Every year, across all grade levels, I hear (and observe) subtraction being a difficult concept for students. Not just a difficult calculation, but concept. I am not talking about reading a context and knowing if subtraction could be a way to solve it, but instead, what is happening when you subtract and how does a change in the subtrahend and minuend impact the difference? I think students can learn a procedure to “operate” with subtraction (as with any operation), but I always question the conceptual understanding behind their work. I also think that we, as teachers, sometimes make some assumptions about student understanding of subtraction when all of their answers are coming out correctly. It feels really nice to see students read a task and solve it correctly with subtraction, but have they thought about whether the answer makes sense or could they explain what would happen to the answer if I increased or decreased one of the numbers in the problem? This could completely be my own wondering because, I admit, I tend to question a lot of my students’ understandings until I hear them talking about the idea or working through it in their journals. To get a better understanding of their thinking and attempt to help them move forward in their thinking, I do Number Talks a lot and most recently have really started to listen and think more about what makes subtraction so difficult for them.
I have a few ideas based on my observations of the students’ conversations and many lie in the fact that we do much relational thinking about addition and subtraction that students assume that the numbers operate in the same manner.
1- Commutativity. When adding, it is so convenient that you could add the tens and ones in either order and still end with the same answer. For example, when adding 34 + 63 I could add (30+60)+(3+4) and still result in the same answer. Even if it changed the context of the problem, it would still result in the correct answer. Whereas, with subtraction if I was subtracting 63-34, I can’t just do (60-30) + (4-3). It now creates a different problem but it is something that students do ALL of the time in order to take a smaller number from a larger one. Which is what I see happening here with the quick subtraction problem I gave students to solve last week before we started looking deeper into decimal subtraction.I just wanted to get a look at what they were thinking, as was not surprised to see this on many papers.
2 – Number Adjustments and the Effect on the Context. This comes out A LOT in our talks. When they are adding, they love to compensate and adjust the addends to make an easier problem. For example, 49 + 33, students would take one from the 33 to give to the 49 to make an easier problem of 50 + 32. Again, it would change the context of the problem they were solving, however not impact the result. Now given 49 – 33, giving 1 to the 49 from the 33 leaves you with 50 – 32 and completely changes the context. Given a removal problem, you are starting with more, but taking away less. Or given a distance problem, you have moved the starting and ending point in opposite directions. There is SO much context in a subtraction problem in just the number adjustments themselves.
3 – Number Adjustments and the Effect on the Outcome. When adding, students understand how adjusting one of the addends affects the solution. If I add one more to this addend it increases the sum by one or if I decrease both addends by 1, the sum will decrease by 2. Again, the context can come into play here, but the students get pretty comfortable with the numbers, stripped of context, in understanding this. Now, subtraction is not so nice in that way. Again, context is SO important. 34 – 12 = 22. If I take one from the 34, making the problem 33 – 12 = 21, it works in the way the students know addition works. However, taking 1 from the 12, making the problem 34 – 11 = 23, it does not. They are so perplexed when they try this and it instead adds to the original difference.
Now, because students do not feel as comfortable with subtraction, I also see less willingness to reach outside of the standard algorithm once they “get it to work”. I appreciate the use of the algorithm, however after this quick formative, I had the feeling that there was some conceptual understanding missing that would really impact our decimal work. Because of this, I decided to start with an Investigations story problem on our grid paper.
“Mercedes had 1.86 grams of gold. She used 0.73 gram of it in a piece of jewelry. How much gold does she have left?
I asked them what this story would look like on grids and I got quite a variety of thoughts but I was very surprised to see students putting all three numbers (the two in the problem and the difference) on three separate grids.
I did have a quick realization of the difference between “Show this problem on the grids” and “Show how this story looks on the grids?”
These showed the STORY….
This student taped the removed part over top of what she had, to leave the answer in purple:
This student set the whole aside because she knew she didn’t need to touch it and dealt with the hundredths.
These involved some taking away of pieces to leave them with the answer.
This student changed the whole to be the tenth, but represented each number in the equation.
To see if they made a connection between what they had done on their grids to the solution process, I asked them to solve it in their journal the way they would have just given the problem (again, most with the algorithm) and then tell if it was similar to what they did on their grids. Many struggled to see any similarities which surprised me, especially with the way some took away the tenths and hundredths on the grids.
This was so interesting to me especially when I saw so many correct answers in their journals but when asked to explain, it was tough! Subtraction is tough…for students and adults. Not the calculation so much, but the concept of what is happening. It is so conceptual and really hard to break away from methods we know that work for us to truly understand the meaning behind them! I know I still have to think harder about subtraction then I do addition, so I want to make it clearer for my students.
So much to think about and I am sure I have so much to learn about subtraction and connecting representations to their thinking, but this is a stepping stone along the way!