Open Ended Questions For Mathematics Answers

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As educators, we want to ask our students questions that are intentional and meaningful in order to uncover their thinking and understanding. Analyzing and examining their responses sheds light on their prior knowledge, what they have learned, what they are thinking, and how they have applied that learning

Asking open ended questions is one such method in which to determine how and what our students are thinking when solving a problem or answering a question. Open ended questions can and should be asked in every subject area from history to science and physical education to mathematics. (see my blog post from 2015 here, https://bit.ly/39zyi5T)

An open ended question is one in which there is no one correct answer; there are multiple right answers depending on the way in which you approach the problem. This is different from an open-routed question , which has one correct answer but many ways to get there. In the last few years, there has been a lot of emphasis and importance placed on open ended questions because they are designed in such a way as to uncover the level of student understanding (e.g., procedural vs. conceptual) and misconceptions of mathematical concepts. However, there is still a place for open routed questions and they are just as important as open ended questions; both should be a part of every mathematics classroom.

Let me give you an example. When I first moved into my condo, there were no window coverings on the door that led from my kitchen to the balcony. I took some measurements, ordered the blinds and made an appointment for the installer to come in and put them up. I still remember the afternoon he came to put them up. I was in the kitchen washing some dishes and as he was putting the blinds in place, I could hear him muttering under his breath so I asked him if everything was okay. He turned to me and said matter of factly, "Ya everything is great. I've been doing this for two years and this is the first time I didn't have to take the blinds back."

I was confused and asked him to clarify. He said that I was the first customer to get the measurements correct. I was surprised to say the least. But he explained that people didn't know how to take measurements accurately and he always ended up taking the blinds with him and having the customers reorder their blinds. We had a good laugh when I told him that I was happy I got the measurements correct since I taught middle school math!

This is a great example of when we need to have one correct answer! Can you think of any others??

There is also value in open ended questions. Not only do student answers reveal what they know and how they think, but it also promotes equity in that there are multiple entry points for students. In other words, all students can participate and answer the question to the best of their ability, knowledge and understanding, therefore, ensuring that students who struggle in mathematics can feel successful. This also allows students to demonstrate a growth mindset in mathematics; they begin to see themselves as mathematicians.

I taught mathematics for many years and used both open routed and open ended questions in my classroom. As I gained more experience with open ended questions, I discovered that there were at least seven categories of open ended questions and I'd like to share them with you here with examples. These are from a workshop I delivered to teachers three years ago.

Category #1: Provide the Answer

CLOSED: Round 37.67 to the nearest tenth

OPEN: Generate three different numbers that when rounded to the nearest tenth give you 37.7

Category #2: Similarities and Differences

CLOSED: Find the LCM of 40 and 35

OPEN: a) How are the numbers 40 and 35 alike? How are they different?

b) How are x 2 and x 3 alike? How are they different?

Category #3: Create a Context/Omit Information

CLOSED: What is 2.5 times the number 12?

OPEN: One number is 2.5 times as much as another number. What could the number be?

Category #4: Provide Conditions

CLOSED: find the mean of the following numbers: 7, 15, 12, 20, 17, 13, 19, 16.

OPEN: produce a set of data that satisfy these conditions:

Mean = median < mode
Mean = mode < median
Median = mode < mean

Category #5: Explanations

CLOSED: Which fraction is bigger – ⅔ or 3/7?

OPEN: Compare two fractions with different denominators. Which one is bigger? How do you know?

Category #6: Create a Question/Sentence

CLOSED: Multiply 1.9 x 5

OPEN: a) Create a sentence with the numbers 2 and 9 and the words "and" and "less"

b) Create a question involving the multiplication of decimals where the digits 1, 5, and 9 appear somewhere

Category #7: Using "soft" words

CLOSED: Create a rectangle that has an area of 28 square inches

OPEN: a) Create two different rectangles whose area is close but different.

b) When you add two numbers, the answer is close to 300. What could

the number be?

Here are some more closed ended questions – how can you turn them into open ended questions?

What is 3 + 4?
Order these fractions from least to greatest: ⅖, ⅓, ⅚, 4/10, 4/7, ⅕
There are 13 strawberries in a bowl and some are on a plate. Altogether, there are 32 strawberries. How many strawberries are on the plate?

As you can see, there are many ways to create open ended questions, which will allow you to assess and evaluate what your students know and the depth of their understanding of a variety of mathematical concepts (e.g., number sense, geometry, measurement, etc).

Good math questions have some of the following elements:

They are open ended
They go beyond computation
They assess skills (e.g., critical thinking, communication, judging, etc)
They start with what students already know
They include more than one mathematical concept

How are you using open ended questions in your mathematics classroom? How are students responding? How are they showing growth? What are you noticing in terms of their attitudes towards mathematics?