Mathematics education has come a long way over the past 10 years. The calculations, of course, are the same (after all, 33 + 20 still equals 53), but teaching mathematics has certainly evolved.
"Back in the day" knowing that the answer was 53 was typically all that teachers expected – it was mostly about getting the correct answer – I honestly cannot remember a single time that a math teacher asked WHY my answer was what it was. Of course, the correct answer is still important, but mathematics education of today requires that learners understand why the answer is 53 – we want our students to deeply understand the patterns and relationships of numbers that allow for greater flexibility and application of the mathematics. In other words, we are teaching our students how to become thinkers of mathematics, not just do-ers of math.
Let's focus on PLACE VALUE. Did you know that "place value" is mentioned 25 times in the Common Core Standards in grades K through 5? Take a look at a brief progression of place value through the grades:
That's good information about the progression of Place Value learning, but WHAT DOES IT LOOK LIKE in the classroom?
Before students even begin adding and subtracting multi-digit numbers in 2nd grade, they should first learn how to be flexible in the way they decompose numbers. Take the number 53, for example. In many 1st grade classrooms, students learn to break 53 into 50 + 3... BUT they also need to understand that 53 can be represented by 53 ones, or 4 ten and 13 ones, or 40 + 13 ➥ this flexible way of thinking about 53 lays the foundation for the work they will do in 2nd grade.
Imagine students are posed with the problem 53 – 27.
What if instead of just recognizing 53 as 5 tens and 3 ones (which poses a problem when they try to imagine taking away 7 ones), they also could easily imagine 53 as 4 tens and 13 ones:
Being flexible with place value gets even more exciting when we subtract larger numbers – this is how those who are mathematically fluent have been solving problems like this for generations (in spite of what they were being taught in school) – it's us educators who are finally catching up to them!
We also saw in the progressions that rounding number relies on an understanding of place value, too. Think about the number 1,742. By understanding place value, we understand that
- 1742 has 1742 ones
- 1742 has 174 tens -- yes, I know there's a 4 as the tens DIGIT, but there are not just 4 tens - there are 174 tens (and 2 ones).
- 1742 can also be thought of as 17 hundreds (and 42 ones) – Let's read that one again so we can process it fully – 1742 has 17 hundreds (not just 7 hundreds).
The ability to think flexibly about place value in this way makes rounding much easier to understand. Imagine we were asked to round 1,742 to the nearest 10. By knowing this number has 174 tens (and a few extras), we understand that this number has a little more than 174 tens (because of the few extras) but less than 175 tens. Of all the numbers in the world, we have narrowed it down to just two specific numbers that this number must round to – either 174 tens (1740) or 175 tens (1750). Then by looking at the "extras", we know that we don't have enough to make it halfway between those two numbers (to 1745), so we must round to the lower value 1740 when rounding 1742 to the nearest 10.
Place value understanding is foundational to building a deep understanding of numbers and supporting fluency. We didn't delve into it this time, but place value should be explored and learned using concrete models. Here are a few images from other educators for inspiration as you continue on the place value journey with your students!
