What are open-middle tasks?
Devan Smith | Bayside High School
March 2023
Have you seen a problem like the one from Robert Kaplinksy (@robertkaplinsky) on Twitter or in one of HCPS' compilation of Number Sense Routines?
This type of problem is called an Open Middle Tasks. Open Middles are rich tasks that allow students to use a variety of strategies to arrive at solutions, of which there can be more than one answer.
Using open-middles promotes students to think more deeply, often at a higher Depth of Knowledge level, about the problem-solving process while also assessing procedural and conceptual understanding. At the heart of our instruction, we want students to be problem-solvers. We want them to use number sense (and all the NCTM processes) to develop solutions to a variety of problems. Open Middles are a great way to promote communication, create connections, and emphasize the traits we want to see from effective problem-solvers.
I often find that these tasks are great for easing students into richer, deeper thinking about mathematics. It not only lets them stretch their brains a little, but it really encourages them to dive down into tying the understanding back to the process. The openness of the tasks take away a lot of the pressure of "being right" and gives students freedom to take risks, something we often see students hesitate to do in mathematics due to the stigmitization of our beloved subject.
have a “closed beginning” meaning that they all start with the same initial problem.
have a “closed end” meaning that they all end with the same answer.
an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
generally have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method. Example
may involve optimization such that it is easy to get an answer but more challenging to get the best or optimal answer. Example
may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it. Example
are generally not as complex as a performance task which may require significant background context to complete. Example
How can you use Open Middle Tasks?
As a classroom teacher, I loved using Open Middles as starters (activating prior knowledge or APKs) to get students talking about concepts. Take a look at the "Fill in the Blank" Open-Middle for Geometry (from HCPS Mathematics).
How much understanding could you assess from students as they work through and discuss the Open-Middle task on the left?
What sorts of answers would you anticipate a proficient student to provide?
Where might you ignite a conversation with a developing student?
How could you facilitate a whole-group discussion that connects the openness of the possible solutions with angle pair relationships' effect on parallelism?
We want to expose students to rich tasks as much as possible, as these sorts of tasks "allow learners to think creatively, work logically, communicate ideas, synthesise their results, analyse different viewpoints, look for commonalities and evaluate findings" (NRICH). As we progress towards preparing students to be future ready, exposing learners to routines like Open Middle tasks helps to deepen understanding of the mathematical process, as well as improve procedural fluency and critical thinking. Open Middles are a great way to ease students into more formal rich tasks, such as these from the VDOE.