this means in all situations? Do they learn about why it’s good to see that:

a = a/1

How about:

a*b = b*a

What are ‘a’ and ‘b’ in

2(x-1)^3

Then there is

x^(-1) = 1/x

Do students know what this means for rational terms made up of many factors in the numerator and denominator? Do students even know what factors or substitution mean for complex expressions?

Some educators love to claim ownership of critical thinking and understanding, but as Barry points out, the reality is quite different. Are those “traditional” kids are just getting by using rote knowledge? They are the ones getting to STEM careers, so dig past pedagogical prejudices to see what’s going on. Don’t rely on rote ed school training.

“From what I’m seeing so far, the implementation of the standards is turning out to be a matter of interpretation, and that interpretation appears to be the same emperor with the same wardrobe.”

The top PLD level 5 (“disguinshed”) defined by PARCC just means being successful in a college algebra course. A PARCC curriculum does not define one that prepares kids for STEM careers. Educators will (continue to) see kids who get there, but they will never ask us parents how that happened. We know more than a little bit about what real mathematical understanding means. Our kids are successes not because we just want what we had when we were growing up. What many schools offer now is worse; talk of critical thinking and understanding that hides low expectations and avoids real mathematics.

]]>Thank you for sharing!!

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