One of my favorite parts of using Twitter has been the exposure to a number of great resources. To me, finding a new resource is exciting. However, it can also be overwhelming, from the sheer number of options that are out there to determining the quality. (This has been especially tough for me since I left the classroom and don’t have my own group of students to experiment with new resources. However, I am lucky to have a team of teachers who are open to broadening their own horizons, and let me join along in the process.) Recently, I had my first experience with an Open Middle problem, and it is one of my new favorite math tasks!

As mentioned, determining what resources to use in your classroom can feel overwhelming. Hopefully, by sharing my experience I can convince a few more people to try an Open Middle problem in their own classroom.

If you haven’t used or seen an Open Middle (OM) problem, they are described on the website in this way:

As mentioned, determining what resources to use in your classroom can feel overwhelming. Hopefully, by sharing my experience I can convince a few more people to try an Open Middle problem in their own classroom.

If you haven’t used or seen an Open Middle (OM) problem, they are described on the website in this way:

* they have a “closed beginning” meaning that they all start with the same initial problem.

* they have a “closed end” meaning that they all end with the same answer.

* they have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

Background

***Task selection:*The teacher and I picked a task that was related to the unit we were teaching (fraction operations). Our goal was to evaluate students’ conceptual understanding of fractions. We decided this task would be perfect to help us achieve our goal.*Implementation*

*:*The teacher and I set a time that it would fit organically into lesson planning - after the students took a quiz. This allowed students to work independently at their own pace while their peers were finishing the quiz. Then when everyone had finished the quiz, they were able to collaborate in partners to continue working on the problem. The whole activity took roughly 20 minutes.

Benefit #1 - Accessible to All (Low Floor)

The beauty in the design of most OM problems is that it can be accessible to any student. Even a student who is completely unsure of where to start can simply “plug and chug” numbers to see what the result is, and move forward from there. Often, especially in the upper grades, we find that students have a sense of learned helplessness. If they don’t know exactly what to do to find the correct answer, they will sit their and do nothing. They would rather not try, then try and be incorrect. The OM problem helps eliminate that, because students are not necessarily going for the “right answer”, but different “attempts” at the same answer. This means that for students who have no strategies, they can simple plug in random numbers to see what answer they get. Then they can reflect to see how they want to modify their next attempt to get a closer answer, and through that process they can develop their own strategies.

Benefit #2 - There’s no “I’m Done” (Productive Struggle)

Often times, students want to complete their math work as quickly as possible, and turn it in so they can be done with doing work. Then they can read, zone out, talk to friends, do whatever they think they can do when they are “done”. This leads to rushing through work and not checking their answers for reasonableness. In OM problems, there isn’t usually just one answer, as in many math problems. Therefore, even if they got

For example, in our problem, the goal was to get as close to ½ as possible. We had a student that after one attempt, walked over to me and handed me his “answer” to signal he was finished. I looked at it (it was not ½), complemented his effort, and handed it back and told him enthusiastically to see if he could get closer! (The look on his face told me how rarely this happened to him in class, and how little he liked it). This idea of the productive struggle is sorely needed in many of our math classes. Math isn’t about just getting a problem right or wrong and being done, it’s about the mental effort required to continuously problem solve.

Benefit #3 - Looking at Student Thinking (not just right or wrong)

A fellow coach and blogger made the following comment about math education, and I thought it was perfectly put. People often say they like math because it’s black or white, right or wrong. But that is such a narrow and outdated way to think about math. However, as math teachers, we often fall into the trap of checking student work for being correct, and not necessarily for what the student is thinking. OM problems force you to look at student thinking, because the answer is already given to the student. This forces you to look at what attempts the students are making, and analyze what those attempts say about the students understanding.

Often times, students want to complete their math work as quickly as possible, and turn it in so they can be done with doing work. Then they can read, zone out, talk to friends, do whatever they think they can do when they are “done”. This leads to rushing through work and not checking their answers for reasonableness. In OM problems, there isn’t usually just one answer, as in many math problems. Therefore, even if they got

*an*answer, they can still continue working to get a*closer*answer.For example, in our problem, the goal was to get as close to ½ as possible. We had a student that after one attempt, walked over to me and handed me his “answer” to signal he was finished. I looked at it (it was not ½), complemented his effort, and handed it back and told him enthusiastically to see if he could get closer! (The look on his face told me how rarely this happened to him in class, and how little he liked it). This idea of the productive struggle is sorely needed in many of our math classes. Math isn’t about just getting a problem right or wrong and being done, it’s about the mental effort required to continuously problem solve.

Benefit #3 - Looking at Student Thinking (not just right or wrong)

A fellow coach and blogger made the following comment about math education, and I thought it was perfectly put. People often say they like math because it’s black or white, right or wrong. But that is such a narrow and outdated way to think about math. However, as math teachers, we often fall into the trap of checking student work for being correct, and not necessarily for what the student is thinking. OM problems force you to look at student thinking, because the answer is already given to the student. This forces you to look at what attempts the students are making, and analyze what those attempts say about the students understanding.

In our lesson, we looked at how students chose fractions. If a student chose ⅞, this clearly showed a lack of understanding of fraction magnitude, as this was already bigger than the ½ solution. Some students were still adding denominators straight across, demonstrating they didn’t completely internalize the idea of equal shares and fraction addition. By analyzing the way students were solving the problem, we had better ideas of where to focus our feedback to students, and how to design lessons tackling these misconceptions.

The Takeaway

I encourage everyone to try this resource, if you haven’t already. If you have, please share ways in which you’ve found it effective in the classroom. I would love to have even more ideas moving forward!

The Takeaway

I encourage everyone to try this resource, if you haven’t already. If you have, please share ways in which you’ve found it effective in the classroom. I would love to have even more ideas moving forward!