Once upon a time, people got tired of writing things like 5+3+3+3+3+3+3+3+3+3+3 and went with 5 + 3(10) instead. And so there was multiplication, as a form of repeated addition.
Later on, people found themselves writing 5*3*3*3*3*3*3*3*3*3*3 so some other short form was deemed necessary. This gave rise to 5*3^10, and the notion of repeated multiplication.
But is the exponent seen as such? Particularly when, in this post, we haven't determined how to make it look like a superscript? Though maybe that's a benefit, it's more clearly not a single multiplication when we use an actual symbol...
Michael Pershan recently ran an experiment looking into Why Kids Mess Up Exponents. The issue seemingly being an intuition that leads towards the wrong places. He had a set of questions, and you should check out that link to see the breakdown.
I mention this as I recently ended up running my own experiment based off of his, though rather more spontaneously.
Quick backstory: My Grade 11U level class started exponents earlier in the week. They would have seen negative exponents in Grade 10, though I started with contextual stuff anyway, which refreshed the notion of negative giving divisions. So too late to gain anything there.
Friday was going to be the day for rational exponents, which they had not seen in a classroom context. So we had a bunch of snow (yes, on April 12), nine people were absent, I still wanted to at least talk about the topic, and had a brainwave. Thus, after going through some review questions based on the previous day's work, I went through a single question three times, collecting in the answers as I did. Here's the results.
|DEPENDING ON YOUR|
DEFINITION OF 'FINE'
What do you think 100^0.5 is? How confident are you? (5 being most confident)
ANSWER: 50 
1- 4 ppl; 2- 0 ppl; 3- 6 ppl; 4- 1 ppl; 5- 0 ppl
ANSWER: 10 
1- 0 ppl; 2- 0 ppl; 3- 3 ppl; 4- 1 ppl; 5- 2 ppl
ANSWER: 0.5 
Confidence level of 1
I then gave some context based on what we'd seen previously, by writing the following table on the board:
|INVERT THE COLOURS, IT WAS|
ON A BLACKBOARD...
NOW: What do you think 100^0.5 is? How confident are you?
ANSWER: 10 
1- 2 ppl; 2- 1 ppl; 3- 5 ppl; 4- 3 ppl; 5- 4 ppl
ANSWER: 50 
Confidence level 3
(Two people didn't give me papers. I wasn't keeping track that closely.)
I indicated that whatever the answer is, it has to be multiplied by itself to get the base back. Demonstrated 100^0.5 * 100^0.5 = 100, along with a couple similar examples, and then outright confirmed the growing suspicion that the prior answer was 10. Then...
What do you think 100^(1/4) is? How confident are you?
ANSWER: root(10) 
1- 0 ppl; 2- 1 ppl; 3- 2 ppl; 4- 0 ppl; 5- 2 ppl
ANSWER: 5 
1- 1 ppl; 2- 2 ppl; 3- 1 ppl; 4- 0 ppl; 5- 0 ppl
ANSWER: 2.5 
1- 1 ppl; 2- 0 ppl; 3- 1 ppl; 4- 0 ppl; 5- 0 ppl
ANSWER: 0.1 
Confidence level 2 (for both)
ANSWER: pi 
Confidence level 2
(This time four papers missing. Also, when I gave the pi guy a curious look after glancing at his sheet, he said 'pi always figures in somehow'. Amusingly, the answer IS 3.16227766...)
There's something more going on than merely treating it like multiplication. Even in set 2, a couple of the 10's were based on students having written down a moving of the decimal to the left. (I don't know if it was the same ones who then came up with 0.1)
My suspicion is that there's a search for patterns going on. Because that's what human beings do, we look for patterns. And after two years of linear patterns, that's what students gravitate to, in absence of any other context. So, provide them context? Even then, we may grab moving decimals before anything else. (If it had been 50^0.5, would the response have been 5? Possibly.)
Trouble is, for this problem, a root is not immediately seen as part of a pattern. A root is irrational. Which is kind of ironic given that it's a rational exponent. Or maybe that too is part of the problem.
For further reading on exponents that came up in blogs this week:
1) Rational Exponents - Third Grade Style
2) Exponentials in Context
Was this exponential week? No one told me!! Hmph, so much for happily ever after.