Beyond Right Beyond Right and Wrongand Wrong

Using Open-Ended Questioning in Using Open-Ended Questioning in the Mathematics Classroomthe Mathematics Classroom

Jeff Mahood, Colegio Bolívar

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

Traditional Math Questions:Traditional Math Questions:

Leave little room for creative thought or Leave little room for creative thought or individualityindividuality

Often test algorithms or processes, and Often test algorithms or processes, and not comprehension or communicationnot comprehension or communication

Are evaluated as “right” and “wrong,” with Are evaluated as “right” and “wrong,” with no grey area.no grey area.

Are we teaching processes, Are we teaching processes, or comprehension?or comprehension?

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

What does “Open-Ended” What does “Open-Ended” mean?mean?

A A closed-ended questionclosed-ended question has a pre- has a pre-determined and expected answer.determined and expected answer.

In contrast, In contrast, open-ended questionsopen-ended questions allow a variety of correct responses and allow a variety of correct responses and elicit a different kind of student thinking. elicit a different kind of student thinking.

What?What?

In a nutshell, open-ended questions ask In a nutshell, open-ended questions ask students to explain a concept, the use of students to explain a concept, the use of a process, or a relationship in full a process, or a relationship in full sentence and/or paragraph form.sentence and/or paragraph form.

The explanation should be stand-alone. ie: The explanation should be stand-alone. ie: the PB&J example.the PB&J example.

Characteristics of Characteristics of Open-Ended QuestionsOpen-Ended Questions

1.1. Involve Significant MathematicsInvolve Significant Mathematics

2.2. Elicit a Range of ResponsesElicit a Range of Responses

3.3. Require CommunicationRequire Communication

4.4. Be Clearly StatedBe Clearly Stated

5.5. Lend Itself to a Scoring Rubric Lend Itself to a Scoring Rubric

1. Involve Significant 1. Involve Significant Mathematics Mathematics

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

2. 2. Elicit a Range of Elicit a Range of Responses Responses

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

3. 3. Require Require Communication Communication

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

4. Be Clearly Stated4. Be Clearly Stated

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

5. Lend Itself to a Scoring 5. Lend Itself to a Scoring RubricRubric

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

Some ExamplesSome ExamplesClosed QuestionsClosed Questions Which of the following Which of the following

numbers are prime? numbers are prime? 7,57,67,1177,57,67,117

What are the next three What are the next three numbers in the numbers in the sequence 1,4,7,10,13,…sequence 1,4,7,10,13,…

Round 37.67 to one Round 37.67 to one decimal placedecimal place

Open QuestionsOpen Questions Pablo says that 57 and Pablo says that 57 and

67 are prime because 67 are prime because they both end in 7. Juan they both end in 7. Juan thinks he is wrong. Who thinks he is wrong. Who is correct and why?is correct and why?

Consider the sequence Consider the sequence 1,4,7,10,13,… Is 100 1,4,7,10,13,… Is 100 part of this sequence? part of this sequence? Explain how you know.Explain how you know.

Give three numbers that Give three numbers that give 37.7 when you give 37.7 when you round to one decimal round to one decimal place.place.

Creating Open-Ended Creating Open-Ended Questions:Questions:

1.1. Change a Closed-Ended Question Change a Closed-Ended Question

2.2. Ask Students to Create an Example Ask Students to Create an Example

3.3. Ask Students Who is Correct and Why Ask Students Who is Correct and Why

4.4. Ask Students to Explain in Multiple Ask Students to Explain in Multiple Ways Ways

1. Change a Closed-1. Change a Closed-Ended QuestionEnded Question

Take a traditional question (things like solving Take a traditional question (things like solving equations and performing other algorithms are equations and performing other algorithms are not usually appropriate) and modify them to not usually appropriate) and modify them to ask a more broad-reaching question.ask a more broad-reaching question.

ie: What is the greatest common factor of 24 and ie: What is the greatest common factor of 24 and 36?36?changes to:changes to:

Why can’t 10 be the greatest common factor Why can’t 10 be the greatest common factor of 24 and 36?of 24 and 36?

2. Ask Students to Create 2. Ask Students to Create an Examplean Example

Creating a question or an example requires a student to Creating a question or an example requires a student to completely understand the ideas underlying a concept. completely understand the ideas underlying a concept. Students must be able to apply their knowledge to a Students must be able to apply their knowledge to a situation.situation.

ie:ie:Make a 4-digit even number using the digits below. Make a 4-digit even number using the digits below. Explain why your number is even.Explain why your number is even.3 6 7 1 53 6 7 1 5

ie:ie:Make an irrational number whose square is smaller Make an irrational number whose square is smaller than itself. Explain why your number fits the criteria or than itself. Explain why your number fits the criteria or argue that such a number does not exist.argue that such a number does not exist.

3. Ask Students who is 3. Ask Students who is correct and whycorrect and why

Explaining who is correct in a given situation allows Explaining who is correct in a given situation allows students to defend an answer as well as demonstrate a students to defend an answer as well as demonstrate a thorough understanding of an underlying concept.thorough understanding of an underlying concept.

ie:ie:Anita and Mariana are trying to decide how to write 5 Anita and Mariana are trying to decide how to write 5 cents as a decimal. Anita thinks it is $0.5 and Mariana cents as a decimal. Anita thinks it is $0.5 and Mariana thinks it is $0.05. Who is right and why?thinks it is $0.05. Who is right and why?

ie:ie:Felipe calculated the tangent and sine of a certain Felipe calculated the tangent and sine of a certain angle and said that the tangent was less than the sine. angle and said that the tangent was less than the sine. Esteban said that this was impossible. Who is correct Esteban said that this was impossible. Who is correct and why?and why?

4. 4. Ask Students to Ask Students to Explain in Multiple Ways Explain in Multiple Ways

This is a more difficult method with which to have success. At 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,” 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 and it is often hard to get students to find a second method because, “I already have one that works.”because, “I already have one that works.”

ie: Describe two transformations that map a square onto itself.ie: Describe two transformations that map a square onto itself.

ie:ie: Using the following names of polygons, write three true Using the following names of polygons, write three true statements of the form, “All _____ are _____.” statements of the form, “All _____ are _____.” Kites, Parallelograms, Quadrilaterals, Rectangles, Rhombi, Kites, Parallelograms, Quadrilaterals, Rectangles, Rhombi, Squares, TrapezoidsSquares, Trapezoids

How Can I Evaluate How Can I Evaluate Them?Them?

Evaluation is going to be different depending on Evaluation is going to be different depending on grade and subject. In high school math, I use:grade and subject. In high school math, I use:

JournalsJournals Specific Test QuestionsSpecific Test Questions Performance TasksPerformance Tasks Homework QuestionsHomework Questions

Remember – you can use Open-Ended Remember – you can use Open-Ended Questions in both Assessment Questions in both Assessment andand Evaluation. Evaluation.

Scoring Student AnswersScoring Student Answers

A general Rubric for these kinds of written responses is as follows:A general Rubric for these kinds of written responses is as follows:

A: Response is correct, the underlying MR is appropriate, and A: Response is correct, the underlying MR is appropriate, and connections may be made to different mathematical conceptsconnections may be made to different mathematical concepts

B: Response is correct and the underlying MR is appropriateB: Response is correct and the underlying MR is appropriate C: Response is substantial in terms of MR, but is lacking some C: Response is substantial in terms of MR, but is lacking some

minor detailsminor details D: There is some evidence of MR but no addressing of main D: There is some evidence of MR but no addressing of main

mathematical ideasmathematical ideas F: Response indicates no mathematical reasoning (MR)F: Response indicates no mathematical reasoning (MR)

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

A Final ThoughtA Final Thought

Students can not be expected to see an open-Students can not be expected to see an open-ended question and answer it if they have not ended question and answer it if they have not been trained to do so.been trained to do so.

Before using these questions as evaluations, be Before using these questions as evaluations, be sure to give students instruction and practice sure to give students instruction and practice assessments during class time. Discuss what assessments during class time. Discuss what makes a good response with them as a group.makes a good response with them as a group.