A mathematician once said that math was "a seamless whole" inside her head.

I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these discrete skills, math is more like a seamless black hole.

This became apparent to me again while teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.

The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!

So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.

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You explained that very well. I tried to explain this to a guy in charge of a math tutoring program for inner city children, why it would be better in the long run to spend a bit of time at the beginning of each tutoring session going over basic skills and concepts, then help students with their current homework.

However, I didn't explain it as well, and he didn't understand. My energy was low at the time because our second child was just a few months old, so I didn't try very hard. But, I'm not sure he was the type to understand even if I had kept trying to persuade him.

It was really crazy to see people trying to do Algebra and Trig with students who had to add up anything greater than 1 + 2 on their calculators.

They learned how to get through 1 night's homework, but were not gaining much true knowledge because of a lack of foundational skills.

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