Preparing to Engage Students in Problem Solving

Dorothy Ann AssadMathematics Department

Austin Peay State UniversityClarksville, Tennessee

Clarksville, Tennessee – 45 miles Northwest of Nashville – Adjacent to Fort Campbell, Kentucky, Home of the 101st Airborne

Turtle and Rabbit are running a race. Suppose Turtle runs at 55 ft/s. Rabbit runs at 80 ft/s, but he gives Turtle a 5-second head start. How many seconds will Turtle have run when Rabbit catches up with him?

According to the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000)

Problem solving means engaging in a task for which the solution is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will develop new mathematical understanding.

Solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with and solve complex problems that requires a significant amount of effort and should then be encouraged to reflect on their thinking. (p. 58)

Piaget – on teaching children

[The] teacher is often tempted to present far too early notions and operations in a framework that is already very formal. In this case, the procedure that would seem indispensable would be to take as the starting point the qualitative concrete levels:

In other words, the representations or models used should correspond to the natural logic of the levels of the pupils in question, and formalization should be kept for a later moment as a type of systematization of the notions already acquired.

The Problem for Mathematics Teacher Educators

Most pre-licensure K-6 teachers are not mathematically sophisticated enough to approach a problem at a less formal level.

Most of our students believe that doing mathematics is doing exercises or practicing skills or applying algorithms. This belief is based on their own school experiences which included few, if any, opportunities to engage in problem solving.

Undergraduate pre-service K-6 teachers at APSU take five mathematics courses, but they often feel uncomfortable when asked to engage in problem solving.

Students in the fast-track Master of Arts in Teaching (MAT) program are less confident about their mathematics and have few experiences with problem solving.

A Model in Development

Components of the Model

• Classroom problem solving experiences. • Investigation of the mathematics underlying

problem solutions.• Teaching a unit to a small group of elementary

students.• Designing and implementing a problem

solving lesson in a K-6 classroom.

Classroom problems solving experiences.

Students solve problems in a variety of ways with a variety of representations, share their solutions in small groups and in whole class settings, and record solutions, strategies, and ideas in a problem solving journal.

Problems must be mathematically rich. They should

• engage students in significant mathematics.• extend the content knowledge of students.• have multiple entry points, multiple solutions,

or multiple paths to the solution.

Many of the problems selected are extensions of problems designed for elementary and middle grades students.

Extension:

If I have two colors of cubes from which to choose, how many different towers 20 cubes high can I make?

If I have two colors of cubes from which to choose, how many different towers four cubes high can I make?

The Tower Problem

If I have three colors of cubes from which to choose, how many different towers four cubes high can I make?

If I have three colors of cubes from which to choose, how many different towers 20 cubes high can I make?

Pizza Problem

How many different pizzas can you make if you have four toppings (pepperoni, peppers, olives, and sausage) from which to choose? Assume that all pizzas already have tomato sauce and cheese on them. Assume that half pepperoni and half peppers is the same as pepperoni and peppers. Answer this question in as many ways as you can.

Suppose you run out of olives. How many pizzas can you make?

Suppose you had 20 different toppings, how many different pizzas could you make?

The Taxman(from Measuring Up: Prototypes for Mathematics Assessment )

There are two players, you and the Taxman. Every time it is your turn, you can take any number in the list, as long as at least some factors of that number are also in the list. You get your number, and the Taxman gets all of the factors of that number that are in the list. 1 2 3 4 5 6 7 8 9 10(The list of numbers can be extended as students gain more experience with the problem.)

Rules for Taxman

• The Taxman must get something every time.That means you can't choose a number if there aren't any factors of that number still in the list.

• When none of the numbers in the list has any factors left in the list, then the game is over, and the Taxman gets all the numbers that are left in the list.

• You win if, at the end of the game, you have more money than the taxman.

Investigation of the mathematics underlying problem solutions.

By continually extending and solving selected problems, pre-service teachers move beyond their comfort level. They begin to look at problems in terms of generalizations rather than immediate solutions.

In the tower problem and the pizza problem, students • Look for patterns, recursive as well as

generalized.• Organize information to help solve problems.• Represent the problem situation in a variety of

ways – models, tables, drawings, graphs, equations.

• Evaluate the effectiveness of various representations and strategies for solving problems.

• Recognize the similar features of the two problems.

In the Taxman problem, students

• Discover and evaluate strategies for successful solutions.

• Explore ideas from number theory, such as factors, multiples, and primes.

• Make generalizations that can be applied to large lists of numbers.

From students’ journals:

I used to feel that problem-solving activities were a waste of time because they frustrated me. Now, after the pizza problem, I realize they can be refreshing. Yes, the pizza problem was still frustrating, but it was fun. It made me aware of the steps a student must go through as he/she solves a problem.

The tower problem was one of my favorite problems. Using the Unifix cubes and creating different towers caused us to be more engaged. If it engaged graduate students, I know it will help to engage children of younger ages!...creating a table helped me to see the relationship between the number of cubes tall and the number of different towers that can be created.

Teaching a unit to a small group of elementary students.

One goal of this experience is to help pre-service teachers understand that problem solving can be integrated throughout the curriculum, that it is not an isolated experience.

Students are given specific goals (aligned with the state curriculum) and resources. They work together to solve problems, develop lessons or revise pre-written lessons, and organize materials. Lessons must involve some level of problem solving.

Students also prepare for the experience by reviewing student work samples and methods for assessing problem solving.

Two sources for student work samples:Open-Ended Assessment in Math (small fee for use) http://books.heinemann.com/math/

National Assessment of Educational Progress (NAEP) http://nces.ed.gov/nationsreportcard/itemmaps/

Alicia used a stick to measure the length of her classroom and got 20 sticks. Danny also measured the length of the classroom and got 30 feet. What can you conclude about Alicia’s stick? Why?

Suggested Scoring Rubric

0 Response indicates no appropriate mathematical reasoning.

1 Response indicates some mathematical reasoning but fails to address the item’s main mathematical ideas.

2 Response indicates substantial and appropriate mathematical reasoning but is lacking in some minor way(s).

3 Response is correct and the underlying reasoning process is appropriate and clearly communicated.

-- From Open-Ended Assessment in Math

Sample Student Responses

Advantages of teaching the unit to a small group:

• Students work cooperatively to plan and implement the unit.

• Several students working in one classroom can compare their experiences and strategies.

• Students have the opportunity to work closely with individual children, reflecting on successes and addressing individual problems.

Teresa reflects on her work with Kayla:

It seems as though Kayla has benefited from using pattern blocks during our sessions. For our next lesson, I will offer them to her (as needed) initially, but after a short while, I will remove them so that she can learn to consistently solve the given problems using paper and pencil.

Designing and implementing a problem solving lesson in a K-6 classroom.

• The whole class lesson is taught near the end of the semester.

• The lesson must be a problem solving experience, preferably based on one of the problems solved in class.

• Students must work closely with a classroom teacher. Often the classroom teacher has little confidence in the ability of children to solve problems.

Jamie posed the pizza problem to a diverse class of 21 second graders:

If you own a pizza shop and you have three different pizza toppings (pepperoni, sausage, and mushroom), how many different types of pizza can you make?

Jamie’s reflection:

I absolutely loved teaching this lesson! The children had such a good time working out the problem. They were having such great conversations pertaining to the problem and what was the best way to solve it!

I was really excited because the teacher told me at the beginning of the lesson that her children would not be able to figure the problem out, but by the end of the lesson she realized that she had not given her own students enough credit!

By the end of the semester, students have acquired more sophisticated strategies, they display more content knowledge and more confidence, and they are beginning to be able to provide similar experiences for students.

They seem to better understand the value of student engagement.

I believe these are skills that will serve them well throughout their teaching careers.

Dorothy Ann Assad

assadd@apsu.edu

http://www.apsu.edu/assadd