Sense-making

The Slinky Task – FAQs

The Slinky Task was dropped a couple of weeks ago and I’ve spoken to a number of people who had a lot of questions about it, so I’ve added some interesting facts (FAQs) about this task below. Enjoy.

  • How did you come up with this idea for the task? I was at a nearby school last fall working with one of the teachers there and a friend of mine, who also works there, mentioned that some students were eating lunch in her room and one of the boys was playing with a slinky that she had. He actually said, I wonder how far this slinky will stretch. She said she immediately thought of me. And then I showed up the next day. It did take me a couple of months to figure out how I wanted to do Act 1.
  • How long did it take to straighten the slinky? The initial straightening (what you see in the beginning of the Act 3 video) took about 45 minutes and the slinky still had a lot of fairly sharp bends in it (see below) . It also gave me several blisters on my thumbs and a couple of fingers. Lesson learned: Wear gloves.
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  • So, how long to straighten the slinky after that? It took about 18 hours. I worked at it for about 2 hours a day, when I could. There was a huge learning curve involved. I made several mistakes that added to the time needed to get this done. It was tedious, but with music playing in the background, it was fine.
  • What tools did you use to get the slinky straightened? As I mentioned, there was a huge learning curve. Some of the suggestions I got from Google searches actually prolonged the work, so I eventually just clamped the slinky in a vise and used pliers and vise-grips to bend it a couple of inches at a time.
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  • Any surprises? Yes. The slinky actually snapped three times as I straightened it. This was not good, but I did come up with a way to hide this in the Act 3 video. I clamped the slinky to the measuring tape using vise grips in multiple places to keep it aligned. Those vise grips are strategically placed to hold (and hide) where the slinky broke.
  • Did anything else not work out the way you thought or hoped? Yes. After I straightened the slinky, I asked my son to help me get it measured. We got it clamped and everything was set, but the measurement wasn’t even close – I think it was about 2 feet off from what I had computed the length should be. I kept thinking that I didn’t straighten the slinky enough, so I kept trying to get more of the bends out. I wasn’t going to even share this task – even after all the time I spent on it – because the numbers just weren’t even close to the real world measurements. Then it occurred to me that maybe my reasoning and computation was wrong. I was so close to this. Maybe I wasn’t seeing something that I needed to see. So, I sent the unfinished task and my mathematical thinking to some math friends and colleagues. I thought some fresh eyes (and minds) might be able to see what I couldn’t. The next morning, I got an email back from @KCwetna, with just the right amount of wondering that helped me reason through my own thinking to find the mistake. And, this mistake she provided me with is now included as a part of the task. It introduces another level of thought for students as they engage in the task – all based on the wonders she shared. Brilliant! So, essentially, my mistake ended up making this whole task better with some help from my friends.
  • What’s next? When the idea for this task was shared with me, I initially thought about doing this for slinky jr and the giant slinky. As of right now, those are on the back burner, but would be great sequels to explore and they are suggested at the end of The Slinky Task. I do have the slinkies for these tasks, but haven’t started the sequels yet.

What is Mathematics?

In my previous post, I shared my personal experience with playing with mathematics as a child. That reflection prompted some digging about when we as humans begin to think mathematically, and I found something fascinating:

Experimental research shows that infants as young as 6 months have the ability to:

  • recognize the approximate difference between two numbers
  • keep precise track of small numbers, and
  • do simple subtraction and addition problems.

And when babies are mathematizing like this, they activate the same parts of the brain that are associated with mathematical thinking and reasoning in adults (I told you this was fascinating). So, before we can even speak, we have the ability to quantify. Our inherent curiosity and ability to think and even reason mathematically is on display here.

The idea that teaching and learning mathematics isn’t just computation, that it involves sense-making through reasoning, is research-based and builds on the natural curiosity, and the mathematical ideas and abilities we possess at a very young age. 

Mathematics helps us make sense of and explain the world around us. It is the science that deals with the logic of shape, quantity, and patterns. Mathematics is a subject created based on the need to solve problems and, in my opinion, should be taught that way. It’s a beautiful, creative, and fascinating subject with applications in every field: teaching, economics, engineering, biology, chemistry, physics, entertainment, shipping, food service, geography, geology, technology, real-estate, and politics, to name just a few.

The common myth is that mathematics = computation. While computation is embedded within mathematics, it is really a very small part of a greater whole. The strong, flexible core of mathematics is all about reasoning and sense-making. The “computation part” of mathematics can be taught with this strong, flexible core in order to make sure that the computation students learn makes sense so that it can be applied to solve problems in the real world. 

Ultimately, mathematics is about sense-making. The mathematics we use today to solve problems was developed by creative thinkers who asked questions like. “What if…?” “Maybe we could try…?” and “I wonder what would happen…?”  This creative thinking is still happening today to solve problems like coastal erosion from tropical storms. You, your students, or your children can be one of these creative thinkers that uses mathematics and mathematical modeling to solve some of the world’s biggest problems. Let’s keep students thinking about mathematics as much as possible!

Learning to Play with Math

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 it 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 the video below to experience it for yourself: 

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 “9” keys 1 time each, 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. 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 – especially about place-value.

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.