Problems with Problem Solving

One of the most important skills students can have, and yet one of the most difficult to teach, is the skill of problem solving. In the Standards of Mathematical Practice, it says that students should be able to “Make sense of problems and persevere in solving them.” The skill of being presented with a problem that has not been seen before and figuring out a solution escapes not just students, but many adults as well.

“Word Problems – I HATE Word Problems!” is a common refrain heard uttered by many students. And yet, the real power and beauty of mathematics comes, not in solving rote problems of the type “3x – 4 = 17”, but in using mathematics to describe a real world scenario in a simpler way, and be able to determine the best solution.

How NOT to teach Problem Solving

The following are things that have been observed in various places and I believe to be ineffective for teaching and assessing problem solving.

Time Limits

True problem solving isn’t something that can be done in 10 minutes. Sure, there may be students who can “see” a solution and solving it quickly, but when we equate doing things quickly with successful problem solving, we do a severe disservice to the majority of our students. We need to recongnize that doing things quickly does NOT make a person a better mathematician.

Mathematics is not about speed – it’s about understanding, accuracy, and checking your solution. When we set kids up to compete against others in timed tests, we demonstrate a lack of understanding about what mathematics is. Although I have no problem with timed competitions as extra-curricular activities, to assign grades based on how quickly someone solves a problem goes against what I believe mathematics is all about. (As a side note – yes, in my career I HAVE had many times where I have imposed time limits on tests or other assessments. And I recognize that sometimes our structures may not allow for anything else. But I do not believe that this is the best way to measure mathematical skill).

Teach solutions

Today I was in a classroom and was looking at a textbook example. This problem was given as a word problem, and telling students the distance by plane from Boston to New York and then from New York to Philadelphia. The problem asked students to find the total distance traveled if the plane made the trip as a round trip.

The problem then proceeded to walk them through the solution. It asked them to complete the following sentences:

  1. The distance from Boston to New York is ________ miles.
  2. The distance from New York to Philadelphia is _______ miles.
  3. The total distance from Boston to New York and New York to Philadelphia is ______ miles.
  4. The total distance traveled in the round trip is ________ miles.

In this problem, we are not teaching students to be problem solvers. We are teaching them to fill in the blank and follow the directions. A student who learns this way is not in any position to be presented with a new type of problem and have any real hope of knowing how to solve it.

How we SHOULD teach Problem Solving

Careful Selection

First, problems need to be selected appropriate for the grade level of the students. Clearly, giving a class of 5th grade students a problem that requires 8th grade content to solve is not going to be a good choice. Additionally, the problem should be a real-world problem. Giving a student a complex equation to solve may be appropriate in some contexts, but that’s not what problem solving should be focused around. Rather, a problem such as presented above is a great example to use as a real-world problem. Thirdly, a problem with multiple strategies to solve is always a good choice, as it allows for more students to attack the problem from different perspectives.

Elicit Student Discussion

We should not consider students working together on a problem to be “cheating”. Students can learn a great deal from each other, if we properly guide discussions. For example, set some guidelines over the types of questions to ask in small groups – not questions such as “How do you solve this?”, but rather questions such as “How will knowing this fact help me with the problem?” Additionally, asking students to explain their thinking to the class is always a valuable thing to do. How about asking “Mary” to explain what she thinks “Donnie” meant by his statement? This allows for further discussion.

This article I wrote earlier references an external blog that provides a possible strategy that can be used as well.

Give students time

This was addressed above but is so important, I’m going to include it again. Do not expect students to be able to finish the problem within a class period. Allow students time to come back to work on it some more, either later in the day or the next day. Give students multiple opportunities to think and work on the problem.

A website that I HIGHLY recommend teachers check out is This people who created this website believe in making math a social activity, but also believe in giving students time to work on the problem, as well as giving them opportunities for multiple attempts. The website stresses the four-step problem solving method to help students become more successful in solving problems.

Conclusion and Resources

Teaching Problem Solving effectively is not something that happens overnight. It takes time and effort on the part of the teacher to do it correctly. We must not delude ourselves into thinking that because students are doing “word problems” we are teaching problem solving. Rather, we must deliberate in our planning and instruction, and we can effectively teach students to become  true problem solvers.

One of my favorite sites as a resource for problems is the Problem of the Week site at the University of Waterloo. There you will find problems for grades 3 – 12 you can sign up for weekly emails of those problems, and you can access the last two years of archives of their problems. I have found several problems from their site that I have been able to successfully use. Additionally, another site I have used is the Illustrative Mathematics website. There you can search for problems by grade level, common strand, and more.

If you have interest in an actual unit plan for teaching problem solving, you can try this unit that I created for one of my Math Leadership courses at the University of Maine at Farmington. I’d love to have feedback on how it worked for you.