Discovery based math

I just picked up a used Miquon Math program for my 2 year old.  Of course, we won't be studying math any time soon, but I saw an opportunity and nabbed it.  I'm new to the idea of "discovery math", but am intrigued with the idea. 

We stumbled upon Discovery based math today with my 10 year old.  It wasn't going to be a lesson driven by student discovery, but that's what it became.  My 10 year old will be taking the 5th Gr IOWA test soon, and like many homeschool moms, I'm using this time to CRAM, lol.  It's not the best method, but it is what it is.  There are many math topics we simply didn't get to this year, and I'm sure many of these things will be the on the test.  I'm hoping to give my 10 year old a heads up on these topics, so she isn't completely lost in the wilderness.  So today, I brought up long division with two-digit going into a three-digit number. Eek!

We pulled out the whiteboard and I put a number on the board.  She immediately said "don't tell me! I want to try doing it myself!"  Her first attempt looked more like a multiplication number, and I simply asked "do you want some direction?" Her answer? No!  So I held my tongue.

When she asked if she was going in the right direction, I explained that she was turning it into a multiplication problem.  I re-phrased it.  The problem was 352 / 44, but I said, "think of it as 44 x what = 352?"  I probably gave a little too much input after that, saying "let's just guess what it would be, but to get a good guess, let's round 44 down to 40..." At this point she stopped me, "Let me try, I've got it".  She went through a new series of processes, but this time thought of 40 as 4, and then tried to multiply by 10s.  This was sort of the right direction, so after she arrived at 88 x 4, I pointed out the error.  So, back to the drawing board, and she was at it again...She went through a series of 40 x a number to get close, and then readjusted by multiplying again with 44.  Of course, this whole series of math experiments took far longer to finally arrive at the correct answer of 8, LOL.  We did one more problem after that, and again, I gave her the marker and let her run with it.  This time, it was still trial and error, but she was much closer to finding the correct answer.  The second problem had a remainder.  I gently assisted when it was welcomed, and watched when she wanted to work it out for herself. 

OK, so this style of math instruction takes a lot longer, brings in a lot more mistakes, and wrong turns.  The teacher could easily show the procedure in 5mins time, which is much quicker and more efficient for both teacher and student. BUT...what is our ultimate goal?  We want to provide the tools for our students to solve problems.  Teaching the standard long division algorithm without exploration is a way to accomplish that.  However, we also want to develop other skills:
Critical Thinking and Problem Solving
Thinking Outside of the Box
Agility and Flexibility
An ability to pull from a variety of different skills to solve NEW problems

As technology advances at a faster and faster rate, our students will need to adapt to these changes in their work environment.  They will see things that they have never seen before.  We hope to graduate students that can say "I don't know the answer, but let's find out!" or "it's never been done before, but I bet I can do it!"

Yes, giving them the marker and saying "have at it" is messy math at its best.  It takes time and a whole lot of patience.  It requires the teacher to bite their tongue and sit on their hands.  The student gets a lot less math accomplished in the same amount of time that they could complete an entire worksheet.  But perhaps the end result - Innovative and Creative Problem Solving - is really what we're trying to teach in the first place.