Lately, for lack of math topics, I've been teaching fractions to a 7 year old. It's going well. I'm basically doing what I did when I taught counting, addition, and multiplication.

We have 3 modes with math, and, like all of my articles, mode #3, which appears at the end, is the most important of all..

Mode #1 is heads down workbook catch up. By catch up, I mean I want the child to catch up to his natural level which is somewhere between 1 and 3 years ahead of school curriculum (in the US). I do this once or twice in the early years, depending on the kid, for 6 months at a time.

Any more than that will backfire, since the child isn't learning anything by becoming an expert at arithmetic or decimals. I could care less if my child has memorized the times table and gets 100's on all tests. I discourage this. This is not learning. Of course, if this is a GAT evaluation year and grades or standardized tests count toward a GAT program, I could care less about learning, because the priority is to pass the GAT requirements and then learning can happen next year.

The math brain is robust to setbacks. A child could take 2 years off of math and, with a bit of work, catch up and surpass his peers. This can't be said about reading. The rest of this article doesn't apply to a 7 year old who has carefully stepped through all of the right steps and is already in the 99th percentile. It applies to every child, especially the ones at the bottom of the ladder just starting out.

Mode #2 is thinking, figuring out, solving puzzles. I used to call this "test prep season", and we spend half a year on it whether there was a test or not. I now call this "cognitive skills season" and my goal is a bigger brain that can teach itself math without my help. I consider 2nd through 4th grade "cognitive skills season" because 90% of the math is useless math facts at this time. If I were in charge of curriculum, math would end in 4th grade (with 2nd grade math) and then start again in the 5th grade. In lieu of actually taking 3 years off of math, I did create my own curriculum and you can find it on Amazon if you google Test Prep Math Cogat. I suppose that the word COGAT ended up in the search criteria because I used the COGAT as inspiration for my methods. You'll see 2 books with graphics of an army ranger on an obstacle course on the cover.

Mode #3 is where actual math learning takes place. Lately we've been doing fractions. Like I did with their math predecessors, adding and subtracting, I provide my child a single problem about every week or so. That's right, one single problem. There's no hurry. Children magically learn things on their own, and when it's one single hard problem, it's a magical learning experience. This is what Mode 3 is all about.

A worksheet is something different. A worksheet requires a set of skills to achieve a different objective - finishing the darn worksheet. It's not so much about the wonder of math concepts, it's about finishing the darn worksheet. Math is an ancillary objective, and not the most important tool when worksheets are involved.

The first problem for younger kids is something like 3 +3 or 2 x 3. Then I just leave them wanting more. Later, we do 5 x 5 or 6 x 2, or whatever I can think of that is easy. Hopefully, I'll get wrong answers, and if I don't, I may do 10 x 3. Something harder, but something with a pattern to it. I'm waiting for light bulbs. The key is a brand new concept that is a stretch for the child. If the child is prepared for multiplication, this step will be incremental and not a leap, and you can just hand them a worksheet. Mode 3 needs a leap, like addition and subtraction for a child who just learned to count.

After we've done about 10 of these problems over the course of a few weeks, we'll try to demonstrate the problem with a picture. I should mention that for smaller children, with addition and subtraction, I start with a picture or 2 stacks of blocks. For older children, I don't provide a picture; instead, I provide a few days to solve the problem so that they have ample opportunity to visualize it first. All math should be drawn at some point. With multiplication, the picture is a the area of a square cut up into units. I have my heart set on graduate level math, and all math needs a picture.

This summer, we did 2/3 + 3/4 and 3/4 x 5/6. I might have given 2 other problems. Last weekend, we did 3/4 ÷ 5/6. That's not a lot of problems to do in a summer, but without going into technical details, it's enough to grasp fractions. By the time my child sees fractions in school, he'll have a mature understanding of what they mean. We dabbled in decimals, but decimals are boring concept I'll leave for school.

Note that working memory is essential to doing advanced math. Multiplication, for example, is not memorizing 6 x 5 = 30, but splitting 6 x 5 into much easier problems and then aggregating the result. In other words, properly done, 6 x 5 = 5 x 5 + 5 or 3 x 5 x 2. That requires working memory, which is why Test Prep Math has such goofy convoluted problems, and a whole ridiculously hard (at first) section that makes the COGAT look like a walk in the park. When you get to 2/3 + 3/4, you have a lot more steps going on than with 6 x 5, and it's all working memory driving the solution.

Next weekend, we're going to draw these problems and work them in a variety of ways. Then we're going to use language to describe them as many ways as we can and related the fraction operator to the division operator. You can do the same thing with all the elements of arithmetic for younger children.

The reason why we do a few problems over a long period of time is that I want triple learning out of my child. I want them to learn the concepts on their own, and learn how to think through things and learn to learn. These are the real goals of math. Arithmetic and fractions are just an excuse to use the brain. At some point, I jump in with lots of questions and different ways of thinking about it, but only after my child is conversant with problem and will readily get what I'm suggesting.

To summarize mode 3: a few problems to spark the imagination and leave a template for the "how to", pictures to explore the actual math concepts, and the language involved to further elaborate the operations. (The "language involved" is also the basis for common core, which is the one thing our education system got right in the last 30 years.)

Most parents see school curriculum and workbooks, and surmise that the goal of math is a lot of work to achieve a big objective, like mastery of operations. Math education researchers used to think the same way, since they developed this curriculum, until they realized that it is a complete failure in the sense that no one is learning anything and no one is choosing math majors in college. Now we know better.

What I know now is that a child who does a few problems thoroughly is going to end up with higher scores in math, and is much more likely to get 100 on every worksheet than a child who just jumps in and does a lot of problems. The road to mastery has a big detour, and I call it Mode 3.

I like your blog :) I helps me learn how to teach my kid.

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