Working our through the unit on decimals in fourth grade, Jeff came up with a good idea to help think through place value while practicing multi-digit addition and subtraction skills.
Thursday:
 |
Pick six cards. Arrange them in any way you'd like to make a three-digit number. Use the ones, tenths, and hundredths places. Try to see how close you can get to a sum of 10.
|
 |
Grid paper helps organize the computation, even if the computation is not always accurate. We hit upon the idea of having the kids place each of their attempts on a number line, because we like number lines. And also because of 4.NF.C.6!
|
And here is what we learned on Thursday:
- The kids were pretty good at adding multi-digit numbers.
- As a whole they weren't really persistent in trying out different arrangements. Many were just "one and done". They had to be encouraged to make multiple attempts.
Friday:
 |
| Choose 6 cards. Arrange them to get the smallest difference possible. No negatives! |
 |
| Nice work! |
 |
Seen this before?
|
 |
| How about this? |
 |
| Really? |
 |
| This is getting bad. |
 |
| One more time... |
Here's what we learned on Friday:
- We've got some subtraction issues!!
This became the lunch discussion topic. Why, over more than two years after first being introduced to subtracting multi-digit numbers with regrouping, were so many still having so much difficulty? Was it underdeveloped number sense? Inability to decompose numbers? Were there still kids not developmentally ready? Or maybe something else is happening. Are kids overgeneralizing the commutative property? Think about how they are taught their "basic facts". I know in our curriculum there is a heavy reliance on "turn-around" facts" (2+3 = 5, 3+2 = 5). Would it then make sense to a kid that 3-2 = 1 and 2-3=1?
Of course when we get to subtraction in vertical columns, teachers often tell the kids, "When the number on the top is smaller than the number on the bottom we have to regroup." Why? "Because we can't take a bigger number away from a smaller number." Which is not true, as Jeff points out, and may make life more difficult for them when they encounter negative numbers.
Regarding this issue, I have never been able to shake something I saw many, many years ago. Our district was between math curricula. We had dropped something like Houghton-Mifflin or Harcourt-Brace (can't remember which) and had not yet adopted Everyday Math. I was among a group of teachers who piloted a program called
CSMP. I was a beginning teacher and did not know much about math education, but there was something about it that was intriguing, and I saved some of the lessons and continued to use them. One lesson described a way to subtract multi-digit numbers that I had never seen:
 |
The relevant piece from a 4 page lesson.
|
Has anyone tried anything like this? Of course for this method to work you'd need to recognize that this is a problem in need of regrouping in the first place, otherwise you'd still get an answer of 456. Any ideas would be helpful, because just today, this happened:
 |
| Not a good sign when the difference is bigger than the minuend. |
I've enjoyed getting to know your work. Thanks for posting these photos!
ReplyDeleteAnd, yes, they do look familiar. I'm teaching fourth grade after 10 years at lower grades than that. I see a lot of those kind of mistakes. Add in another one: the re-grouping that didn't need to happen, and I wonder if part of the issue is the vertical format coupled with a new routine learned mechanically makes the problem not really subtraction at all, but a...let's see...what do I do...maybe this? kind of problem.
I've wondered if all this contributes to a misunderstanding about which number is being reduced. If so, then a possible solution might be to help see the structure of vertical subtraction better?
Thanks for getting me thinking. This is a soooo common!
Thanks Steve. I had never thought of looking at the structure (but I'm going to now). Have you played around with it at all? I'm going to start. Maybe it's that they moved out of the concrete (base 10 blocks) too fast?
ReplyDeleteI wondered about moving the concrete out, too, before a clear mental image was established. I know that there is a lot of pressure to get on with things...which seems really counter-productive. :(
ReplyDeleteHere's what I've tried, but don't really have any great data about it, though.
I've tried to reinforce the "structure" of a vertical subtraction problem by 1) noting that it looks different and we have to figure out that that means!; 2) Creating a "story" that the kids can use to help them remember that there is some sort of "reduction" of one number by another. (I know that sometimes it's not reduction but comparison...yet...hmmm...maybe I'm contributing to another problem down the road!)
The story goes something like this (and it involves donuts.)
The minuend has packages of donuts, singles in the 1s place, 10 packs in the 10s place, 100 packs in the 100s place. Say, for the probem 92 - 57, the student starts by holding out his hand pretending to have two donuts. Then he can imagine another person saying: "Hey, can you share 7 donuts with a friend? I'm hungry!" (I am that friend at first, but transfer both parts of the story to the student as soon as possible.)
The minuend says, "Sure! No problem!" ('cause that's what you always say when a friend is in need!) Meanwhile, the minuend can "freak out" because he doesn't have enough for my request.
So he turns to his neighbor, the 10s, and says: "Can I you share some donuts for a friend?" The answer is, you guessed it, "Sure! No problem! But all I have is a package of 10! You can break them apart and keep the rest."
As the donuts arc through the air (I initially have the students draw a little arc from the 10s to the 1s to trace the donut trajectory) the 10s lowers by one pack, the 1s breaks the pack apart and gives out seven imaginary donuts. Same happens with the 10s, etc.
The nice thing is that the conversation drives the "story":
1. Hey can you help a friend?
2. Sure, no problem.
3. Freak out or not?
4. Toss a pack over if needed.
5. Adjust to show how things changed after the toss.
6. Reduce minuend by the appropriate amount.
I think that stories are a nice way to make the abstract more real, and I'm not saying this is rocket science (or good dietary practice!), but the story does seem to help students understand that the minuend is the one with the donuts to give out.
Sorry to be so long winded, but this really is a vexing problem.
Steve,
DeleteI agree that stories are a good way to make the abstract real. I like the donuts, and I also like how you have your kids draw an arc to represent the trajectory. Do they tell themselves this story as a kind of self-talk when they do multi-digit subtraction? That would be really cool.
Your response has me thinking about a few things:
1. Are we also bumping up against something to do with conservation of number? We know that 9 packs of ten and 2 single donuts is the same as 8 packs of ten and 12 singles, but do they really understand that nothing has changed about the number of donuts we have?
2. I think you're right to reflect about how making it all about reduction and not comparison may lead to problems. How about the fact that, in your example, 2 donuts - 7 donuts (could) = -5 donuts. Does that lead to integer problems later?
3. I think the most important point you raise is the rush to have children master algorithms before they've done the necessary conceptual groundwork. It happens in multi-digit subtraction, in multi-digit multiplication, and also in long division, which we're working through right now in grade 4 (look for a post about this soon). I hate the arbitrary "By the end of grade ____, students will be able to..." as if all students in the grade level are at exactly the same place along the cognitive and developmental continuum. So yes, in our panic to have them meet these standards we push them all along, ready or not. What are the consequences of that? I have to wonder whether or not the folks who wrote the standards thought about that.
Thanks for such a thoughtful response. Yes! The story becomes a way to encode a self-talk script. The story become a scaffold for the operation.
DeleteEven though the story is not a comparison story but a story of reduction, I think it actually helps students see the differences (and the similarities) between the two kind of subtraction. A visual comparison can help.
And yes, the -5 issue you raise could be a problem, although...maybe not so much if we also introduce the idea you "you owe me--debt", which they usually understand pretty well?
For #3, I couldn't agree more. How much fun would it be to introduce subtraction as simple columns (as one kid I taught did all on his own). He just added or subtracted the difference within columns depending on which number he was starting with. A complete invention.
225
- 132
--------
100
- 10
--------
90
+ 3
------
93
We're working through something similar right now with long division. We have an "equal sharing" scaffold that does not make sense in an equal grouping context. We figured once the scaffold was removed and they could just do the algorithm correctly they could then apply it regardless of the context.
ReplyDeleteYour last example plays to the obvious desire for kids to find the difference between 2 and 3 ('m looking in the tens place now) as long as they understand that that as a -1.
I am very interested to read the post on division. Thank so much for the conversation. I've enjoyed it and learned a lot.
Delete