# Problems vs. Exercises

My high school mathematics teacher, Mr. Davis, classified all learning tasks in our classroom into two categories: problems and exercises. The distinction between the two is pretty simple. Problems set up a non-routine mathematical conflict. Once that conflict is resolved once, problems cease to be problems – they become exercises. Exercises tend to develop content skills or application of knowledge – problems serve to develop one’s habits of mathematical practice and understanding.

I tend to give a mixture of the two types to my students. The immediate question in an assessment context is whether my students have a particular skill or can apply concepts. Sometimes this can be established by doing several problems of the same or similar type. This is usually the situation when students sign up for a reassessment on a learning standard. In cases where I believe my students have overfit their understanding to a particular question type, I might throw them a problem – a new task that requires higher levels of understanding. I might also give them a task that I know is similar to a question they had wrong last time, with a twist. What I have found over time is that there needs to be a difference between what I give them on a subsequent assessment, or I won’t get a good reading on their mastery level.

The difficulty I’ve established over the past few years learning to use SBG has been curating my own set of problems and exercises for assessment. I have textbooks, both electronic and hard copy, and I’ve noted down the locations of good problems in analog and digital forms. I’ve always felt the need to guard these and not share them with students so that they don’t become exercises. My sense is that good problems are hard to find. Good exercises, on the other hand, are all over the place. This also means that if I’ve given Student A a particular problem, that I have to find an entirely different one for Student B in case the two pool their resources. In other words, Student A’s problem then becomes Student B’s exercise. I haven’t found that students end up thinking that way, but I still feel weird about using the same problem multiple times.

What I’ve always wanted was a source of problems that somehow straddled the two categories. I want to be able to give Student A a specific problem that I carefully designed for assessing a particular standard, and student B a different manifestation of that same problem. This might mean different numbers, or a slight variation that still assesses the same thing. I don’t want to have to reinvent the problem every single time – there must be a way to avoid repeating that effort. By carefully designing a problem once, and letting, say, a computer make randomized changes to different instances of that problem, I’ve created a task I can use with different students. Even if I’m in the market for exercises, it would be nice to be able to create those quickly and efficiently too. Being able to share that initial effort with other teachers who also share a need would be a bonus.

I think I’ve made an initial stab at creating something to fit that need.

## 3 thoughts on “Problems vs. Exercises”

1. Following this [https://www.youtube.com/watch?v=zhNqhBltoDQ] now I classify activities in “when” (real-life, application, etc.), “how” (procedural, rutine based) and “why” (conceptual understanding). I hope it serves you

2. I use ‘problems’ to set the stage for productive struggle. I think that parallels what you are saying. Exercises are practice. I also need variations of exercises to check for mastery. I go one step further in that the answers tell me (sometimes) what the child’s misconception may be, and I can address it. I have found a really interesting resource for problems that highlight misconceptions in the GCSE emails I am getting from Craig Barton craig@diagnosticquestions.com

1. I think we are in agreement. I like the distinction you made on practice – problems can be opportunities to demonstrate mastery and understanding, while exercises are more focused on developing fluency. There is a place for all of these in our classrooms.