Beyond Right and Wrong Using Open-Ended Questioning in the Mathematics Classroom Jeff Mahood, Colegio Bolívar

“I can explain it for you, but I can’t understand it for you.”

Traditional Math Questions:  Leave little room for creative thought or individuality  Often test algorithms or processes, and not comprehension or communication  Are evaluated as “right” and “wrong,” with no grey area.

Are we teaching processes, or comprehension? Traditional math evaluation holds that an answer is either right or wrong, with no middle ground. However, teachers already evaluate some questions using partial credit. Black and white marking determines whether an answer is correct – not whether the student understands the process or the concept.

What does “Open-Ended” mean?  A closed-ended question has a pre- determined and expected answer.  In contrast, open-ended questions allow a variety of correct responses and elicit a different kind of student thinking.

What? In a nutshell, open-ended questions ask students to explain a concept, the use of a process, or a relationship in full sentence and/or paragraph form. The explanation should be stand-alone. ie: the PB&J example.

Characteristics of Open-Ended Questions 1.Involve Significant Mathematics 2.Elicit a Range of Responses 3.Require Communication 4.Be Clearly Stated 5.Lend Itself to a Scoring Rubric

1. Involve Significant Mathematics The math that is being evaluated must be important. Open-ended questions often have several objectives, allowing students to demonstrate an understanding of connections in math.

2. Elicit a Range of Responses Items that require a student to explain their thinking are likely to encourage different answers, because not all students think alike.

3. Require Communication When students are required to communicate their reasoning process, we have a better chance to see what they know and whether they can apply it to a given problem

4. Be Clearly Stated Questions should have a clear purpose even though there may be many possible responses, and students should know what is respected of them.

5. Lend Itself to a Scoring Rubric The purpose is to give students the chance to communicate understanding in a scenario other than right/wrong. It should therefore be possible to think of a response that is worth some credit.

Some Examples Closed Questions  Which of the following numbers are prime? 7,57,67,117  What are the next three numbers in the sequence 1,4,7,10,13,…  Round to one decimal place Open Questions  Pablo says that 57 and 67 are prime because they both end in 7. Juan thinks he is wrong. Who is correct and why?  Consider the sequence 1,4,7,10,13,… Is 100 part of this sequence? Explain how you know.  Give three numbers that give 37.7 when you round to one decimal place.

Creating Open-Ended Questions: 1.Change a Closed-Ended Question 2.Ask Students to Create an Example 3.Ask Students Who is Correct and Why 4.Ask Students to Explain in Multiple Ways

1. Change a Closed-Ended Question Take a traditional question (things like solving equations and performing other algorithms are not usually appropriate) and modify them to ask a more broad-reaching question. ie: What is the greatest common factor of 24 and 36? changes to: Why can’t 10 be the greatest common factor of 24 and 36? Why can’t 10 be the greatest common factor of 24 and 36?

2. Ask Students to Create an Example Creating a question or an example requires a student to completely understand the ideas underlying a concept. Students must be able to apply their knowledge to a situation. ie:Make a 4-digit even number using the digits below. Explain why your number is even ie:Make an irrational number whose square is smaller than itself. Explain why your number fits the criteria or argue that such a number does not exist.

3. Ask Students who is correct and why Explaining who is correct in a given situation allows students to defend an answer as well as demonstrate a thorough understanding of an underlying concept. ie:Anita and Mariana are trying to decide how to write 5 cents as a decimal. Anita thinks it is $0.5 and Mariana thinks it is $0.05. Who is right and why? ie:Felipe calculated the tangent and sine of a certain angle and said that the tangent was less than the sine. Esteban said that this was impossible. Who is correct and why?

4. Ask Students to Explain in Multiple Ways This is a more difficult method with which to have success. At primary ages it is difficult for a student to “think outside the box,” and it is often hard to get students to find a second method because, “I already have one that works.” ie: Describe two transformations that map a square onto itself. ie:Using the following names of polygons, write three true statements of the form, “All _____ are _____.” Kites, Parallelograms, Quadrilaterals, Rectangles, Rhombi, Squares, Trapezoids

How Can I Evaluate Them? Evaluation is going to be different depending on grade and subject. In high school math, I use:  Journals  Specific Test Questions  Performance Tasks  Homework Questions Remember – you can use Open-Ended Questions in both Assessment and Evaluation.

Scoring Student Answers A general Rubric for these kinds of written responses is as follows:  A: Response is correct, the underlying MR is appropriate, and connections may be made to different mathematical concepts  B: Response is correct and the underlying MR is appropriate  C: Response is substantial in terms of MR, but is lacking some minor details  D: There is some evidence of MR but no addressing of main mathematical ideas  F: Response indicates no mathematical reasoning (MR)

“Tell me and I forget. Teach me and I remember. Involve me and I understand.”

A Final Thought Students can not be expected to see an open- ended question and answer it if they have not been trained to do so. Before using these questions as evaluations, be sure to give students instruction and practice assessments during class time. Discuss what makes a good response with them as a group.