These days, half of Willow's math for the school week involves problems and projects that utilize math concepts--building mathematical models, completing word problems, playing a game, etc. On another day, she plays on a Grade 3 math app on our ipad--some review, some new work, but all quick-format, question-and-answer style. That leaves one day a week for pencil-and-paper calculations. It's a slower math education than I'd originally desired--we haven't moved into fractions yet, for instance, or much multiplication and division--but it's certainly a lot broader, is a better overall way to internalize math concepts and strategies, and I'm coming to see that Willow has a lot more patience for our pencil and paper calculations when they only take place once a week.
This week, for instance, I'm pretty sure that we got through a week's worth of subtraction understanding in this one session together. Willow came to the table already understanding multi-digit subtraction without borrowing, and having worked through subtraction with borrowing with me before, without really getting the concept down. She also has her money concepts down, so this time, we used dollar bills (hundreds), dimes (tens), and pennies (ones) to model breaking down a number for borrowing, and then Will worked some problems on the dry-erase board:
When she had the method down fairly well, I told Willow that she was going to make a video tutorial to teach others how to subtract with borrowing.
At this, the entire tenor of the lesson changed.
I did, amazingly, have a willing and focused, though fairly unenthusiastic, kiddo. I now, incredibly, had a willing, focused, highly enthusiastic kiddo who suddenly adopted a calmly pedantic tone (tell me this is NOT how I sound when I teach!) and happily filmed two takes of her video tutorial:
I had assumed (correctly, for a change) that Willow would enjoy filming a tutorial. I already knew that being required to teach the method would help her further understand the method, herself, and that's why I chose the activity.
What I did not anticipate is how valuable this video would be for me. This video shows exactly what Willow understands about subtraction with borrowing at this exact moment; how wonderful that I can look at it at my leisure, away from the lesson, and evaluate it!
Here's what I see:
- Willow understands the procedure (she has an excellent memory for things like this), but she doesn't understand the activity beneath the procedure to my satisfaction--if she did, then she wouldn't have modeled taking a "one" away from the two in the tens place; she would have explained that you take away a ten. In another take of this tutorial, Willow jokingly read the number that is created from the subtrahend after you notate your changes--211113! This shows, I think, that she doesn't yet understand that subtraction with borrowing requires "playing" with Base Ten, so that your work isn't in Base Ten format, although your answer is.
- Although I'm glad that Willow has a handy strategy to call upon for computation, it's clear that she STILL doesn't have her subtraction math facts memorized! To me, this just makes learning a higher level of math more difficult--not only do you have to struggle to understand the concept of subtraction with borrowing, and memorize the procedure, but you also have to painstakingly calculate all the numbers, too?!?
So thanks to this one four-minute video, I know exactly where Willow is in math, and I know exactly how to proceed:
- Willow needs more practice using manipulatives to break down the subtrahend. Next week, I bet that she'll have a ball making a video tutorial for that!
- Willow needs more practice with worksheets of subtraction problems--if she can't erase the digits to replace them, she may better understand the purpose of the notations during the procedure.
- Willow needs to learn the subtraction facts! I'll be replacing one of her math project days with a day of entertaining math drills--cute Halloweeny puzzles, matching games, etc. This will actually be great, since it will be one more activity that she and Sydney can do together.
We've delved far from the third-grade Singapore math workbook that I had thought would be our scaffold for math (I think multi-digit subtraction without borrowing was where we got off the bus), but after Will had mastered these subtraction concepts and can work them mentally, we'll check back in and see what Singapore wants us to do next!