As a private math tutor, I am called upon for a number of reasons.  Some parents contact me out of concern that their child is falling behind.  Others reach out to me in the hopes that I can help get their kid to the top of their class or achieve the scores they need to get accepted to the prestigious school of their choice.  Lately, however, I have been approached to simply help students with their homework.  It is not necessarily that the student is behind or looking to expand their knowledge and jump ahead.  Instead, the parents see the methods being used and come to me in frustration to answer the questions that their children have.

While it is true that this is good for business, and it is nice to make a living doing what I love helping people better understand math…I would feel too guilty charging for something that I can remedy in less than an hour.  That is why I am hoping to simplify some of the “crazy stuff” that kids are bringing home these days so that parents aren’t left scratching their heads wondering what is is that a teacher is looking for.

First of all, let’s start with the basics.  If you have a kid who’s in Kindergarten through Second Grade, they are most likely working on adding and subtracting numbers to begin with.  Now many reading this might scoff and say, “Addition? Subtraction?  Why can’t they figure this stuff out?”  Well, let me show you a few things.

Let’s look at an example of a fairly basic problem:

I would hope that this looks simple to the majority of us.  Now consider this, how do you know that 4 added to 8 results in 12.  In the beginning we were given pictures; one group of 8 somethings then a group of 4 somethings and told to count them all up.  Of course, we would count 12 somethings in total.  The problem with this is that counting is slow and we would like to eventually move into memorizing this as a basic addition fact.  How then can we help a young student to internalize this idea that 8 plus 4 equals 12?

This is where some of the Common Core strategies come into play.  Now, let me just preface this with a disclaimer.  I like the methods for helping students to learn concepts that are having trouble with more “old-school” techniques, but these are not useful for everyone.  I believe I made my stance clear on the matter in my other post here (Common Core Rant).  We good?  Ok, back to delving into some of these things.

If a child is having trouble quickly remembering that 8 and 4 make 12, then we have to come up with another way to come up with the answer that is more efficient than drawing objects and counting.  The first step would be to recognize which number is larger.  In this case, 8 is bigger than 4, so we would begin with 8 and count up by 4: 9, 10, 11, 12.  So 12 is our answer.  This is still slow, but it is slightly quicker than counting up to 12 with two groups.  The reason this step is important is for problems like this:

Though this is the same problem, initially, students don’t realize this.  Showing young children that a group of 4 and a group of 8 is the same thing as a group of 8 and a group of 4 is another building block that helps with fact families and more difficult problems later on.  For a student to realize that he or she can start with the larger number will also save time as counting up 8 from 4 will take longer than counting up 4 from 8.  This idea of addition being Commutative is a concept that we get to show students even though they may already have memorized 8+4=12, which is one reason that a teacher may spend extra time on some of these facts.

After establishing that we can count on from the bigger number, we can extend this further.  Knowing something called Ten Facts, we can help students to break problems down into easier pieces.  For instance, we know:

Since we are adding 4, we need to add two more to get to our answer 12.  Ok, there is a lot going on here.  Our number system is based on tens, so being able to recognize how to get to ten, and later to multiples of ten, can help improve students’ efficiency later on in life.  So recognizing all the tens facts,

will help with knowing how to make it to ten for the first step in addition problems involving answers over ten.  Again, notice that both 8+2 and 2+8 are listed as facts.  This is a chance to reinforce the idea that addition can be performed in any order so that with extra practice, students are more likely to internalize the concept.  The second important skill necessary here is to know that 2+2=4.  With the tens facts, a child can see how to get from the larger number up to ten, but that won’t do any good if he or she doesn’t know how much left to add.

Being able to decompose numbers into parts or groups is not only a way for children to perform the rest of this problem and others like it, it also helps to lead into subtraction later on.  Knowing doubles like 2+2=4, or being able to see facts like 1+3, 2+2, and 0+4 all equal 4 will help a student to figure out how much more must be added after figuring out the distance from 8 to ten.  All these methods are explained in school, and the student decides which is easiest or which he or she remembers the quickest to know that 2 more must still be added.

This brings us to the final step of adding 2 to 10.  Being able to add single-digit numbers to ten is useful because it is one of the more quickly learned skills (which builds confidence in students who are struggling) and can carry forward to adding to multiples of ten like 20, 30, 40, and so on.  One of my favorite ways to show students how to add numbers is to say them close together.  For example with 3 and ten, if you say them quickly, you get “threeten,” which sounds a lot like thirteen.  The same with “fourten” and others.  With 11 and 12, the sound doesn’t play out so well, but after practicing the others, the student usually pics up the pattern (and finding a pattern is the best we can hope for our students as a teacher).  Whatever the path, the student will find a way to finish up to see that 10 +2 =12.  So let’s review:

hmmm… I know:

and