The power of estimation in mathematics is well documented. Estimation gives students the outer limits, or the inner limits, in order to frame unfamiliar quantities. Publishers of math curricula know this and will almost always begin a topic of instruction (such as adding fractions or dividing whole numbers) with estimation.
I find, though, that many students struggle with estimation. For example, some become confused when rounding a five-digit number to the tens place because they don’t remember place values. Some don’t remember the rules for whether to round up or down. Others are simply turned off by a perceived lack of relevance—what’s the point of estimating numbers?
The best way I’ve found to both formatively assess students’ abilities to estimate and help them understand how to do a better job of estimating is to start with them. By talking through their solutions, sharing their personal anecdotes, and creating their own strategies, my students generate ways to understand estimating numbers. I share a few examples here.
During number talks, students explain the rationale behind their problem solving. During a class with my fifth graders, I asked them to estimate the answer to 56,901 - 43,098. Most could round the first number to 50,000 or 60,000 and the second to 40,000 or 50,000 and then estimate 10,000 to 20,000 for an answer.
This makes them happy and surprisingly satisfied, and why shouldn’t it? Why does a child need to know how to round these numbers in a pattern yet? Isn’t it more important that they arrive at something reasonable first? As past classes had talks like this one, I began to notice something.
Kids with better number sense would sometimes round to the second digit instead of the first. For example, these more proficient students might round to 57,000 and 43,000 and come up with an estimate of 14,000. That satisfied them as they had a stronger number sense to understand it.
Oftentimes students approach solving problems through ways that make sense to them but that are not necessarily theoretical or practical. They see it as a way to minimize the time and effort involved. And while they are learning, sometimes a very rough estimate is just fine.
Getting on the Board
During one class, a student shared that he thought of rounding as being like a bull’s-eye. He played darts and the idea sprang to life as he spoke. He said the answer was in the middle of the board, and he was just trying to get on the board.
After a fun talk, our classroom decided there were times when just getting on the board was probably OK. When students feel comfortable and confident, as was the case here when my student explained his rationale in rounding or estimating numbers, they feel empowered. Now in my classroom, we value getting on the board.
But Sometimes Precision Is Important
I value my students’ problem-solving rationale. But they still need to know the “real world” relevance of precision. I need to support them in becoming more skilled in reasoning and problem-solving—it’s my job to help them learn how to aim for the bull’s-eye. To do this, I have a perfect scenario that a student once gave me.
He told me, “Sometimes Mom lets me buy things at the grocery store. She gives me money and tells me I have five minutes to buy as much as I can with it. Whatever I don’t spend goes back to her.”
“But how is that like rounding?” I asked.
“I don’t have paper or pencil, so I have to round anything I want to buy, then add it up to make sure that it’s less than five dollars. But if I round too fast, then I probably won’t get to use as much of the five bucks as I can.”
I present that to students with a scenario like this: “Your mom gives you five dollars. She tells you that you can buy whatever you want, just not go over five dollars. There is no tax. You’ve only got a few minutes because Mom is in a hurry, and you are not allowed to write anything down. Estimate what items you can purchase (you have no pencil).”
- Gummies: $1.63
- Soda: $1.52
- Toy: $1.72
Some students conclude that they can buy only two items, while another suggests that they can purchase all three. This usually prompts a classroom discussion around the idea of rounding to different place values. Kids now discover that there are merits to precision—it makes all the difference between buying just two items or all three.
Extending Student Learning
Often students will round numbers depending on the context. But to assess how they generalize estimation, try these suggestions. Give them problems to solve without stories behind them, or ask them to create their own problems—you can even challenge them to create their own contexts.
These suggestions could lead to great discussions illustrating how students rationalize their answers, get on the board, or bring in real-life examples to solve math problems. Once they feel validated and empowered in their ability to estimate, they have in their hands powerful tools that can put them on the board to solve a problem that a test, or life, gives them.