There are a lot of good math curriculums that teach the mechanics of fractions. I’ve seen step by step diagrams to add fractions with different denominators and add mixed fractions. With a thorough explanation and lots of practice, a young child can do fractions without any increase in academic skills or knowledge of math whatsoever.

So we’re not going to learn fractions this way.

The MAP test distinguishes kids who are ahead in math from the rest in the early grades. In later grades, it distinguishes kids who can figure out new math on their own. That’s what we want.

The starting point for fractions is for the child to tell me what they know about fractions. Some kids have not learned to articulate math, so we can work on this gap, It is most likely going to take some time for their brain to digest fractions on its own WITH NO HELP so I’m willing to wait. Plus, i need to find out where they are. Plus, they need to figure out what they already know because they are going to have to use it.

Start with ½, ⅓,¼ etc. What are these? Order them biggest to smallest? Can you draw it? If we put 2 in the numerator position, what do we get?

If you wanted me to teach fractions to your 7 year old who has never seen fractions before, we wouldn’t do more than 1 or 2 problems a day. Each problem is on par with a really good science experiment that spurs the imagination. Doing a bunch of problems is pointless to the learning process. Once the imagination is engaged, we’re learning, and during the thinking process WITH NO HELP learning skills are being generated that I’ll need in 3 years when I plunk down an SAT book.

What is the difference between ⅖ and 2 divided by 5. I want to know. Let’s do it. Suppose we divide 2 by 6 and then by 7. What’s going on? I want to know. Tell me, or we can figure it out together.

By the way, mathematicians never use the “divided by” sign. We always use ⅘ and say ‘4 divided by 5’ when we mean divide by or four fifths when we mean fractions, because these are the same and the divided by sign is lame.

Over the next few days or once a week, we’ll continue forward or repeat this conversation while it sinks in. If this kid is learning fractions now, then we’ll be decomposing 2nd degree polynomials soon and I won’t be in the mood to help. That’s why I won’t assign a fractions worksheet. Instead, I’ll ask them to decompose every number 1 through 100 and circle the prime numbers. When they need this, they won’t know it so I’ll have to tell them, but they are just kids.

From experience, the most important thing the kid needs to know is the answer to this question: If i add 3 pieces of cloth to 2 T-shirts, how many T-shirts do I have now? (10 minutes later) It’s the same with fractions. Either you make a T-shirt out of the 3 pieces of cloth and add it to the 2 T-shirts to get 3 T-shirts, or you rip each T-Shirt in half and add it to the 3 pieces of cloth to get 7 pieces of cloth. But you can’t add T-shirts and pieces of cloth without doing something.

Then I would take a single question of each type and we’ll do it together and look at it. By ‘together’ I mean I’m not going to help at all. Maybe I’ll give hints. Once they get it, we can do a harder version of that question type later. Or we try a different one. Ore we draw pictures, try an easier version, split it into 2 problems, or sometimes just iterate through all integers with that version of the question, starting with 1/1 and ½, ⅔, etc until patterns emerge. Or turn it into a word problem that is relevant to their world. Or all of the above.

Can you imagine what a little child who wants to be a piano expert does to become better? They practice the same piece over and over and over again. They drill and drill and scales and scales over and over.

Math is not like the piano at all. Math is learning to think, to analyze, to find patterns, to impute and make logical deductions, inferences, leaps. To put 2 unrelated things together. Drilling teaches none of this. Doing a single hard problem for 15 or 30 minutes while the parent is silent or asks questions is the prerequisite of thinking.

If I were starting from scratch with your child, I’m guessing this might take 1 to 2 months, maybe more to get to the really hard fraction problems. It would require very little effort on either of our parts. Just a lot of staring, questions, and thinking.

Where did I get the ability to teach fractions? We were doing fractions for the first time and I had 25 minutes of silence to stare at the problem while the work was in progress. I asked ‘what are fractions anyway’ and started to look at them anew.

At some point, you might want to assign a workbook page or the whole thing to get the ball rolling. When and how is your preference. I would never assign a fractions worksheet ever because a 7 or 8 year old doesn’t need fractions, and they will get smart enough by doing fractions to determine that math is useless, boring, and lame. This is my personal opinion. What I do instead is assign material that has lots of problem types, including fractions, and I assign that. It’s more sneaky. I just download tests of all kids and we do the problems that are appropriate. On these tests, either the problem is within reach, we skip it, or I’ll do it because they won’t see it again for a year.

The big issue to keep in mind is what your child did in the last few years. By the time we got to fractions, we had already been through this type of experience a few times and had done material that was less math topic and more hard core thinking. If you have less practice with this, then fractions will be your boot camp.