Relishing Our Unexpected “A-ha” Moments

There are times when homeschooling one’s children can be the most frustrating, stressful, and, quite frankly, unsuccessful endeavor you will ever attempt. Through the years, I have had my share of days where I felt completely defeated and absolutely unqualified to teach my children even the most basic of subjects.


These days would lead to sleepless nights spent relentlessly analyzing why things had gone so terribly wrong. These nights would, in turn, lead to days filled with exhaustive research on curricula, lesson ideas, learning styles, and any other thing I could find to help us. Eventually, I would feel like I understood what my boys needed and a small ray of hope would appear. If I just tried X and Y, everything would work out, and most of the time it did – at least for a little while.

All this analysis and research yielded some good results and was definitely worth all the effort, but I have found that the greatest success and joy in teaching comes when your child has an “a-ha” moment, and those moments rarely come from your carefully crafted lessons. They have a habit of sneaking up on you in the most unexpected ways.

Not long ago, Archimedes had a bit of a meltdown over the value of pi. He understands that it represents the ratio of the circumference of a circle to its diameter. We physically proved it with string and a bunch of mixing bowls. Lots of mixing bowls, lots of times. He gets how it’s calculated and how it’s used throughout mathematics, but he doesn’t like its value. Why isn’t the ratio 4.21 or 5.89? Why, in the grand cosmic scheme of things, does this ratio turn out to be 3.14 specifically? Oh… well… um. Yeah, he had me there. Time for a sleepless night and a day of “googling” theoretical mathematics.math-pi

As it turns out, Archimedes was sitting beside me while I was scouring the internet in hopes of finding some way to explain pi’s specific value (other than, “Just accept it!” or “Because Mommy said so!”) when he noticed an odd entry for “Japanese multiplication” in the search results.

Multiplication is a particularly painful process for Archimedes. While he has mastered the standard “western” procedure for completing multiplication problems, he has never felt like he understands why this process works, and he is extremely frustrated by the amount of writing it requires. So, just for fun, we clicked on the link and watched a video in which “Japanese multiplication” was demonstrated. And the “a-ha” moment occurred!! Just like that, out of the blue, without any of my meticulous planning!

Japanese multiplication is a visual method of completing multiplication problems. Though many people refer to this method as Japanese, it is not actually known who originally devised this technique. Japanese, Chinese, and Indian cultures have all been credited with its development a one point or another. The method uses the visual intersection of lines representing the ones, tens, and hundreds places in numbers to calculate the product of a multiplication problem. A quick search on YouTube will reveal many good videos on this subject.

As Archimedes watched the video, his eyes lit up and he exclaimed, “That’s how it works! That’s so cool!” My son who will do anything to avoid using any type of writing implement promptly pulled out three colored pencils and began “drawing” multiplication problems just for fun! I used to watch his shoulders sag as he resolutely wrote out multiplication problems on his dry erase board, but now he actually gets his colored pencils out for calculations without hesitation or reminders and happily draws his lines and counts his dots.

Source: Su, Francis E., et al. “Visual Multiplication with Lines.” Math Fun Facts. <;.

It was such a pleasure to see Archimedes understand something that had long been a mystery for him and to see him actually apply that new understanding to different situations. And, it all came about from an impulsive click on a random link that showed up in an unrelated search! While I still spend many sleepless nights and active days trying to build that perfect homeschool experience for my children, I have learned that sometimes it’s best just to let things happen naturally and let the boys explore on their own. Sometimes following a tangent or a different line of questioning leads to awe-inspiring revelations in a completely different topic. These discoveries have proven more valuable than any planned academic study has ever been for us.

I am a Type A, ultra-organized, have-to-know-everything-in-advance type of person so following a path without knowing where it goes is actually quite difficult for me. But seeing the happiness and growth that comes from your children suddenly grasping an elusive concept is an unbelievable joy. I truly do relish our unexpected, but welcome, “a-ha” moments.

Have you or your kids had some unexpected “a-ha” moments, too?

2 thoughts on “Relishing Our Unexpected “A-ha” Moments

  1. That’s wonderful! How nice archimedes picked up a pen of his own volition.
    I was just ‘chatting’ to a mother of an autistic boy and I thought of your two boys, I hope they are well.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s