Down with Math Rules! part 2

This is part 2 of our “Down with Math Rules!” series. In this post I will focus on the subtraction poem.  I’m sure you’ve heard it. It goes something like this:

More on top?

No need to stop!

More on the floor?

Go next door…

And get 10 more!

Numbers the same?

Zero’s the game!

This poem is supposed to help students remember when to regroup when doing multi-digit subtraction.  I’m not implying that the poem is incorrect or that it won’t assist students, my problem is that most teachers start with the poem and only teach the traditional standard algorithm.

Students should be comfortable with counting to 100, counting using hundreds charts and number lines, basic addition and subtraction facts and be able to compose and decompose numbers through 20, especially the number 10. If students have had regular exposure to rekenreks, number bonds, hundreds charts, and ten frames, they should be comfortable with these skills and be ready to explore multi-digit subtraction. Students who are still struggling with these concepts should receive intervention before moving on.

I like to start my second grade student exploration of multi-digit subtraction with hundreds charts and rekenrek 100’s. Students should have gotten practice counting by 10’s and 1’s on both of these manipulatives while exploring multi-digit addition. After exploring a variety of 2 digit subtraction problems on cards students should have noticed that sometimes they have to go back up to the previous row when counting back the ones and sometimes they don’t.

During the next day of exploration, students use the same cards and look at them again. As they go, have them sort them into two piles: the ones that require you to go up to the previous row when counting back the ones and in the other pile the ones that don’t.  Write some from each category on the board and see if they students can discuss what they problems in each category have in common.  I will be honest with you that this sometimes takes more than 1 lesson for students to figure out what you want them to figure out.  Don’t rush it!! Let them make the discovery, it is much more powerful than if you tell them.

Once they make the generalization, talk about why when the first number’s ones places is bigger than the second number’s ones place you don’t jump up to the previous line. Have students talk about it with a partner and share their thoughts. Then do the same discussion about the other category.  We don’t ever teach the traditional algorithm in second grade, so this generalization is just a concept for them to think about as they subtract. Will they need to jump lines on this problem or not?

I then follow up with practice on subtracting on paper and beaded number lines and with manipulatives, such as, unifix cubes or base-10 blocks. I encourage students to use multiple strategies to double check their answers.  I allow them to use these manipulatives throughout the entire unit on 2 digit addition and subtraction and even on the assessment.  Part of their assessment is a math interview, during which they may use any strategy to solve 2 different subtraction problems (one that would require regrouping and one that wouldn’t) and a discussion on why looking at the ones places is helpful when subtracting.

Students will eventually be exposed to the traditional algorithm is future grades, but their conceptual understanding of subtraction and what regrouping means will be essential for their learning of why we regroup. It is so much more useful than a poem that most  of them will forget over the summer.

Math Fact Fluency: Not Memorization part 1

When I was in elementary school fact memorization was ingrained into our thinking. First it was our addition and subtraction facts; later it was multiplication and division. We were victims of “drill and kill” and teachers did it without remorse. This was what educators thought was best. In fact, I still hear it among many teachers still today.

I do not subscribe to this theory and I know that many of the biggest names in math education don’t either. I believe that as long as a student has a quick, comfortable strategy to get to the answers, memorization is not necessary. If instant answers are what you are looking for (although I don’t think they are needed); I will say that they more they use their strategies to answer the facts the more they will memorize the ones they use the most.
In this post I will just cover the basics of developing fact fluency. True fact fluency is composed of 3 things: Accuracy, Efficiency, and Flexibility. (Explained in the following article: http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm) Students who just have the answers memorized may have accuracy and efficiency, but they don’t have flexibility. Our students who are still counting on their fingers or using pictures or manipulatives only have accuracy. Our students need all 3 in order to have true fact fluency.
Before we can start with facts our students need to have subitizing down. It is important that students recognize numbers 1-10 in different patterns; such as, dice/domino patterns, finger patterns, 10 frame and/or rekenrek patterns. This will help students with conservation of number that will go a long way to helping them with addition and subtraction facts. Let’s be honest, we have all had those students that had to count their fingers every time they put them up. For example, the problem 4+2= may go something like this. They count the 4 on one hand, the 2 on the other hand and then they go back and start at one and count the total. It takes kids much longer than if they could just put up 4 fingers and 2 fingers without counting them and then start with 4 and count 2 more to get the answer.

Let’s start with the addition and subtraction facts. Students start learning facts to 10 in kindergarten and early first grade. Most of the time they are using some sort of manipulative to solve the problems whether it is their fingers or counting cubes, bears, etc… I would encourage you to use 10 frames and rekenreks along with any other manipulatives you are letting the students use. This will help students in the long run by letting them see how numbers can be broken apart and how they relate to the benchmarks of 5 and 10. If you start teaching addition and subtraction facts using 10 frames and rekenreks it will be much easier for students as they move into facts to 20.

Part 2 of this series will focus on specific strategies that will help students become fluent with their addition and subtraction facts to 20. Part 3 will focus on multiplication and division fact.

Counting: Not as simple as it seems

The more and more I work with struggling math students the more it becomes clear to me that part of their number sense issues are that they can’t count. I’m not referring to starting at 1 and counting, but really understanding numbers enough to start at any number and count on or backwards.
I was working with a fourth grade student last year and the following problem came up: 43,046 – 42,958=. She wrote down the problem vertically and started regrouping. Because following steps and procedures is very difficult for her, she made a careless mistake and ended up with the wrong answer.

As a teacher, I’m sure you can quickly figure out what she did wrong. I know that you have seen the same mistake time and time again.
Instead of going back over the procedure with her I asked her if she thought the 2 numbers were close together and she answered yes. Then I offered the alternative of counting up like she was taught to do in first grade. Although she remembered the strategy, she said that they numbers were too big to count. I simply asked her what number would come next after 42,958 and she couldn’t tell me.
I was amazed and started asking some of my other 3rd, 4th, and 5th grade students if they could count on from larger numbers and most of them struggled with it. The few who could do it confidently could only go forward and not backwards. I couldn’t believe it. I went back in the Virginia Math Standards and saw that counting forward and backwards is only mentioned in K, 1, and 2; forward to 100 and backwards from 30.
Because it’s never mentioned in the standards again, we forget that it is an essential skill that should be reviewed and expanded upon all the way through the elementary grades and beyond. As students learn numbers with more places they should be counting those kinds of numbers. They should be able to count forward and backwards from any number that is given to them.
Once activity I started doing with my students is writing random numbers (up to the same number of digits they are required to work with in their grade) on index cards. As they were getting ready for group or packing up I would pick a card and a student. They would have to start counting forward from that number until I said stop. I would then call on another student to start counting backwards from that same number. I would ask another student for 10/100/1000/etc… more or less than the number. It was a simple, quick activity that helped my students develop a better sense of numbers and counting.
I continued working with the same student using open number lines to solve addition and subtraction problems.

As she became a more confident counter she stopped drawing a number line and started counting on using her fingers to keep track of how many jumps she was making. A few months later I brought up the same subtraction problem from the original story and asked her to solve it. This was her answer, “ 42,958. (while keeping track of her jumps on her fingers) 68….78…88….98…43,008…18…28…38….48, nope, too far. 38 is 80 (and she wrote down 80 on her board); (again keeping track on her fingers) 38..39…40…41…42…43…44…45…46… that’s 8 more. 80 plus 8 is 88. The answer to the problem is 88.”

No regrouping, no struggling to remember steps. She could confidently count the numbers and find the answer. This helped all my students with their mental math skills. Keep on counting with your students, don’t assume that they can do it.