Hangin' It All Together

In a previous post, I referenced Dr. Wickelgren's comparison of the traditional approach and "new math's discovery" approach to teaching math concepts. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he made the following observation about the traditional approach:

"The traditional approach, in which classes are arranged by topics such as arithmetic, fractions, algebra, and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

I've been blessed to teach a very structured, incremental math program. Students who are used to struggling in math tell me over and over that they "get math" for the first time. Now, I see why. Closely related pieces of knowledge "hang together" easily and make sense to them. The slow, incremental steps mean that they practice and master the small pieces, one at a time. Then those pieces are carefully linked to other small pieces and the students see the logical relationship and how they are connected.

Contrast that to a sixth grade book I was using recently to tutor a student. All of the definitions and all of the formulas were thrown together in quick succession, so that my student was thoroughly overwhelmed and confused. (The students are given only that chapter to memorize them all, while trying to learn how to use them, all at the same time. ) It's madness!

And sadly there's not usually much practice of each small piece of knowledge, isolated from all of the other pieces. (Remember lack of practice is a characteristic of "new math.") Here's the line up: Teach one formula, and maybe do a little work. Then the next day, along comes the next formula, and then the next day, another formula. The students are soon doing problems of each formula, before they have mastered and are comfortable with the first one. This is NOT how kids learn.

There was nothing to "hang" the second , or the third, on. It was as if there were pieces floating around in the air and the student saw them all and had no idea which of them was related to what he was being expected to to next. There were no hooks to hang anything on because there had been no time for mastery.

Teachers probably feel that they don't have time for mastery because of the number of chapters they know they must cover. This book was huge!

And if smart kids have trouble with it all hitting them at once, imagine the weaker student's response. They are overwhelmed.

This is not how kids learn!!

Students must see how the pieces fit and hang together. And traditional math helps them do that.

Traditional math is wonderful! It's beautiful!! It's joyful! It's exciting!! It's liberating!! It's confidence-building! It's knowing I can succeed!!

It's like a kid who can finally ride a bicycle by himself!!

Traditional math helps kids do what they all want to do -- learn! I've never seen a kid who didn't want to learn. Learning new things is fun. Learning how to do something hard is even better!

Perils of Discovery Learning, Part III: 'Interdisciplinary' Activities

One of the big "buzz words" in mathematic circles today is "interdisciplinary" activities and projects.

If I'm teaching a unit in Science, students might do research in my class or the Library, write papers using skills taught in Writing class, and then generate the paper in the Computer Lab. Or if students are learning about Indian Villages in History, they might work in groups, or individually, to make a village in Art Class.

I often feel that teachers are evaluated (unofficially perhaps) on how much content crosses over into other disciplines, although no requirements have ever been made of me in my private school. It has been "suggested" that I find ways to involve other disciplines, but that's been the extent of it.

I have seen instances when I feel an interdisciplinary project has been very effective and where students are completely immersed. If the other discipline is a favorite of a student, if he loves art, or if she loves to write, he or she will really be engaged. I just don't like the pressure of forcing the project where it doesn't naturally go, where time is lost, all for the sake of "show". We can now brag at how many other disciplines were involved!!!

Dr. Wayne Wickelgren has made studies of interdisciplinary projects and I respect his opinion. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he contrasts the traditional approach with the interdisciplinary approach:

"The traditional approach, in which classes are arranged by topics, such as arithmetic, fractions, algebra and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

[That is just beautiful! And it makes such good sense.]

Hangs together in logical ways.

Now for Dr. Wickelgren's assessment of the interdisciplinary approach:

"Teaching across subject boundaries lacks depth. It may be fun for the students, but it doesn't help the mind organize the knowledge in a logical way, making it harder to remember."

There is also the likelihood that a teacher will overlook an important basic fact or principle that would usually be included in an incremental, structured approach.

I just have to say this one more time:
Hangs together in logical ways.