Open-Ended Maths Questions: How to Turn Closed Ones Around

Sarah Keating starts with a simple closed question: I have 10 red apples, 5 green. How many do I have? "We all can see the answer is 15," she says. "It's a quick answer and move on question." Then she shows how the same question can open up, one step at a time.

So how do you write open-ended maths questions? Start with a closed question you already use and open it up in two steps. First, give students options on how they reach the answer. Then take away the fixed total, so the question has many possible answers. Along the way, ask students to explain, prove or justify their thinking. An open-ended question has more than one approach and more than one solution, so every student can find a way in, and you see how each one thinks as well as what they got.

Sarah is Co-Founder of Little Loves Teaching Resources, and these ideas come from her course on numeracy in the upper primary years. She has a Bachelor of Education (Primary), a Master's in Clinical Teaching and 11 years of experience teaching across all year levels, with roles including e-learning leader, unit leader and learning mentor. Her module on questioning opens with this line: "As teachers, one thing we're fantastic at is asking questions."

The ideas

1. Keep closed questions for when a set answer is the goal.Sarah says closed questions have a place in the classroom, and numeracy uses them often. The thing to watch is letting them take over from open-ended questions.
2. Give students options on how they get there.This is the semi-open step. The apples question becomes a tree with green and red apples, a pick of 15, and a question about which combination could be in the basket.
3. Then take away the fixed total.The fully open version leaves the number out, so students can extend themselves, show their thought process and find their own entry point. As Sarah says, "It also sets students up to be successful."
4. Ask students to explain, prove or justify.Sarah's example is asking students to "prove that 367 is larger than 234". She describes one student answering with place value, another drawing it and another reaching for MAB or counters.
5. Ask about the method as well as the answer.In Sarah's words, "There's a powerful difference between great, you got the answer compared to how did you come to that answer." Follow it with "Tell me about it."

Creating Engaging and Meaningful Numeracy Lessons in the Upper Primary YearsStart this course

One question, three ways

Here's how Sarah's apples question changes at each step.

Each step keeps the same apples and gives students more room to choose. Sarah sums up the example this way: "We can see through this example how easy it can be to take a simple closed question and turn to something much more beneficial for our students learning."

The fully open version also lets every student start somewhere. Later in the course, Sarah builds a whole module on this idea of multiple entry points, where all students complete the same task with "different goal posts" and each one can feel successful.

Why the explaining matters

When students are working towards an answer, Sarah wants the questioning to focus on "how they got an answer, what was their thinking process and what their understanding of the learning actually is."

That's why "explain, prove, or justify" sits alongside the apples example. For the 367 and 234 question, Sarah gives the range of ways in: place value knowledge, a drawing, or MAB and counters. As Sarah puts it, this "allows students to demonstrate their thought process while showing the teacher the level of their understanding."

She lists the benefits for teachers too. Open-ended questions give you specific information about how students understand and apply their learning, and they show you how flexible their thinking is. Students get to practise problem solving, reasoning, critical thinking and communication along the way.

An article on open questions on the Mathematics Hub makes a related point: including them provides opportunities for students to explain their reasoning and articulate their thoughts clearly (Mathematics Hub).

How to use this in your classroom this week

• Add a "prove it" question. Try "prove that 367 is larger than 234" and let students choose their method, whether that's place value, a drawing, MAB or counters.
• Change what you say when a student gives you an answer. Ask "How did you come to that answer?" and then "Tell me about it."
• Keep your closed questions for the moments when a set answer is what you need, and watch that they don't crowd out the open ones.
• If you're a casual relief teacher, Sarah flags one task as "a great one to have up your sleeve": The answer is .45. What could the equation be? It works as a warm-up, for early finishers or as a small group task in rotations, and it can grow into a whole lesson. Change the number to suit the class you have that day.

Take it further in your maths lessons

The apples question is one small part of Sarah's approach. In Creating Engaging and Meaningful Numeracy Lessons in the Upper Primary Years, she goes much further across four modules. Inside the course, you'll find:

• her full list of open-ended questions and prompts to use in conversation with students during a lesson
• a quiz to test whether you can tell open questions from closed ones
• an activity to practise turning closed questions into open ones
• how to use multiple entry points, including the "forced entry points" to avoid and a times tables fact rebuilt as an open task
• how to link maths to students' interests and real life experiences
• a collection of lessons Sarah has used in her own classroom, built around real world situations, open-ended questions and multiple entry points

For more maths ideas across the primary years, try K-6 Numeracy Strategies and Activities. You'll find more courses for your stage in primary teacher PD.

Sarah calls herself "a big fan of open-ended questions", and she uses them often in her own numeracy lessons. The course shares her favourites from her time teaching.

HBWritten by Helen Bryant, Director of Education, ClassCover Learn. Helen has been teaching for over twelve years in both government and independent schools, across all stages from K-6.