I have been teaching for about 29 years. Most of those years in elementary. Math has always been my most favorite class to teach, but as most teachers did in the past, I used the I-do-you-do approach. I did my best at explaining how to solve problems using manipulatives and pictures, and quickly moved into abstract strategies using tricks, and helping my students memorize steps. I remember how subtracting with re-grouping was always a “hot topic” in my classes.
I thought I was doing justice to my job by taking extra time to review the steps to re-group when solving subtraction. I was patient and used small group instruction to review the steps. I offered extra sessions during recess to review steps for subtracting using the algorithm efficiently. I thought this was best practice. This was back in the early 2000’s.
It wasn’t until several years later, when I started using a student-centered-approach to teaching and learning math. That was one of the best years in my long journey as a teacher. That was the year when my students dicovered on their own, what re-grouping really means, and why we use it. That was the first year, that re-grouping made sense!
It is ideal to teach addition and subtraction strategies at the same time. Math problems can be solved by adding or subtracting. Use concrete manipulatives for students to see what re-grouping looks like.
When students can use concrete manipulatives, like these base ten blocks, they can make sense of what “take-away” looks like. For instance, when taking away 128 from 274, students will start by taking away the hundreds, to then take away the tens.
Then they have 154 left. When they want to take away 8 from the ones, they will realize they do no thave enough tens, so they organically will “exchange” a ten for 10 ones, and take the 8 ones from there. At this point, there are no rules or tricks. Students know that they have enough to take away 8, they can SEE that they do not have enough ones but they have enough tens.
It is important to allow students to work in pairs, so they discuss this, so if one of them is not thinking of a strategy, the other student will probably come up with it, and all of a sudden, no one knows who came up with it, it was their work combined!
After that students just see what they have left. They have 1 hundred, 4 tens, and 6 ones.
It is important to allow students to use manipulatives for a few days, so they really learn to trust the process of using these base ten blocks to find the answer. A lot of students do not trust this process and quickly use an algorithm to find the solution, an dthen just represent the answer with the base ten blocks. That is fake modeling. That is not using manipulatives to build understanding.
After students get use to using manipulatives, and understanding how to make sense of place value, they can explore other strategies like “subtracting by parts” where they decompose the number, and subtract part by part.
When students develop strong number sense, and use numbers flexibly, they could come up with other strategies like “subtracting by place.” It is ideal to allow this strategy to organically emerge from one of the students, instead of teaching it. When they use their own words to explain a new strategy, it suddenly makes sense.
I have come to realize that when your main focus on the lesson the problem, with engaging context, students really care about the context, and feel free to explore different strategies. They are not focused on using one strategy efficiently, they care about the math that is involved to figure out that good problem.
Today, I do not have to teach the algorithm strategy, nor do I teach any tricks and steps to solve a subtraction problem, because when students build understanding of using place value strategies with base ten blocks, and using decomposing strategies, they make sense of how the algorithm works. They figure out that algorithm tricks like “borrow from the “neighbor”, is nothing but realizing that if you don’t have enough ones, well, then you take it from the tens!
Making Sense of Mathematics 