Open-ended math problems are problems that have more than one possible answer. These problems might present an end result and then ask students to work backward to figure out how that end result might have been achieved or they might ask students to compare two concepts that can be compared in a variety of different ways. But whatever way they are presented, the purpose of open-ended math problems is always to encourage students to use higher order thinking skills to solve problems and understand that some questions/equations can be solved in many ways, with many outcomes.
Example of open-ended math problems:
A Year 2, an open-ended math problem might be: “I’m thinking of the number 8. What two numbers could work together to make the number 8?” You could provide the students with manipulatives they normally use for composing and decomposing numbers, like counters, small erasers, counting bears, cubes, or even play-doh. The extra bonus about this kind of problem is that it’s extremely easy for students to show their math skills. Some might use addition, others will use subtraction, and you may even run into a student or two who can use multiplication to find the number. However students choose to explore all the possibilities for answers, be sure to give them a few options to show their thinking. This might include simply writing equations, drawing pictures with the equations, or even building the number with manipulatives and then taking a picture of it with an iPad.
As students get older and move onto more abstract thinking in year three and four, you might incorporate more word problems like: “The difference between the temperature on Monday and Tuesday was 13 degrees. What could the temperature have been on each day? Find and explain at least 3 different answers.” Or “Penelope sees 37 children playing in a corn maze. If those children split into four groups, how many children could be in each group? Find and explain at least 3 different answers.” As always, be sure to provide students with manipulatives, paper and pencils, dry erase markers and whiteboards, or whatever you normally use to help them solve problems and then let them go to work! By presenting these kinds of word problems, you’ll expose students to a variety of math concepts (such as division in this example) just by allowing them to think about how to solve the problem on their own. Then, when these concepts are formally introduced, they will hopefully feel more familiar to some students.
There are many benefits to incorporating these kinds of problems into your students’ daily routine, here are a few obvious and effective ones we use at Astor:
Open-ended problems build confidence in your students. Once students recognise that there are many possibilities for correct answers and thinking, they begin to participate more readily. Students who normally struggle with math might solve the problem on a very basic level, using a basic strategy, but they’ll be correct! And your advanced students can solve it on their advanced level and be just as correct as the student who struggles. Simply knowing that the way that they specifically thought and solved the problem was considered correct builds confidence for students.
Open-ended problems are engaging! Students are immediately engaged in these kinds of problems because they recognise that there are so many different ways to solve it. Whether students are working in small groups or independently, there is the possibility for so many different ideas and answers to be correct that everyone wants in on it. This engagement, in turn, encourages collaboration among students and soon, they’re sharing their thinking and learning from each other to solve problems.
Open-ended problems encourage creativity. Students are capable of using so many strategies that they’ve learned over the years to solve problems and, given the space and time, they can even come up with some of their own strategies for solving problems. Open-ended problems give students permission to be creative in their thinking and problem solving.
Open-ended problems make it easy for teachers to see what levels students are working at. Simply by walking around the room while students are working to solve an open-ended math problem, you’ll be able to informally assess what kind of level they are independently working on. This can be extremely beneficial as you are collecting data, forming groups, or simply getting a feel for what kind of skills each student is working with.
Creating a space that is safe for your students to take chances and risks with their learning is one of the greatest gifts you can give them. By incorporating ways for your students to express their individual ways of thinking, like open-ended math problems, you’ll foster a love of creative thinking and confidence in problem-solving skills, something we really value at Astor International School.