Yesterday I worked with Robert on changing customary units of length:inches, feet, yards, and miles. It was not Robert’s first encounter with requests to switch from inches to feet or yards and vice versa. He also heard before that a mile had 1760 yards or 5280 feet. He heard but vaguely remembered. Not once I prepared for Robert worksheets of the form:

1 foot = …………..inches

1 yard= ……………feet

4 feet =……………inches

……………feet= 36 inches

4yards=………….feet=……………………….inches

………..yards= 15 feet

Those exercises went rather smoothly, although sometimes Robert needed a short reminder of what to do before he started answering questions.

Yesterday, however, I used not my worksheets but pages from 4th grade Spectrum Math workbook. Those pages presented similar problems but with a huge range of numbers.

For instance:

27feet=……………………..inches

………..feet= 180 inches

132 yards= ……………….feet

132 feet= …………………yards

Robert was lost. As before he had to either multiply or divide, except previous operations he did easily in his head, almost automatically. Now, he had to do a chosen operation on paper. Somehow **choosing the operation** became much more confusing.

I realized that without being specifically taught Robert solve “easy” problem by comparing two sets of equations:

1 foot=12 inches

X feet= 48 inches

He made an easy proportion.

When the number became large, they, somehow complicated everything:

1foot = 12 inches

X feet= 192 inches

Now, Robert needed a method, an algorithm that would withstand menacing character of big numbers. The simple method that would replace confusion about which operation to choose with clear step by step process.

Yesterday, I did not think about such method. I was observing as Robert was half guessing, half understanding the choice between multiplication and division. I was trying to understand why Robert seemed to almost “intuitively” solve problems with simple numbers and could not do the same with larger ones.

I think that Robert knows those smaller numbers much better. When he sees 60 he also sees 12 times 5 behind. On the other hand, in the number 192, he doesn’t see 12 times 16.

Two things struck me yesterday:

What looks like almost intuitive ability to solve easy problem might be the results of very well-practiced/mastered skills.

Ability to solve easy problems “without” consciously applying any algorithm might not help Robert in understanding the mechanism behind finding solution.

As I stated before, last evening, I did not use any new strategies to extend Robert’s skills to larger numbers. Today, I will attempt to practice solving proportions, and later introduce them as a way to help with changing units. I worry, however, that doing so, would make the whole process more abstract and artificial. In the end, it might lead Robert to loosing his present understanding of units of lengths and the way they change into each other.