This is the first in a series of posts about learning to think like a mathematician. This is my first memory of playing with numbers, questioning my own thinking, and making sense of new ideas. As my brother Peter said, when I shot the video clip below, “and this is how it all began!”

When I was around 5 years, my older sister showed me an adding machine that was in my grandmother’s closet. It was large (to a fiver year old) and very heavy. It was completely mechanical and had 81 numbered hexagonal keys – 9 rows of 9 keys (see image, above). Each column of keys was numbered with the digits 1-9. Pressing a digit in a given column would display that digit in a corresponding window along the bottom row. There was also a lever on the right side that could be pulled to reset all of the column windows to zero.

My first experience with this incredible machine, thanks to my sister, Susie, was to press the “1” in the lower right corner (the ones place) continuously.  The tenth press felt different and the display at the bottom returned to zero, but the place just to the left turned to display a “1.” Pressing the same “1” button felt the same again for 9 presses, then on the tenth press if felt different again, and again the window below turned to a “0.” And the window to the left turned to a 2. I could make the second window count to ten, too? By only pressing the one button? This was fascinating! After a long time, and lots of presses of the same key, when all of the bottom windows displayed nines, I would press that same “1” button and it would “feel a little different” and something amazing and extremely satisfying happened. Watch what happened. 

All of the 9s would alternately flip to 0s, like a row of dominoes flipping over. It was a sensory explosion! I could hear the dials flipping. I could see them flipping, and I could feel when it should happen. It seemed magical at the time. 

I (we) were told not to play with this because it was an antique and I (we) might break it. I was a pretty good rule follower, but the allure of this machine was too much. I would sneak into my grandmother’s closet and lug that heavy machine out to play with it – a lot – even when I was a little older, just to get to see, feel, and hear that domino-like effect of numbers flipping over. Even now, it makes me smile to think about it. 

I would spend a lot of time pressing that “1” key until it got to 9, then press it one more time and a “1” would pop up in the place to the left and a zero was in the spot below the key I was pressing. It never got old. I kept pressing that key, not just to see what I had come to know would happen, but to figure it out. I began to make predictions and ask myself questions, like when the display read “249” one more press and the digit in the place to the left of my key changed to a 5. “I bet it changes to a six next time.” When it did, I was hooked. This continued and every time the next column got to a 9, I’d quickly press my “1” key one more time to get it to click over. What I noticed, though, is that it took a lot longer to get each row to 9, but that wasn’t enough. I wanted to know how much longer. I kept going because the more nines I had, the cooler it sounded when they all flipped over! Eventually, I figured out that it took ten presses of that “1” key to make the next column change, and that column had to change 10 times to make the next column change. I discovered a pattern of tens. 

What I didn’t realize, initially, was that I didn’t have to keep adding ones to get the full row of nines (I was still only about 5). I could just press each of the nine “1” keys 9 times, then add 1 more by pressing the far right “1” key. Then, satisfaction and amazement came much sooner! Once I figured this out, it was a much quicker experience but, frankly, a little less satisfying. After a while, I really took note of the other keys and realized that I could use some of them to my advantage as well. For example, I could press the “9” key in each column once and then press the far right “1” key. Still satisfying and much more efficient to get to the end result, but not as enjoyable as seeing this one button do so much. Watch the video below to see and hear what I loved so much.

This was one of my first experiences with playing with mathematics and the effects it can have in the sense-making and building deep understandings of mathematical ideas. I believe it is one of my earliest mathematics learning experiences, and I believe it had a huge impact in how I think, mathematically.

My hope is that this and some future posts may cause you to reflect on some of your own, similar, experiences. If so, please share. I’d love to hear your experiences. Stories like this, I think, have the potential to bring to light just how beautiful mathematics can be and the connections that can be made by studying this amazing subject! 

Side note: I really wanted to take this amazing contraption apart to see how it worked. My parents are thankful that I never did, but I still wonder what the inner workings of this adding machine look like. Unfortunately, I never got to find out, but I am still very curious. Perhaps there’s a video out there that I can watch so I don’t ruin this antique with my tinkering.