The other night my 10-year old son came to me for help with a word problem involving volume. It was late at night and I was tired. I could’ve given him a hint about how to solve it, or even worse, solve it for him. Instead, I ended up spending half an hour with him as we then expanded upon his newly constructed knowledge. It was worth it!
I wanted to make sure that my son understood the problem he was trying to solve, so I asked him to explain what he was trying to do and why. Realizing he didn’t understand the problem, I encouraged him to read it again and explain the different components as he progressed. Once he understood the problem, he immediately saw what he needed to do to solve it.
When I started teaching almost two decades ago, someone mentioned how knowing when to help a student during problem-solving was “an art that requires much experience.” He was concerned that giving a hint at the wrong time could destroy those important “aha” moments. He was right about the latter; my experience, however, informs me that there are simple signs we can use to know when to give a hint. It’s more (pedagogical) common sense than art.
Let us put this in the context of the 3-stage problem-solving cycle.
The first assistance students should receive, if any, should be directed at helping them understand the problem. Experience tells me this is often the missing (and ignored) link, even when students seem engaged scribbling and doodling.
Once we have verified that a student understands the problem, avoid giving hints about how to solve it until the student is on the verge of frustration (or already frustrated). Offer a hint that doesn’t give away the problem and observe how the student reacts. At this stage, we should evaluate whether the student is ready for this particular challenge. If not, we need to decide whether to change the problem altogether or offer problems that will address the prerequisite knowledge and skills.
In a nutshell, give a hint only after it is clear that a student understands the problem and is stuck. A hint given at the wrong time, either too soon or too late, can ruin the creative process, not only for the student, but for everyone in the room.
About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.
A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us. During two decades of observing myself and others in the teaching and learning process, I’ve noticed that the most neglected phase is often the first one—understanding the problem.
3-Stage Problem-Solving Cycle |
The Three Stages Explained
I have produced videos explaining each one of these strategies individually using problems we have solved at the Chapel Hill Math Circle.
The Cycle
As we implement our strategies, we might not be able to solve the problem, but we might refine our understanding of the problem. As we refine our understanding of the problem, we can refine our strategy. As we refine our strategy and implement a new approach, we get closer to solving the problem, and so on. Of course, even after several iterations of this cycle spanning across hours, days, or even years, one may still not be able to solve a particular problem. That’s part of the enchanting beauty of mathematics.
I invite you to observe your own thinking—and that of your students—as you move along the problem-solving cycle!
[1] Problem-Solving Strategies in Mathematics, Posamentier and Krulik, 2015.
About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.
This PDF is an article Ana Quintero and I published in the British journal Creative Teaching & Learning. It furthers a constructivist approach to mathematics teaching and learning based on Freudenthals’ Realistic Mathematics Philosophy. Here’s an excerpt.
“When selecting problems for children to solve, we should remember that the ‘realistic’ concepts students bring to the classroom involve aliens flying across galaxies; wizards, gnomes and elves inhabiting enchanted forests; dragon-fighting knights rescuing princesses from mystical castles; superheroes saving the world; and much, much more! We can use these elements and concepts to create problems and puzzles that capture students’ attention and feed their imagination.”
Here’s another one.
“Yes, the learning of maths must follow children’s cognitive development, but this must take place in a community that promotes genuine interest in problem solving—in making conjectures, and in sharing, discussing and arguing those conjectures with their peers. This environment must be alive and support pupils’ participation in the process of using mathematics with understanding, in order to comprehend situations that are of interest and relevant to them. Only then will maths truly make sense.”
Here’s a picture of the cover, as published. My daughter was validated.
About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.
About the author: You may contact Hector Rosario at hr111@caa.columbia.edu.