Yesterday Michael Pershan tweeted the following:

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```The word "discovery" can mean literally anything. // via @misterpatterson pic.twitter.com/uIGLJYFR0J

— Michael Pershan (@mpershan) May 5, 2015

He was pretty critical of this worksheet, which is definitely not as strong as Kate Nowak’s trigonometry investigation (or my own if saying that doesn’t sound too conceited). But what interested me was that he was also critical of the idea that anything that is worksheet-driven or teacher-led could be classified as “discovery learning.” He seems to suggest that it’s only discovery if the student comes up with it completely on their own.

I’ve already looked at discovery learning on this blog once before, and I think I can say that I’m really not for this narrow idea of discovery learning and think that it can actually be detrimental to a student’s mathematical development. I am, though, very much a fan of **investigation**.

Which brings me to the real reason for this post:

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```#MTBoS – I'm Looking for ideas for introducing rational exponents to Int. Algebra 2 students. Anyone blogging on that? #mathchat

— Jim Pardun (@JimPa23) May 4, 2015

First, this is an investigation I did into Zero & Negative Exponents with MYP3 (US 8th grade age). The students did the first part (worksheet included on third page) under test conditions in class. Upon receiving their graded work back they wrote the essay (fourth page) where they not only used their understanding of a pattern to write a rule but verified that their rule was correct by using the rules of algebra.

A year later, in MYP4 (US 9th grade), I do the following investigation on Rational Exponents with the students, though it is more informal. They work in groups and are not graded on their work for this.

*Scribd is doing some weird things with my document…there are no red question marks in the original. So here’s a picture of what it really looks like for me:*

These investigations are absolutely not “discovery learning,” because there are specific steps (suggested by me) that the students follow to ultimately “discover” the rule. But the steps ensure that the students have a focused approach, and it models for them what I think is an essential chain of mathematical inquiry:

Finally, not an investigation per se on exponents, but more of a hat-tip on the many wonderful ideas I steal from this community, here’s something we spent a class discussing and delving into, which I culled from various blogs and twitter feeds:

*Again with the red question marks, bah! (anyone know why it’s doing that?) Here:*

This is all about notation. In fact maths is in large part a slave to its notation.

2 to the power 3 only has meaning because we give it meaning, namely 2 times 2 times 2.

So likewise 2 to the power of -3, but the meaning to be ascribed is not quite so simple.

What is more easily accepted is 1 divided by 2 to the power of 3 ( 1/2^3 ). ( I can’t do latex or whatever).

Then we can see that it works to write 1/2^3 as 2^-3

One of the problems with exponent notation results from the original definition

2^n = 2x2x2x2x….. n times

It is more satisfactory to see it as ((1×2)x2)x2… ie start with 1 and keep multiplying by 2 the specified number of times. Then 2^0 is 1 times no 2’s at all, which is 1

Regarding 8 to the power box times 8 to the power box, 8^(1/6) times 8^(1/2) = 8^(2/3)

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