The Power of Open Middle Problems

Last week, I wrote this problem. I personally found it interesting that a thought that I had about it’s possible validity actually turned into something that worked quite nicely.

I didn’t actually write this for a lesson or class, I was simply curious about whether or not the problem could actually work. After solving it once, I shared it with the Maths Teaching community (#MTBoS) on Twitter. My Adelaidean friend, David Butler (@DavidKButlerUoA), took an approach to this problem I would dream my students to one day take. Check this out below by clicking on David’s first tweet of many below:

The way David was able to commentate his way through the problem allowed those reading his tweets to take part on his journey, following his every thought and attempt to understand, predict, and test his thinking. David’s freehand drawings illustrated how he saw the problem and how he was planning to attack the problem, which gave readers of the thread an insight into how he made sense of the information given to him and the implications of the restrictions placed by the problem. Annotations of these drawings revealed the emotive responses that David had with the results of his conjectures as he tested them, reinforcing the idea that it’s how you bounce back from an attempt that is the really important step when something doesn’t go as you predicted or would have liked it to. The underlying level of excitement in discovering more about the problem gave me, as the creator, an unbelievable sense of joy and gratefulness that someone was willing to throw themselves into solving a problem purely for the joy of it.

This is what I’d love to inspire my students to do.

About a week earlier, I put out another Open Middle style problem on the Maths Twitter Blog-o-sphere, and it caught the eye of A LOT of people.

Unlike my most recent problem, this one was actually uploaded after I gave it to my Year 8 class. My Year 8s found this problem very challenging and, to be totally honest, I was surprised with how easily some of them gave up or didn’t begin to try it at all. Upon sharing it on Twitter, I was inspired to think more creatively about how to build a more psychologically safe space for students to immerse themselves in challenging mathematical problems. A few teachers, particularly in the US, shared photos and videos of their students doing just this.

Ian Fischer’s (@MrIanFischer) pan of his classroom demonstrated how NPVS (non-permanent vertical surfaces) can be used to get students up and working, while also promoting discussion about their thinking with peers (check out Sara Van Der Werf’s thoughts on Stand and Talks).

While Chelsea McClellan (@marvelousmcc) shared multiple glory shots of her students standing proudly next to a solution they’ve clearly battled with and won.

My Year 8s are not normally difficult to engage nor are they particularly reluctant to take on problems they are faced with, which got me thinking, “What on Earth did I do wrong?”. Was it too “mathsy”? Were there too many things they had to first make sense of before being able to take an educated guess to start? Did I introduce it at the wrong time? Should I have given them a simpler problem in a similar style to start off with? It could have been one, none, or all of those things – I’ll probably never know. I chose to go with my gut and tried to build up their confidence with some similar problems that were somewhat simpler. These were what I used:

Some students got an answer straight away and because I had anticipated some possible responses prior to the lesson (a rarity, I must admit), I had a few questions up my sleeve to challenge the robust-ness of their answers and stretch their thinking. There were, however, some students just sitting there like they couldn’t see the problem on the board at all. For these students, I brought in a hint and BOOM! Up came the tide like the Moon moved out of turn. A harmonious “a-ha” sound echoed around the room as the eyes peered towards the ½ I wrote on one of the fractions of the problems. Did I take away an opportunity to discover the key part of the problem? Maybe. Did I invite those who were struggling to get started to come and take part in the key learning intended from the problem, I hope so.

Although we don’t always get things right the first, second, or even third time we attempt things, just like David solving my linear equation problem, it is how we bounce back to try something in a different way to what we first thought might work that is key. For now, I need to find ways to increase the amount of NPVS in my class and get students out of their seats!


For more problems of this style, check out the Open Middle website: openmiddle.com.

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