The Nature of Inductive Reasoning

What if you saw 2, 4, ___ and were asked to fill in the blank? What would you think goes next? To me, there are two different answers that both would seem straightforward, and right. Counting by 2’s gives us a 6 next. Doubling gives us an 8 next. (If it were doubling, it could have started at 1, though, so I’ll vote for counting by 2’s.)

In the first 2 lessons, Jacobs gave us puzzles where we were trying to see the pattern. But sometimes the pattern we see isn’t the ‘right’ one, so we have to ask how we know a pattern is right. Inductive reasoning doesn’t always lead us to the ‘right’ pattern. This section is meant to help us think about that issue.

In science, once a pattern is found, we still want to think about why that pattern would be there. In the 1st edition, Jacobs gives a pattern related to the planets, called Bodes’ Numbers. In the teacher’s guide to the 3rd edition, he mentions that it’s still not known whether there’s a reason for this pattern or whether it’s just a coincidence. (Wikipedia says it’s mostly the latter.)

I hope you can spend some time with the circle problem in set III. (Draw the circles big enough to draw lots of lines in them. Be prepared to do it over if it gets too messy.) I would recommend dong the circle with 7 points, too. I remember doing this one when I was young, and being fascinated. I’ll say more later… (In the 3rd edition, this circle problem has been moved to the 4th lesson. I’m assuming it’s in either lesson 3 or 4 in the 2nd edition.)

I’m curious whether this lesson will be harder or easier than the first two.

- Notes by Sue VanHattum