Putting the “Problem” Back in Problem Solving

“Most, if not all, important mathematics concepts and procedures can best be

taught through problem solving.”--John Van de Walle

2009 Educators Summer Symposium

Sue McAdaragh

What is Problem Solving?

“Problem solving means engaging in a task for which the solution method is not known in advance.”

--Principles and Standards for School Mathematics

It encompasses exploring, reasoning, strategizing, estimating, conjecturing, testing, explaining, and proving.

What is a Problem?

A problem is a task that requires the learner to reason through a situation that will be challenging but not impossible.

Most often, the learner is working with a group of other students to meet the challenge.

Problem or Exercise?

An exercise is a set of number sentences intended for practice in the development of a skill.

A problem is what we commonly refer to as a “word problem.”

But beware! Problems can become exercises!!

Common Characteristics of a Good Problem It should be challenging to the

learner. The task must allow the students to treat the situations as problematic, rather than as a prescription to be followed.

Common Characteristics of a Good Problem It should hold the learner’s interest. The learner should be able to connect

the problem to her life and/or to other math problems or subjects.

It should contain a range of challenges.

It should be able to be solved in several ways.

Tasks Should Allow Students to Use Tools* Students learn to use tools by actually

using them. Tools are used for a purpose—they are

not an end in themselves. The learner must choose the tool and

discover if it was a good choice. Tools should be readily available.

*Tools are defined as things the student already knows and materials that an be used to solve problems.

What Does It Mean to Be Successful at Problem Solving? Having success means that the child

has discovered a way of thinking about mathematics that he had not experienced before he came upon this problem.

Success will involve the process of problem solving as well as understanding the content presented.

How many rectangles appear in the figure below?

Success with “How Many Rectangles” Do the students resolve the question

about whether to include the squares in their count of rectangles?

Do they understand that squares meet all the criteria to be considered a rectangle?

Do they recognize that there are many different sizes of rectangles in the drawing?

Success with “How Many Rectangles” Have the students devised a way of

counting the rectangles they find? Do they find patterns in the number

of different-sized rectangles? Do they think about the concepts

embedded in the problem differently than before?

The Teacher’s Role in Problem Solving

“The more regularly that teachers make it part of the curriculum, the more opportunities students will have to become successful problem solvers.”

--Children Are Mathematical Problem Solvers

Choosing Problem-Solving Tasks

The teacher must …make sure that the problem is

meaningful to the students. …sometimes adapt the problem to

make it more meaningful. …work the problem to anticipate

mathematical ideas and possible questions that problem might raise.

Presenting the Problem

It is better to begin with problems (not exercises) that allow students a chance to develop methods for solving those problems, and recognize that what they take away from this experience is what they have learned.

The task must offer students the chance to use skills and knowledge they already possess.

The teacher has to decide whether students will work individually or in groups.

Group Work or Individual Work? In groups, students don’t give up as quickly. Students have greater confidence in their abilities to

solve problems when working in groups. When in a group, students hear a broad range of

strategies from others. Kids enjoy working in groups! Students remember what they learn better when they

assist each other. If students are less productive, arrangements can be

made for them to work alone. There will be a heightened noise level—but conversation

is an important part of the learning process.

Once the Kids Are Working… Allow students to “wrestle” with the

problem without just telling them the answer!

If we are just telling them what to do, the students are not engaged in the process.

Finally, teachers have to determine how to assess what the students are learning and what they need to learn next.

Questions While They Work Why did you_____________?

--make a ten? --use doubles? --take that apart? --start with the tens? --change your mind?

Questions While They Work Is it OK to change your mind? Why do you think that? What is alike and different about

your answer and hers? How does this relate to…? Does that make sense? Why is that true?

Questions While They Work How did you think about the problem? Would you ask the rest of the class

that question? Will that always work? Do you see a pattern? What ideas that we have learned

before were useful in solving this problem?

Questions While They Work

Does anyone have the same answer but a different way to explain it?

How could you prove that? Can you explain his/her strategy?

Assessing Understanding

Listen in on and record the students’ conversations as they solve the problem.

Have students explain their solutions in writing.

Give them another problem that requires them to use what they learned in the first problem.

Listen carefully as they present their solutions to the class.

The Challenge…

Thinking of understandings as outcomes of solving problems rather than as concepts that we teach directly requires a fundamental change in our perceptions of teaching.

Using traditional methods, students often have trouble connecting the concepts they are learning with the procedures they are practicing.

The Challenge…

Students learn procedures by imitating and practicing rather than by understanding.

It’s hard to go back and try to understand a procedure you have practiced many times!

How Can We Start?

Begin with problems! If students develop their own

procedures for solving problems, then they must use what they already know, including the understandings they have already constructed, even if the methods they have constructed are not the most efficient.

What Else Can We Do?

Most all of the content in current curricula, as presented in popular textbooks, is appropriate as long as students are allowed to make the mathematics problematic.

Consider This Problem…

How many tiles are in the 25th figure of this pattern?

Now Consider This One…

Investigate and report all you can about this pattern.

What do you notice about this pattern?

What’s the Difference?

In the first problem, students are not encouraged to take time for reflection and sense making. They move from the task to a focus on the product.

In the second problem, students experience several points of teacher-prompted reflection as they discuss observations and generalizations.

While both invent solution strategies, only students in the second situation are asked to explore multiple strategies and analyze these strategies.

So What Should We Try to Do?

Instead of always focusing on helping students to “find the answer…”

…try to be prepared to see where the students’ observations and questions may take them. Ask questions!

What Else Could We Try?

Instead of providing solution strategies right away, try not to “rescue” the students…

…instead, try to encourage multiple approaches and allow time for communication and reflection about those strategies.

And Finally?

Instead of expecting specific responses…

…be ready to ask questions that uncover students’ thinking and press for the students’ reasoning behind the process. You might be surprised at what they come up with!

Consider this fourth grade class…

The students have already spent time reviewing addition and subtraction computation and have been working on multiplication concepts and facts.

Some students know all of their facts, some know a few.

The students have been exploring rectangles to show how repeated addition can be related to an array of squares in a rectangle.

Each child’s unique collection of ideas is connected in different ways. Some ideas are well understood, others less so. Some ideas are still emerging.

The Fourth Grade Task:

Teacher shows a 6 x 8 rectangle of “squares” on which the bottom 8 squares are shaded.

Students quickly agree that adding up 6 eights will tell how many squares are in the rectangle.

“But if we didn’t remember that 6 sets of 8 is 48, could we slice the rectangle into two smaller pieces where we know the multiplication fact and use that to get the total in our heads?”

Four ideas are offered:

Slice one row off the bottom. The top part is 5 by 8 or 40; 40 and the 8 on the bottom is 48.

We cut it in half top to bottom. Each side is 6 x 4; 24 and 24 is 48.

You can also slice it in half the other way and get 3 x 8 = 24, doubled.

If you take two columns of 6, that’s 12. the double that will give you four columns, or 24. And then double 24 is 48.

Now for the actual task…

The teacher passes out some centimeter grid paper and on the board sketches a large rectangle which she labels 24 by 8.

“I want you to sketch a rectangle on the grid paper that is 24 by 8. Your task is to figure out how many squares are in the rectangle. But since we are very lazy, I don’t want you to count the squares. Instead, slice the rectangle into two or more parts—sort of like we did with the 6 x 8 example—and use the smaller parts to figure out the whole thing. See if you can find ways that make it really easy to calculate.”

Your turn…

…see how many different ways you can think of to solve this problem (24 x 8) without using the traditional algorithm. Draw a sketch for each way you can think of.

Work with a partner and then share with another pair.

Group #1

We know that 8 times 8 is 64, so we made three squares that took up the 64. We added 64 + 64 + 64.

8

8

8 8

Group #2

We used tens. We sliced two sections of 10 and then there were 4 left at the end. Ten times 8 is 80. that makes 160 in the two big sections and then the last section is 8 x 4. We added 160 and 32 in our heads.

10

8

10 4

Group #3

Our method was sort of like that but we just used 20 times 8. Since 2 times 8 is 16, you can add a zero and get 160. then you add the 32 at the end.

Group #4

We didn’t really slice the rectangle. Instead, we added a row at the end and made it 25 by 8. then we knew that 4 rows of 25 is like a dollar, or 100. that makes 200 squares in all. But then we had to take off the 8 in the row that we added.

25 x 4 is 100. Double that is 200. 8

Which of these 21st Century Skills did you (or the students in the example) use?

Communicating•Conveying ideas graphically•Reading and writing with understanding•Speaking so others understand•Listening actively•Observing critically

Collaborating•Learning cooperatively•Valuing contributions of others•Negotiating and resolving conflict•Guiding others•Working together as a team

Analytical Thinking•Identifying similarities and differences•Using cues, questions and advance organizers•Planning•Classifying•Prioritizing

Problem Solving•Defining the problem and its variables•Generating and testing hypotheses and predictions•Summarizing and note taking•Determining relationships•Making decisions

Finding/Evaluating Information•Recognizing need for more information•Developing a strategy to find information•Using multiple sources of information•Determining credibility, reliability, accuracy and relevance

Creating and Innovating•Originality and inventiveness in work•Developing, implementing and communicating new ideas•Being open and responsive•Acting on creative ideas

Compare to a Show and Tell Method:

Teacher has kids draw a 24 by 8 rectangle on centimeter grid paper. Teacher draws a similar rectangle on the board and labels it with the multiplication problem.

Students are directed to count over 20 squares and draw a vertical line in the rectangle as the teacher demonstrates.

What is 8 x 4? (Points to small section of the rectangle)

We want to record the 32 in our problem. (The teacher demonstrates how to write 2 beneath the line in the problem and “carry” the 3. she also writes “32” in the small portion of the rectangle.)

32

3

24

x 8

2

What is 8 x 2? (Attention is directed to the 8 by 20 portion of the rectangle).

So here we are really multiplying by 20 or by 2 tens. Eight times 2 tens is 16 tens. (The teacher writes “16 tens” in the large portion of the rectangle.)

16 tens

24

x 8

2

We already have 3 more tens. How much is 16 and 3?

We record the 19 tens below the line. The final answer is 192.

16 tens

3

24

x 8

192

Consider this…

The teacher tells students how to divide the rectangle. Only one way is suggested and no reason is given.

The teacher receives no information about the ideas that individual students may have. She can only find out who has and who has not been able to follow the directions.

The age-old assumption is that those students who solve the problems correctly also understand.

However…

Many students (including some of those who do the problems correctly) will not understand and will be reinforced in their belief that mathematics is a collection of rules to be learned.

Students are not given the opportunity to found out that their own personal ideas count or that there are numerous good ways to solve the problem.

Some students become disengaged. Those who need more time to develop their understanding don’t get it and those who could easily find other solutions never get the chance.

Students are likely to use the same method to solve 4 x 51 when they should do this problem easily in their heads.

What Are Good Questions Again?

They require more than just remembering a fact or reproducing a skill.

Students learn by answering the questions and teachers learn about the students from their responses.

There may be several acceptable answers.

Here’s one…

I want to make a garden in the shape of a rectangle. I have 30 meters of fence for my garden. What might be the area of the garden?

Turn and talk: What makes this a good question?

Here are a couple more…

The average of five numbers is 6. What might the numbers be?

After five games, the goalie had averaged blocking six goals per game. What might be the number of goals he blocked in each game?

How to Create Good Questions: Method 1

Step 1: Identify a topic. Step 2: Think of a closed question

and write down the answer. Step 3: Make up a question that

includes the answer.

For example…

Step 1: the topic for tomorrow is “averages.”

Step 2: The closed question might be The children in the Smith family are aged 3, 8, 9, 10, and 15. What is their average age? (9)

For example…

Step 3: The good question could be

There are five children in the Smith family. Their average age is 9. How old might the children be?

See if you can make up a good question for each answer… Step 1: Identify a topic.

Rounding

Step 2: Think of an answer. 11.7

See if you can make up a good question for each answer… Step 3: Make up a question that

includes the answer.

My coach said that I ran 100 yards in about 12 seconds. What might the numbers on the stopwatch have been?

See if you can make up a good question for each answer… Step 1: Identify a topic.

Counting

Step 2: Think of an answer. 4 chairs

See if you can make up a good question for each answer… Make up a question that includes

the answer.

I counted something in our room. There were exactly 4. What might I have counted?

See if you can make up a good question for each answer… 1. Identify a topic.

Area

2. Think of an answer. 6 square centimeters

See if you can make up a good question for each answer… Make up a question that includes

the answer.

How many triangles can you draw each with an area of 6 square centimeters?

See if you can make up a good question for each answer… Identify a topic.

Fractions

Think of an answer. 3 and 1/2

See if you can make up a good question for each answer… Make up a question that includes

the answer.

Two numbers are multiplied to give 3 and ½. What might the numbers be?

See if you can make up a good question for each answer… Think of a topic.

Money

Think of an answer. 35 cents

See if you can make up a good question for each answer… Make up a question that includes

the answer.

I bought some things at the grocery store and got 35 cents change. What did I buy and how much did each item cost?

How to Create Good Questions: Method 2

Adapting a Standard Question

Step 1: Identify a topic. Step 2: Think of a standard question. Step 3: Adapt it to make a good

question.

For example…

Step 1: The topic for tomorrow is measuring length using nonstandard units.

Step 2: A typical exercise might be What is the length of your table measured in handspans?

Step 3: A new question could be Can you find an object that is three handspans long?

See if you can adapt the question… Identify a topic.

Space

Think of a standard question. What is a square?

See if you can adapt the question… Adapt it to make a new question.

How many things can you write about this square?

See if you can adapt the question… Identify a topic.

Addition

Think of a standard question. 337 + 456 = ____

See if you can adapt the question… Adapt it to make a new question.

I was working on my homework last night and spilled some water on my paper. Now I can’t read what I had written! Can you help?

3?7 + ??6 = 79?

See if you can adapt the question… Identify a topic.

Subtraction

Think of a standard question. 731 – 256 = ____

See if you can adapt the question… Adapt it to make a new question.

Instead of solving this problem, rearrange the numbers and subtract so that the difference is between 100 and 200.

Go Slow to Go Fast

This is the “motto” of SD Counts. We recognize that these changes

do not happen overnight.

Take your time and trust yourself as a professional!!

Sources

Children Are Mathematical Problem Solvers by Lynae E. Sakshaug, Melfried Olson, and Judith Olson, 2002.

Making Sense by James Hiebert, et. al., 1997. “Fostering Mathematical Thinking and Problem Solving: The

Teacher’s Role.” by Nicole R. Rigelman; Teaching Children Mathematics, February, 2007.

Good Questions for Math Teaching: Why Ask Them and What to Ask (K-6) by Peter Sullivan and Pat Lilburn, 2002.

Good Questions for Math Teaching: Why Ask Them and What to Ask (5-8) by Lainie Schuster and Nancy Canavan Anderson, 2005.

Teaching Student-Centered Mathematics, Grades K-3 by John Van de Walle, 2005.

Teachng Student-Centered Mathematics, Grades 3-5 by John Van de Walle, 2005.