Enriching Mathematics Instruction

Opening up our Questions

TLQP 2013-14

Thomas F. Sweeney, Ph.DThe Sage Colleges

Today’s Goals:

•Classify open and closed problems and tasks;

•Understand how open ended problems promote student learning;

•Generate open-ended tasks from closed ones;

•Learn the importance of good questioning.

CCSSM New level of rigorDeeper understanding

A broadening in our teaching styles

APPR is attempting to measure this

Promotes higher level thinkingFacilitates productive discussion in the classroomStudents are better able to make sense of ideas, demonstrate understanding, and reflect on their thinking.

Why are the questions we ask our students so important?

Transition to: students working in small groups to solve open-ended mathematics problems

The problem tasks should provide:

(i) opportunities for students to generate several options and solutions;(ii) opportunities for students to discuss together; (iii) opportunities for students to make decisions and justify their decisions.

Successful implementation of CCSSM

Recall: Border Problem

Traditionally “closed” questions : •involve recalling a fact or performing a skill;

•a single answer;

•content specific - often routine;

•a prescribed solution technique.

Features of Open-ended Problems:

• No fixed method

• No fixed answer/many possible answers

• Solved in different ways and on different levels (accessible to mixed abilities).

• Offer pupils room for own decision making and natural mathematical way of thinking.

• Develop reasoning & communication skills

Open-ended task/ problem

They can include tasks that ask students to explain, justify, and make predictions beyond the information they are given.

Good Questions Are Self-Differentiating

Many questions worth asking can be answered on various levels according to a student's conceptual understanding. Thus, they are self-differentiating. Students with stronger skills or a deeper understanding can probe the question in depth or uncover all the possible answers, while other students might respond on a more basic level.

The teacher also can prepare one or more follow-up questions to extend the challenge.

Examples

Closed question: "What are the factors of 36?“

Open question: "Find a number under 100 that has lots of factors."

Closed question:

Find the mean of 8, 10, 12, 12, and 18.

corresponding Open-ended question :

The mean of a set of 5 scores is 12. 'What might be the scores?

Closed:What is the area of a rectangle 8 meters long and 3 meters wide?

Open: A rectangle has a perimeter of 30 m. What might be the area? Explain.

Closed- EndedFind the value of n if 3 x n = 12

Open-Ended

Solve the riddle using the clues and numbers in the table.

1. It is not 3 x 4. 2. It is not > 56.3. It does not equal 4 tens.4. It does not equal 2 x 7.5. It is not the missing

number in 3 x n = 12.

The number is ______.

40 4

14 12

59 37

I want to paint the wall but not the door. What area of wall do I need to paint?

The area of the shaded part of this diagram is 8m2• What might be the dimensions of the large and small rectangles (Give at least 3 answers)

closed

open

Closed question:

What is 8 x 6?

Open question: Suppose you forgot what 8 x 6 is, but

you remembered that 5 x 6 is 30. How could you use this fact to figure out what 8 x 6 is?

Open-ended questions help students relate math to the real world.

Students are often required to make decisions and be creative in finding ways to answer the questions they are given.

When teachers can assess all these things they can be sure that the students really understand the mathematics they are being taught.

Why open-ended tasks?

Students might be asked to bring together and synthesize a range of mathematical concepts and skills in solving one problem.

The question might trigger a student-driven investigation or exploration.

The teacher learns about the students' conceptual understanding and problem-solving skills.

Why open-ended tasks?

1). Students participate more actively in lessons and express their Ideas more frequently.

2). Students have more opportunities to make comprehensive use of their mathematical knowledge and skills.

3). Every student can respond to the problem in some significant ways of his / her own.

4) Teachers can better assess students understanding and misconceptions.

Advantages

What Makes a Task Rich?• Significant mathematics• Mathematical Practices• Multiple layers of complexity• Multiple entry points• Multiple solutions and/or strategies• Leads to discussion or other questions• Students are the workers and the decision

makers• Warrants reflection - Paired with discourse

Making questions open ended• Method 1: Working backwards

1. Indentify a mathematical topic or concept.

2. Think of a closed question and write down the answer.

3. Make up a new question that includes (or addresses) the answer.

Method 1: Working backwards

Example: 1. Multiplication

2. 40 x 9 = 360

3. Two whole numbers multiply to make 360. What might the two numbers be?

Round this decimal to one decimal place: 5.7347

can become …. A number has been rounded off to 5.8. What might the number be?

Method 1: Working backwards

Method 1: Working backwards

Find the difference between 6 and 1

can become ….

Method 1: Working backwards

The difference between two numbers is 5.

What might the two numbers be?

Measurement Example• Outcome – Recognize and demonstrate that objects

of the same area can have different perimeters.

• Typical Question (closed task, no choice):Build each of the following shapes with your color

tiles. Find the perimeter of each shape.

Which shape has the greatest perimeter?

Measurement Example (continued)New Task (open, choice in number of tiles):

Using 8, 16, or 20 color tiles create different shapes and determine the perimeter of each. Record your findings on grid paper. – What do you think is the smallest perimeter you can make?– What do you think is the greatest perimeter you can make?– Prepare a poster presentation to show your results.– Sides of squares must match up exactly.

Allowed Not Allowed

Compute .

Your turn. Open up the following.

Method 2: Adapting a standard question

Indentify a mathematical topic or concept.

Think of a standard question

Adapt it to make an open ended question.

Making questions open ended

What is the time shown on this clock?Can become…

My friend was sitting in class and she looked up at the clock. What time might it have shown? Show this time on a clock

Method 2: Adapting a standard question

Method 2: Adapting a standard question

731 – 256 =

Can become…

Arrange the digits so that the difference is between 100 and 200.

Ten birds were in a tree. Six flew away.

How many were left?

Can become…

Method 2: Adapting a standard question

Method 2: Adapting a standard question

Ten birds were in a tree. Some flew

away. How many flew away and how

many were left in the tree?

Method 2: Adapting a standard question

Your turn. Open up the following.

What are the next three numbers in the following sequence? 1, 4, 7, 10, 13, ___, ____, ____

Method 3:

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.

Sample Elementary-Level Questions Make a 4-digit even number using the digits below. Explain why your number is even. 3 6 7 1 5 _____ _____ _____ _____

Give an example of an event that has a probability of 0. Explain how you know the probability is 0.

Draw a rectangle and label the sides so that the perimeter is between 19 and 20 units. Explain how you know the perimeter is greater than 19 and less than 20.

Draw a triangle that has the line of symmetry below.

Sample Middle School-Level QuestionsIdentify three numbers whose greatest common factor is 5 and whose least common multiple is 180. Describe how you found the numbers.

Create a set of data that would satisfy the following conditions:

The set includes 7 data points.The range is 10 units.The mean is greater than the median.

Show that your data set satisfies the conditions.

Fill in values for a and b to make the equation below true. Explain why your equation is true.

Draw a quadrilateral ABCD that has one and only one line of symmetry. Explain why your quadrilateral satisfies the given condition.

Ask Students to Explain Who Is Correct and Why

Method 4:

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.

Of the coins made by the U.S. Mint in one year, 73% were pennies and 6% were quarters. Suppose you could have all of the coins of one type. Alex says you would get more money if you had all the pennies. Austin says you would get more money if you had all the quarters. Jenna says it depends on how many of the two coins were made. Who is right and why?

Sample Elementary-Level Question

Sample Middle School-Level Questions

Casey claims he has divided the rectangle below into four equal areas. Terrell disagrees. Who is correct and why?

Creating Open Ended Questionsfrom Closed Ended Questions

But first . . .

When you create open-ended items, make sure they are really different from traditional items. For example, the following items is really no different than simply asking students to solve the equation:

Johnny solved 2x + 4 = 8 and got 2. Susie solved the equation and got 6. Who is correct and why?

Rani said the solution to the inequality (x - 2)(x - 4)< 0 is 2<x<4.

Susie solved the inequality and got x<2 or x>4. Who is correct and why?

The created item should require students to explain their reasoning, not simply to reproduce an algorithm.

Open up the questions on the handout ;

or

Create several open questions of your own.

Probably best to work by grade level

Some important considerations

•The mathematical focus

•The clarity of the task / question

•That it is open ended

Thoughts from Teachers

Teacher X: “A lesson such as an open-ended problem solving was indeed a fruitful one for my pupils and me. There was another avenue to motivate pupils to learn mathematics. ….When the pupils gotstarted to work in pairs, I could see that they were thinking very hard as to how the work could bedone. I heard comments like’ How can a person be so heavy?’ and ‘It is not possible to have so manypassengers on a bus?’. I was glad that they tried to explain and justify to their peers…” (primary four)

Singapore Teacher Comments

Teacher Y: “From my pupils’ work, I feel that critical thinking can be taught. These pupils werefrightened at first to take the first step, for fear of getting the answers wrong. But after seeing theirfriends’ answers and reasons, these fears were lifted…..Lastly, the pupils do include their ownpersonal experiences and knowledge. I was quite impressed.” (primary four)

Teacher Z: “ ….Another advantage of open-ended investigation is that learning is more active.Pupils had a great time making wild guesses, trying to defend their procedure. However, our pupils are still not accustomed to accepting other pupils’ alternative answers, it will take them some time to recognize the fact that there are sometimes more than one solution to a problem…… Open-ended investigations here no doubt require more preparation and time on the part of the teachers, but judging from the positive learning outcome of the activity carried out from most of the pupils, the pupils will begin to see mathematics can be meaningful and interesting. (primary six)

What do you think?• When: When would you provide students

opportunities to engage in open or parallel questions? – near the beginning or end of a lesson or unit?– before or after closed problems?

• How: How would you use open and parallel tasks?– assessment for, as or of learning?– in groups or individually?

• Conditions: What classroom conditions are needed to be to be successful engaging in open and parallel tasks? How would you build this capacity?

Building open ended tasks into a lesson

It is important to plan two further questions/ prompts:

•For those children who are unable to start working (enabling prompts).

•For those children who finish quickly (extending prompts).

Why are the questions we ask our students so important?

• Promotes higher level thinking• Facilitates productive discussion in

the classroom• Students are better able to make

sense of ideas, demonstrate understanding, and reflect on their thinking.

Generic Set of Questions?

• Why do you think that?• How did you know to try that strategy?• How do you know you have an answer?• Will this work for every number, every

situation?• When will this strategy not work? Can you

provide me with a counterexample?

Generic Set of Questions?

• Jake, do you have a different strategy?• How is your answer alike or different from

Alyson’s?• Can you repeat Brian’s idea in your own

words?• Do you agree or disagree with Courtney’s

idea? Why?

Answering and Asking Questions – Students Asking

• What are some things that you (the teacher) can do when asked a question other than directly answering it?

• Think of ways that you can help students answer their own questions…

Answering and Asking Questions – Students Asking

• 1. Repeat the question, paraphrasing it…

• 2. Redirect the question

• 3. Ask probing questions

• 4. Promote a discussion among the students

• Plan questions in lesson design•Choose a variety of questions•Video some lessons to assess level of questioning•Focus questions on student understanding; remove focus from right/wrong answers.•Assume all answers are meaningful.•Allow multiple opportunities for interaction centered around math ideas: questioning, discussion, and reflection.•Increase wait time.

Strategies to Improve Questioning

Pauses and Silence

• 1. Wait, pauses and silence are appropriate class behaviors…

• 2. Wait, give the students time to think…

• 3. Wait, or you will establish an undesirable norm…

References:

Designing open-ended question Louise Hodgson 2012http://pt.slideshare.net/evat71/designing-quality-open-ended-tasks