Closed- & Open-Ended Problem Solving
What’s the difference?
Close-ended: Many of the questions we traditionally ask students call for a single number, figure, or mathematical object. These kinds of questions are closed-ended because the expected answers are predetermined and specific. There are many examples of closed-ended problems online. These can be very valuable practice, and excellent homework support.
Examples of closed-ended questions:
1) Jennie wanted to buy some flowers for her mom. Each flower costs $3. If Jennie buys 15 flowers, how much money will she have spent?
2) The total number of people at Monday’s football game was
50,000 + 9,000 + 300 + 1. What is this number, written in standard form?
In contrast, Open-ended questions allow a variety of correct responses and elicit a different kind of student thinking. The open-ended nature of the question allows students to demonstrate their own ways of solving the problem.
Examples of open-ended questions:
Type 1: Ask Students to Explain Who Is Correct and Why: These types of items present two or more views of some mathematical concept or principle and the student has to decide which of the positions is correct and why. Example: Ian says that when you find the sum, you have a lot of choices for a common denominator. Frank says there is only one choice for the common denominator. Who is correct and why?
Type 2: Ask Students to Create a Situation or an Example That Satisfies Certain Conditions Questions of this type require students to recognize the defining characteristics of the underlying concept. Students must take what they know about a concept and apply it to create an example. Example: At Friday’s game, there were more people than there were at Monday’s game and fewer people than at Saturday’s game. How many people could have been at Friday’s game? Explain how you know your answer is right.
I’d like some more sample problems!
There are many math problems online - and many are close-ended. You can easily convert a close-ended problem into an open ended problem (and visa versa). Both are excellent practice, and we use both open- and close-ended problems in our math classroom.
Original Closed-Ended Item | Revised Open-Ended Item |
Find the Lowest Common Multiple of 18 and 24. | Why can't 48 be the LCM of 18 and 24? |
What are the next three numbers in the following sequence? 1, 4, 7, 10, 13, ___, ____, ____ | Consider the following sequence: 1, 4, 7, 10, 13, ... . Is 100 a member of this sequence? Explain your reasoning. |
Round 37.67 to the nearest 10th. | Generate three different numbers that when rounded to the nearest 10th give 37.7. |
The following sites have great close-ended word problems organized by grade or strand
http://ca.ixl.com/math/grade-4 http://math.about.com/od/wordproblem1/Worksheets_for_Word_Problems_Various_grades.htm
http://www.dadsworksheets.com/v1/Worksheets/Word%20Problems.html
http://www.mathplayground.com/wpdatabase/wpindex.html
Some open-ended (others closed):
http://math.wiki.cvsd.k12.pa.us/Open-Ended+Questions
