I had a message from a parent of a student in a New England school district who recent asked about the posts on Facebook about the Common Core Math. I can certainly understand why she and other people are confused. If I were looking at the problems shown and did not have a mathematics background, I too might find the examples frustrating. Along with the example problem there is a demonstration of the more traditional algorithm that most people know and understand. But, let’s pause for a moment and think about a few things.
1. The standard algorithm is understood by adults who were taught math in that format. Makes perfect sense to me. However, most of these same adults cannot understand he example of the teaching algorithm.
2. The teaching algorithm appears to be more complex and takes (possibly) more steps or time to complete. Just from the cursory view, I would again say that makes sense to me. Yet, someone who understands it and can teach it probably has a deeper understanding of mathematics.
Why would anyone want to teach something that breaks something down into additional steps and takes more time to do? Well, from my perspective, that is the role of a teacher. To take the complex and break it down. Remember, if you know how to do the problem already you don’t need to learn how to do it again. But if you have not learned how, then it is the role of the teacher to break it down and have the student learn step-by-step.
For example teaching subtraction. Many of us were taught that subtraction was “taking away”, which in some cases is correct. But many people do not realize that subtraction is looking for the difference between two numbers. Look at the picture in the picture there are two different problems. The first, at the top, is X+8=13 which is the same problem as 13-8=X. The difference is the operation that is being used to represent the equation. In the top example it is the operation of addition and the second example is subtraction. In both of these examples we are looking for the same information, but how we go about gathering that information may be done differently.
In the top example we would read the problem as “What when added to 8 is the same as (or equal to) 13?”
In the second example we would read that problem as “13 minus 8 is the same as (or equal to) what?” In this format traditionally students who have been taught Place Value (which I strongly recommend) would look at this problem and typically say, “I can’t subtract 8 from 3, so I need to regroup (borrow) the one from the tens place.” thus making the new problem 13-8… just what they started with!
So, what if we helped the students actually understand subtraction as an operation in a way that truly explains what is happening in a subtraction problem? Allow me to show you.
Using the manipulatives you see in the picture, you can see that the left column is a blue 5-bar and a brown 8-bar, totaling 13, which represents the top problem.
The right column is made up of a pink 3-bar and a blue 10-bar, also totaling 13. Both columns together help to explain the second problem. What we are trying to do in a subtraction problem is to find the DIFFERENCE between the two numbers. This is why the answer in a subtraction problem is called the difference.
So, looking at the 8-bar compared to the 10-bar. What is the difference in these two bars? The difference is 2.
Now, how far is it from the end of the 10-bar to the end of the 3-bar? The difference is 3.
If we were to add 2 (the first difference) with the 3 (the second difference) then we have a total of 5. We did this without any regrouping (in the traditional sense).
Now let’s take that to another whole level of problem.145 – 98
Just looking at that problem and thinking about using the traditional algorithm it is clear that the student will need to regroup (borrow) multiple times in order to complete the problem. This creates several opportunities for error.
But what if we looked at the problem from the perspective of the difference? Ask yourself these questions.
What number is rounded to a place value (tens, hundreds, thousands) that is in between 145 and 98? 100!
Okay so using 100, ask “How far is it from 98 to 100?” 2
“How far is it from 100 to 145?” 45
“What is 2 + 45?” 47 done.
And yes this works on all sorts of problems and is incredibly helpful in long division!
The Math-U-See curriculum is designed to help teachers to learn how to break down complex algorithms into concrete, visual representations. If you have questions please call us at 800-454-6284!