Carman and Sudoku are in the thick of the teenage years. This morning we were all reminded of why we manage their education in our special quirky way.

It happens that Doodle (10) was struggling a bit with "Converting multi-digit repeating decimals to fractions", and I thought it would be a good exercise for all of us (including Milkmaid, but not Rosebud) to jump in and work a problem with him.

So, I prepared a little lesson to warm up the older minds for a problem of this sort. The warm-up was much needed, it turns out, and after about 20 minutes we arrived at the presentation of the problem and I turned everyone loose on it.

Unsurprisingly, Doodle finished in a few minutes and his answer was right. Some of the rest of us needed 30 minutes or more to get the right answer, mostly because of little math and algebra slips along the path of the multi-step procedure.

After it was all over, we gathered to sum up the deeper lessons from what took place.

Clearly, Doodle was quick at the problem because he had just been doing them on Khan Academy. In spite of my own math history rising to Differential Equations, I needed to get help as I was preparing the lesson. So, it's easy to be good at something you just learned but hard to be good at it later.

Also, we can see that there is precious little overlap between this kind of math knowledge and anything practical. While I have used trigonometry and mid-level algebra in my years of building physical things and internet things, I've never needed to convert a repeating decimal into a fraction. Sudoku took a break from writing some killer php/ajax back-office code in order to do this little math exercise, and wound up being sorely tempted to feel like a chump. In the end, the best we can say is that this little exercise/method is just a way for us to stretch our minds in a different way.

Also, it seems clear that one can succeed at such problems in a "plug-and-chug" way, without having a grasp of any deeper meaning.

Getting to my point, what is the best kind of mental heavy lifting for teenagers? What do they really need?

In my opinion, the foundation of everything (is character development, but let's not get too broad here, okay) is for a teenager to acquire, as quickly as possible, an unlicensed trade which will buy them their intellectual freedom and keep their curiosity alive into adulthood. This allows them the time and funding to go about any further education in their own time and at their own risk, without which said education often turns out badly.

The second thing is for them to have 10,000 occasions to think deeply about their world and connect the dots between as many spheres as possible. In other words: theory.

And this is where I see today's exercise fitting. It was a chance to look again at the "forests" of decimals vs fractions, algebra, and the minefield of multistage problems.

As a parent in 2015, I find it hard to justify pushing my teenagers into a high proficiency with:

• math techniques that will be forgotten and are better handled by computers
• names and terms from biology/chemistry/etc

On the other hand, I feel that there will not be nearly enough time to pound home the key concepts of the natural and social sciences so that they may apprehend those fields in a deep way.