Over the last few years, there has been a shift in mathematics instruction and learning as we move away from the more traditional forms of teaching (e.g., using textbooks, sitting in rows, and unit tests at the end of a chapter) to more engaging and innovative methods in order for students to understand the big ideas and concepts of mathematics. One such strategy is the use of open ended questions in the math classroom. As an AQ Instructor, teachers in my courses often express the value in having students work through these types of problems. Open ended problems have many advantages. The first being that these types of problems allow for multiple entry points as well as students recognizing that there isn’t one conventional method of solving a problem. Students can work on the problem with strategies that make sense to them. They realize there isn’t only one way to solve a problem and hence their self confidence increases because of their “aha” moments. I struggle with the issue of the most “efficient” method or the strategy that is the “fastest”. The entire objective of open ended questions is allowing students to recognize that their way of solving a problem is valued. Not every student’s brain works the same way and that in itself needs to be celebrated. Just because it’s the most efficient or fastest way does not mean it will work for every student. It’s about the understanding and learning; its not about the achievement, speed, or performance. We want students to succeed in life and by allowing them to solve problems that make the best sense to them will allow them to transfer those skills to real life, when they are adults. You could pair up the student who is using the “more efficient” strategy with the students who is using the “less efficient” strategy. Give the pair some guiding questions to keep the conversation focused, have both of them reflect on their conversation in a journal, and then have a follow up conversation with the student who is using the “less efficient” strategy and ask them about what they though of the “more efficient” strategy – does it make sense? why or why not? If a student is not ready to use other strategies, don’t force it. It needs to come naturally when they are ready.
The biggest challenge in terms of using open ended questions is creating these questions in the first place. Marian Small’s books are a great place to start to gain some knowledge and experience with these types of questions. After reading her books, I then turned to the math textbook and started turning their close ended questions into open ended questions. After gaining experience and understanding with open ended questions over the last couple of years, I created the following mini-guide:
Open Ended Questions in Math: Four Main Categories
|Provide Conditions||Find the mean of the following numbers: 7, 15, 12, 20, 17, 13, 19, 16||Produce sets of data that satisfy these conditions.
a) mean = median < mode
b) mean = mode < median
c) median = mode < mean
|Give the Answer||Round 37.67 to the nearest tenth||Generate three different numbers that when rounded to the nearest tenth give 37.7|
|Similarities and Differences||Find the LCM of 40 and 35||How are the numbers 40 and 35 alike? How are they different?|
|Create the Context/Omit Information||What is 1/3 + ½?||You add two fractions and the sum is 5/9. What could the fractions be?|
Analyzing the Task
- What prior learning does the student need to understand and respond to each question
- What scaffolding can we provide struggling students to engage in the problem?
- What scaffolding can we provide competent students to go deeper into the problem
Points to Ponder
- When would you engage students in these types of questions (beginning, middle, end of unit; after closed problems)?
- How would you use these types of questions (assessment of, for, or as learning)?
- What type of classroom environment is needed to promote these types of questions and this type of learning?
It is also helpful to know the difference between these two terms:
Open Routed: A question with more than one strategy to obtain one correct answer
Open Ended: A question or problem which has more than one correct answer and more than one strategy to obtain this answer
- WHAT? Open-ended and parallel tasks allow for multiple entry points and multiple answers
- WHY? Questions engage learners at all levels
- HOW? Allow students to choose numbers, the context, generate the question
- WHO? All students as parallel tasks allow for the removal of potential barriers
Example for you to try
Create a question that uses the words double, triple, 3 and 8.