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

Phew, that’s a lot of steps, but look at all the good things we learned to get there.  In the end, memorization is the goal.  However, simply memorizing isn’t always the best.  As I stated in Common Core Rant, students who have a deeper understanding of more basic concepts are better prepared to extend that knowledge when working on more difficult problems later on.  Yes, this takes more work, but it could be the only way that your child is capable of getting to the answer if he or she just isn’t remembering the fact initially.

Ultimately, if you prefer that your child learns in a different way, that’s what tutors such as myself are around for, to supplement the education that they are receiving.  However, my goal here is to show the method to the madness and help explain away some of the confusion that many parents have come to me with.  Hopefully, next time your young student comes home with this “crazy new math,” you’ll have an idea of what is going on and what the teacher is trying to convey through these longer processes.  Again, not every student learns the same way, which is my problem with Common Core, since teachers are required to teach the entire class the same methods even though it may not be the most effective.  The techniques are useful, however, and we should appreciate the thought that has gone into helping all children better understand math.

I will continue on with some of the subtraction techniques in my next post, and from there I will go on to cover multiplication and division.  Thanks for reading, and be sure to like or subscribe as this will most likely become a series I am writing!

Common Core Rant

As a private tutor, I am incessantly met with complaints about the current method of educating students in mathematics.  In many cases, when I meet with a student and parent for the first time, the topic of the dreaded Common Core methods come up.  Time after time I hear the same frustrations.

“Why do they make it so much more difficult than when I was in school?”

“What’s wrong with the old way they taught it?”

“If I can’t understand it, then how can my kid?”

“Why do all these steps when you could just do it like this?”

These concerns are very real, and it is never a bad idea to be involved in your child’s education.  Taking an interest in how your son or daughter learn lets them know that you care, that you’re there to help them should they get discouraged, and helps them realize the importance of practicing their academic skills.  However, I take the unpopular position that the Common Core methods are not inherently bad.

What many people do not realize is that math builds on itself.  It is very difficult to comprehend a topic without the foundation beneath it and fully grasping the necessary prerequisite skills.  What many Common Core strategies address is not only how to arrive at a correct solution, but also how best to build up the techniques that will be used later on in math.  Let’s look at an example.

Say we want to multiply 364 by 12.  One of the Common Core methods for this is to use something called partial products.  Instead of brute forcing it with the algorithm that many of us know, we do it in pieces.  One way would be as follows:

364 x 10 = 3640

and then,

364 x 2 = ?

Here again we break into pieces:

300 x 2 = 600

60 x 2 = 120

4 x 2 = 8

All together we have:

600 + 120 + 8 = 728

So our final answer would be 3640 + 728 = 4368.

I know what you’re thinking (because I’ve had it yelled at me before), “That’s so many steps!  Why bother?”  Well, what do you really do when you line up the numbers and follow the algorithm?

364

x   12

You would multiply 2 by each part on the top, and get 728, then you would go down, put your zero because it’s in the tens place, and then come up with 3640.  Hmmm.  Hopefully it seems a lot more similar now.  It’s the same concept, just the way it has been written is different.

Unfortunately, people are creatures of habit.  I will be the first to tell you that if you know how to do something in math, good for you.  By all means stick with it, and keep practiced!  Where we run into a problem, however, is when a student does not understand how to do the algorithm the way you do.  What then?  Should I simply keep telling them to do it until he or she might finally accept the process.  How many of you knew why to put the zero in the second line when you first learned multi-digit multiplication?  How many of you just thought about it now?

Is it important to be able to simply arrive at a correct answer?  Of course it is!  Wouldn’t it be far more useful later on to have the knowledge of why a process is happening?  You be the judge.

The reason I tutor is so that I don’t have to stick to one method only.  (More on that in a little bit.)  I get the opportunity to find out which method makes more sense to a child rather than only staying stuck, nailed, and riveted to the same techniques regardless of what helps a child learn.  Because of this, I have taught both ways to multiply on numerous occasions.  Here is what I’ve noticed.  When I teach the algorithm to someone who is struggling and it takes hold, they can usually get it and very soon are having no problem.  Then I ask them to multiply by a three-digit number and….they get stuck.  No idea what to do on the third row with very few exceptions.

The kids who happen to like the partial products method more, however, rarely get tripped up when moving into three digits.  Why is this?  It’s simple really.  The students who do the longer steps don’t need to learn a new rule.  Instead they are prepared to extend their knowledge and tackle a larger problem.  With the algorithm, it must be taught to put two zeroes on the third line as you are now in the hundreds place.  Usually at this point, the child sees a pattern and can move into 4 and 5 digit numbers with little trouble.

Does this mean that one way is superior to the other?  Nope.  Whichever way makes more sense is the way to teach it.  The ability is more important than the method for sure.  Once understanding has taken place, then other methods can be taught to help improve speed and efficiency.

But that brings us back to our initial complaint.  Why does the school force everyone to do the first way then?  Now we are asking the correct questions.  There is nothing inherently wrong with the Common Core methods.  In fact, they are better for preparing students for more advanced topics later on, if any comparison needs to be made.  Where we run into trouble is that the school system has made the mistake of assuming that every person needs to perform calculations in the same manner.

Every person learns differently, and the key to education is figuring out the best way to convey a topic to a student.  When the schools only teach one way in order to attempt to reach the majority, there will always be those who are left out.  The problem with the new Common Core system is not necessarily the methods being used to teach, but rather the ideology behind when and how to use those techniques.  So the next time you have trouble with a topic, or you see your child working through a method that you question the usefulness of, try another approach instead of complaining.  I think you’ll be surprised how it takes away the frustration and helps kids grow in leaps and bounds.