The term open-ended is frequently applied to problems that don’t actually have an open end, but rather an open middle. Here are two problem-based lessons I’ll discuss to make the difference clearer:

First, let’s consider how each problem ends. With the hybrid car problem, depending on the assumptions you make (cost of gas, miles driven, price of car, etc.), there are multiple correct answers for how long it will take to make a hybrid car worth the extra cost. That makes it open-ended. However, with the In-N-Out Cheeseburger problem, there is only one right answer, so the problem is closed-ended.

Now, let’s consider both problems’ middle. With both problems, there are many ways to solve them, so the middles are open. In contrast, something like “Use the standard algorithm to find the answer to 475 ÷ 25” is closed-middled because there is only one acceptable way to solve it.

Personally, I prefer problems that have open middles and closed ends. These problems all have the same answer (closed end) and allow for great discussions around the strategies students used (open middle). When the problems have open ends, it can be challenging to have discussions because you can’t readily compare how different approaches ended with the same result. Accordingly, the website Open Middle is a collection of problems like this.

To be clear, I am not saying that open-ended problems are bad and open-ended problems are good. What I am saying is that each have their own purpose and we should acknowledge the differences between them.

What do you think? Let me know in the comments.

Robert — I like this distinction a lot. I find that problems with an open middle but a closed-end are most helpful in generating meaningful peer-to-peer discussion discussion because they give students a target to aim for while they are learning. Thank you for clarifying this distinction.

– Elizabeth (@cheesemonkeysf)

Yes, I agree Elizabeth. That tends to be my goal more often so it’s helpful to know what types of problems are more likely to encourage those discussions.

I find this an interesting and important distinction. Both open-ended and open-middle problems have a place in math instructional practice. I agree that open-middle problems are well-suited to promoting mathematical discourse. This discourse can deepen student thinking as they explore alternate solution paths. However, I think it is important to also find a place for open-ended problems. Once students leave an educational setting and are asked to use mathematics to solve real problems, they discover that much of their work is in fact open-ended. They must learn how to frame and define the parameters of the problem as well as finding an appropriate solution. I want to give my students opportunities to develop those skills as well, when I can.

Great point Cheryl. Clearly open-ended problems have their place and I really like the way you articulated it.

Thanks!

A nice taxonomy. It really got me thinking and, when I noticed my comment was getting too long, I decided to make it into a separate blog post: Math Make-overs

For what it is worth, the main/fun points are:

(a) probably any problem can be converted between types and there are simple methods to make-over the presentation to get the type you want.

(b) How can 475/25 become open-ended? Well…we didn’t specify which base we are in. In base 10, of course, we get 19. In base 8, though, the answer is 17 with remainder 2! I’m sure cheeky students out there could find other innovative ways to interpret the question, if given a chance.

Thanks for pushing my thinking Josh. I agree that there is probably flexibility in opening or closing problems’ middle and endings. I would say that I am currently less concerned with that though than simply knowing how the attributes of the problem I’m using affects how students interact with it. Thanks again.

Great post and the timing is impeccable for me! This arrives right on the heels of a debate I had with my colleague a few weeks back about “The right answer” to a specific 3 act math problem used. I was okay with not knowing the absolute answer to the problem but it was making her crazy to not know. I think this speaks to many of our students. She could simply not get past it and felt the experience was a waste without the solution. The open middle concept is wonderful, but at the end of the day, not having a resolution will leave many students unsatisfied. The question I asked my colleague (the topic was probability) was, well, does everything actually have a solution? What are your thoughts?

Merryl, your idea about the experience being a waste without a solution speaks to the long-standing notion in mathematics teaching that finding the solution is the only purpose to doing math. Moreover, the right solution is all that is to be valued, so for those who don’t often get the right solution, math was out of reach, pointless.

I have delivered lessons that contain both formats without really realizing it. I tend to do more open middles but at the beginning of the year when I’m demonstrating to students the value of just engaging in math and that all of us have skills and ideas to contribute, I use open ended. How would you measure a puddle? It also becomes a team building activity. They have so many questions (I will not answer) and are forced to think mathematically, delivering a unique solution that can never be right or wrong. Thank you, Robert, for outlining this distinction. Being aware of distinctions like these help me teach more purposefully.

Trish, I could not agree with your comments more! I am actually giving you a standing ovation (in my mind) as I type this. Thank you! Our job is to make math reachable for all and the one answer concept…well, that does not help that happen.

Interesting thoughts Merryl and Trish. First, let me make sure we are on the same page about a “right answer” / “absolute answer” / “resolution”. Consider this problem: http://robertkaplinsky.com/work/drug-money/. This problem has an open middle and closed end, even if the answers are not particularly accurate because the information given is not as well-defined as someone would want. For example, I don’t have info on how many bills are in each pile. So, this could be a time where it is closed-ended but still not giving an “absolute answer”.

Going back to your story, Merryl, I have had similar experiences. I once had a teacher say that she wouldn’t use Andrew Stadel’s file cabinet task because the handles made it have one less post-it note and so the answer was 935 and not 936. She didn’t want students to think they didn’t understand math so she wanted to avoid the problem.

Generally, I think it is worth using these problems and then having conversations about sources of error if the answers are different than what was expected.

Trish, I think your comment echos my own thoughts. I also understand your comment of “I have delivered lessons that contain both formats without really realizing it.” If I’m being honest, I think that 6 years ago when I was still in the classroom, I was more doing closed middles and endings: basically making little math robots.

It’s good to reflect on our past to see how much our efforts have helped us progress. Thank you both.

I like both types of problem. I think there is a high degree of usefulness for problems with an open end in that they take into account forecasting. Being able to explain where your answer came from is a higher order thinking process (assuming there is some logic of course) that is often greater than the thinking that takes place in a problem that contains exclusively open middle problem solving.

Yes Mark. I definitely think it’s worth incorporating both. The process of being able to define what an acceptable answer is for an open-ended problem is very much needed.

I would mention though that I see the skill of “being able to explain where your answer came from” is independent of whether the problem is open-ended or closed-ended. Regardless of the type, students should be able to do that.

Your blog post really made me think! I just had to right a post myself. https://jboninducharme.wordpress.com/2016/05/24/convergent-or-divergent-problem-solving-mtbos30-23/

Thanks Jules. I’m glad the post resonated with you and I like what you wrote up. I’m glad you shared it.

Hi Robert,

Today we did the following open middle problem: __x + __ = ___ Using the digits 1-9 each for the largest value.

We did it again with __ * ___ x + ____ ____ = ____ ____ Again using the same rules.

The Geometry students wanted to solve it way one, except they wanted to use any symbols.

Another girl asked, “does x have to be positive?” ( she was thinking a negative/negative = positive!)

Cheers

Fun Amy! Lots of great conversations out of the generally procedural land of two-step equations. Glad your students got a lot out of it and thanks for sharing.

I appreciate the distinction between open-middle and open-ended. However, I disagree with your characterization of open-ended. You describe that when the inputs can vary (e.g. cost of gas, miles driven, etc.), the problem is open-ended. Even if the parameters are different for different people, though, each person will still get a single, no-two-ways-about-it, answer; put another way, each person is funneled to their respective closed answer.

I think a better distinction has to do with whether, once the student solves the problem, the conversation is finished or not. Here, I’d argue that the car problem is still open-ended, but for a different reason. Once students agree on an answer — for instance, a gasoline car will save $500/year (or whatever) — they can move on to the larger question: Is it worth buying a hybrid? Since there’s no right or wrong answer, the end is totally open. Indeed, there is no real end.

Hi Karim. Thanks for pushing back. It’s been my experience that when I disagree with you, I often have lots to learn about your perspective. Let me try to explain my thinking in another way. Consider the video game Ms. Pac-Man.

It always begins in the same way: you start towards the bottom middle facing to the left. So, the beginning is closed.

It also ends the same way: you either get the last dot and beat the level or get touched by a ghost and die. So, the end is closed.

What’s different every single time is the middle of the level where you can go any way you want and have different experiences every time. So, the middle is open.

I do see what you mean about relatively closed endings versus absolutely closed endings. That is a useful distinction. However, in terms of facilitating problems with students, it is often useful to incorporate problems where all students have the same answer but used different methods to get that answer. So, I describe that as closed ending and open middled.

What are your thoughts on this?

Love the open middle problems.

Can’t wait to use this for tomorrow morning’s warm-up

(which I call my COD – Challenge of the Day).

Thanks.

I think as students are working on learning a concept an open middle but closed end is necessary so that students really grasp the concept. However, I think once the concept is grasped, it is important to have open ended problems. These are the most common problems in life and too often we don’t show the life side of math.

This sounds like a very reasonable path to me, Telannia.

I am blushing. Thanks for confirming my thoughts.

A very interesting post indeed, like you, I prefer the closed end with open middle simply because there is a clearcut answer most times in the end. It’s so crucial to let students see that although each of us are going on the same path, with the same goals, what that path looks like can really differ. That being said though, I can also see the value of the open ended answer as these experiences really help us provide us with multiple opportunities that in life, not everything is so clear cut and a definite answer doesn’t exist. Does that make sense?

Yes, Lisa. My intention isn’t so much to say that open middles are better than open ends, but rather to help articulate the differences so we can more intentionally choose problems that meet our needs.

Thank you Robert for all that you share. So often I catch my fuller developed adult brain reaching back into the days of its concrete existence when the reality is my elementary aged SE students depend on my abstract abilities to keep things interesting. Open middles have really helped me to incorporate more abstract learning models in all areas of my differentiated lesson planning. Open middles to me tell my kiddos that there are alternative paths to the same outcome, and in a world where so much is deemed black and white, it’s sometimes nice to give them the opportunity to use higher thinking skills to come to a solution when also knowing that parameters are set for where to begin and logically where to end up. We appreciate your insight!

Thanks for unpacking this, Stephanie. I really like this line: “Open middles to me tell my kiddos that there are alternative paths to the same outcome, and in a world where so much is deemed black and white.” I agree completely.