Learning Without Understanding

Turns out my own 8-year-old daughter, Skyla, is too good at memorizing things and also good at pattern matching.  She can get through some of my MathScore topics with a rating of 100 without achieving the deep understanding that I actually desire.  Problem is, once she hits a problem that is too hard, she gives up because she didn't understand what she was doing when she worked on the easier problems.

Take for example, a new math topic I created, called Fraction Multiples.

Here is one of the hardest questions from that topic:

Skyla can do these problems fairly easily.  But now here's what shows up in Fraction Multiples 2:

Now she's in trouble!  The difference is that the numerator in the first fraction is not 1.  You have to have a better grasp of fraction equivalence to solve these.  Skyla was smart enough to hack her way through the first topic and halfway through the second topic without a true understanding.  So how do I get her to the next level?  I could of course tutor her directly, but that's cheating.  If I'm going to really change the world in math, I have to accomplish this through MathScore, so here's what I'm going to try next:

From now on, I'm going to make a strong effort to include math questions that quiz the process itself.  So rather than only ask for numeric answers, sometimes I will make my students select from a choice of approaches.  Here's an example of a new type of question I intend to add to Fraction Multiples 2:

Given the fraction 3/5, how would you compute the numerator for an equivalent fraction that has a denominator of 30?
- The new denominator of 30 is 6 times greater than the old denominator, so the new numerator is 6.
- The new denominator of 30 is 6 times greater than the old denominator, so multiply the old numerator by 6 to get 18.

I may include other answer choices, but you can see the direction this is going.  Perhaps I could potentially make a question as hard as this one:
Given the fraction 3/5, how do you calculate the numerator for the equivalent fraction whose denominator is x?
- Divide x by the denominator, 5, then multiply by the old numerator, 3.
- Other answer choices would be included

Also notice that I can't go fully algebraic because that would confuse the heck out of elementary school students, so it's not proper for me to ask for a final expression like 3x/5.

In the past, I generally taught understanding through explicit math lessons and solution explanations when you get something wrong.  But lots of students will only gloss through lessons and also gloss through the explanations.  The only way I can guarantee that I can make a student think about the process is to directly quiz it, so that's the direction I'm taking.

Anyway, it was eye-opening for me to come to this realization.  It's clear that I learn differently from others.  When I learn things, I focus on the underlying process or algorithm by default, so I kind of assumed that most students would at least make an attempt to do the same.  But if the majority of students did that by default, teaching math would be easy.  So now I'm going to assume that students are trying to actively avoid a deep understanding (in other words, cut corners when possible), so the only way to teach those students is to throw it directly in their face.

I had a similar problem, by the way, when I asked Skyla to try some new perimeter questions I developed.  For example, given that the perimeter of a rectangle is 22 cm and the length is 5cm, what's the width?  I asked Skyla how to solve this, and she told me that she doubles the length, subtracts from 22, then divides by 2.  I asked why she divides by 2 as the last step and she couldn't give me a proper answer!  Argh!  See?  She's good at memorizing things, which means she might get an A on a math test, but for the wrong reasons.   This is very frustrating for me, but at least I now have a perfect test subject for MathScore!