To help students fluently subtract across zeros, I wanted to find out if there were misconceptions I had seen in the past. We have been practicing speaking place value language regularly in our classroom for a couple of weeks now and I was hoping to have it transfer into this sticky part of subtraction.
I usually hear students explain using language like this: cross out, turn into, and make it a...
My goal in the end is to hear: regroup ten tens...etc. because that shows place value understanding when they are working an algorithm, truly mastering the standard at the level it is intended.
I began by writing a subtraction problem and asked a volunteer to subtract and do a "think aloud" explaining what was being done.
I chose someone I thought would know fairly well how to do it correctly.Student explanation of subtraction across zeros.jpg I expected there would be some small errors or at least the language I expected; cross out, this changes into that, turns into a..
I got a surprise! It was that language, plus no understanding of what to do with those tens!
Why do zeros freak kids out? Why do students who seem to have mastered a standard algorithm struggle with regrouping each place value when there are mostly zeros? I want my students to fluently subtract with zeros using place value understanding as demanded by 4.NBT.B.4.
After my student finished explaining his solution, I complimented him on how well he did to get through such a large subtraction problem and deal with all those zeros. I told him that we would make a math mountain, add the two addends and see if it added up to the total ( or what we started with). (We have not been using minuend or subtrahend words...but Start, Change and Result to clarify rolls of the numbers in an equation.)
Everyone tried it in their notebooks and discovered that the result was way off. They seemed amazed and could see the difference in the place value of their answer.
"Hmm," I said. 'There must be something wrong with that regrouping."
He offered up a few explanations to what he thought he did wrong. Several students chimed in trying to figure out. One student knew exactly what happened up until the thousands place.
We all agreed that not understanding what we regrouped and "gave" to the next place value was why it was wrong.
I asked: How about if we used something that would help us sort out the place values first and help us think about how the zeros will work within the problem?
I turned the page on the Smart Board Lesson.Subtraction Using a Place Value Chart.pdf
We then reworked the problem step by step using the place value chart that organized the zeros for us very well. As we worked the algorithm, I stated exactly where each new value for the zero was coming from by talking in place value language.
I said: It sounds like this: Since I can't subtract 8 ones from zero ones, I need to look at the tens place.
I then continued until I get to the 400,000 value where I can find a group of hundred thousands to regroup,modeling each regrouping in my language and with my pen.
I kept talking and repeating the process in that manner.
I asked: What did you notice?
" I noticed that you started subtracting after you were done borrowing."
" I noticed you never said " cross out." ( That made me happy.)
" I noticed all the numbers are really neat on top."
I asked them to subtract with me and join me in my talk. I then went back and had students repeat my language for each step. I think this modeling helps it settle in their brains and hopefully it will recall the sounds, giving way to logical thinking through the process.
I asked: What do you usually say when you subtract with zeros?
Two students raised their hands, one being the student that volunteered. I chose the other boy this time.
I usually say, " go over to the number, cross off the four, it turns into a three and make a ten, then..." And he continued using my algorithm in the place value chart.
Common Mistake: I told my students that the most common mistake was that students just start crossing off numbers and changing them into other numbers without thinking what the values should be. (I demonstrated it.)
I also told them not to put the ones used in borrowing tens between the digits because it gets in the way of showing clear understanding of values. It gets messy. I showed them. ( In the past, teachers have used "short cuts" with subtraction by placing the tens regrouped as part of the number being subtracted. This practice in both addition and subtraction does not help them demonstrate their place value understanding. So, I avoid it and correct it when I see it, making them write clearly and place new values neatly about the number.
We practiced another problem to do together and we talked through it again.
It got smoother with practice.
After the fourth time, I felt that students could reteach each other. Student explanation
In this movie, we see my student trying to independently explain the process and why we are regrouping to subtract accurately. This peer teaching really does two things: it helps the student hear themselves understand and also helps them pick out what might be going wrong. The second thing it does is helps others see that everyone can struggle, and practice to become better.
I noticed that often, this simple practice is what gets my students over a hump.
*He stumbled a few times, but was getting it as I coached. It's really hard to master, but I think if I continually demand that place value language is spoken, the mastery of place value understanding is really understood! Before CCSS, this in depth understanding was missing in our pedagogy.
*If notebooks are difficult for students to use, I have included a place value chart to copy.
After several rounds of practice on the Smart board, we returned to the original problem and talked through it together again.
In reading, I use "thinkmarks" from Fountas and Pinnell's Guiding Readers and Writers. Their idea about "thinkmarks"is a simple book mark type piece of paper that has places for notes on thinking about reading. It could be: I predict, I inferred, I wonder about....etc. Why wouldn't this idea support thinking in mathematics and therefore Common Core?
I modified this idea for math. My Math Think Mark.doc. This little tool is a way of organizing small bits of critical thinking. Students could choose four of the six questions on the Smart Board to answer on their Math Think Marks. I am hoping that these will be good tools to help students study for their test.
On the last slide...
Regrouping with a purpose!
Think Point: Answer these questions on your think mark.
How does place value language help you think about what you are doing?
How does place value language help you think about why you are doing it?
Did subtracting on the place value chart help you and why?
Why is saying "we cross off and this becomes a _________ or saying it "turns into a" not work with the thinking about place value?
Have we mastered the goal of fluently subtracting across zeros yet?
Rather than practicing the skill today, which will turn up in estimation anyway, I was more interested in their thinking and recording their thoughts about place value talk. It's no easy task because the conceptualization of place value was not important before Common Core. They seemed to like the "Thinkmark for Math" and understood why it worked just as well for math as it did for reading.