# MiddleSchoolPortal/Big vs Little Problems: What Is Problem Solving

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### Big vs Little Problems: What Is Problem Solving? - Introduction

Problem solving is at the heart of today's goal of teaching and learning mathematics for understanding. It is the first of the five process standards addressed in the NCTM's Principals and Standards for School Mathematics (NCTM, 2000). More than just how to solve word problems, a problem-solving approach can be applied to mathematics found in either big multiday interdisciplinary projects or, more typically, to small problems that require deep thought.

In this publication, we focus on those problems—big and small—that encourage students to think, ask questions, try a variety of problem-solving approaches, and discuss their strategies and solutions. Built on the ideas found in the Standards, we feature in the Background section pedagogical examples and resources to expand teachers' understanding of mathematics and problem solving. Through the selected resources in the Activities sections, we recognize that there is a vast difference between numerical problems dressed up as episodes of everyday life and real mathematics used every day to facilitate understanding and decision making. The Standards document stresses that problem solving should not be taught in isolation, but rather used as an approach to develop concepts in all of the five mathematics learning strands. The problems and activities suggested here are based on the belief that "an item is an exercise (not a problem) if a learner knows exactly how to approach it." (Rubenstein, Beckmann, & Thompson, 2004, p. 18)

## Contents

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### Teacher Background

"Teachers' actions are what encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions." (NCTM, 2000, p.18)

Problem solving and creative thinking come naturally in some settings, but sometimes not so naturally in math class. We selected each resource below to help teachers broaden their understanding of problem solving and to facilitate using a problem-solving approach with students. The first set of resources illustrates actual teaching in the spirit of the NCTM Standards, where problem solving, reasoning and proof, communications, connections, and representation are the essence of mathematics learning. The second set features examples of teaching strategies that support problem solving.

Background for teaching with a Standards-based approach

Private Universe Project in Mathematics: Workshop 4: "Thinking Like a Mathematician" This 25-minute video offers a peek into the lives of two professional mathematicians as one engages in real-world problem solving, and the second turns the familiar Tower of Hanoi puzzle into a mathematics problem-solving investigation with a sixth grade class. This is a must-see!

Petals Around the Rose Game This simple number game played with dice can be used as a student problem solving activity. The resource below features ideas about how to set up a critical thinking activity that is accessible to all students, regardless of ability. Students can expand the activity by sharing the game with families and friends. Check out this online resource below to get the thinking and fun started! Petals Around the Rose: Identifying Patterns in a Dice Game

Learning math: geometry Feel like you need a little review in problem solving? This site offers a college-level online mathematics course designed to teach geometry content to teachers of elementary and middle school. The site uses geometric reasoning to highlight problem-solving methods as it engages teachers through video lessons, problem-solving activities, and online demonstrations. The course contains 10 sessions, each providing video lessons, activities, and homework problems. The final session furnishes case studies that teachers may use to examine problem-solving processes in their own classrooms from the perspective of a student.

Classic Problems Teachers can use these problems to show how math can answer questions about everyday life, including questions about birthday probability, the old favorite about the missing dollar, and trains leaving a station. The illustrated problem solutions are very helpful.

### Challenging Student Thinking and Creativity

We can look at mathematics instruction as question and answer or as an opportunity to examine, think, and draw conclusions. Is it strictly one or the other? No—students need to develop not only skills with straight-forward problems, but also understanding with problems where the approach is not stated and the answer is not a forgone conclusion.

If you want to nurture students who are problem solvers, you must make them comfortable with the practice of explaining their solutions and methods. Converting a typical mathematics problem into a problem solving activity can be challenging work for teachers. But any problem—big or small—that the student does not know exactly how to approach can be an opportunity to foster problem-solving strategies. A teacher who encourages students to explain their methods and solutions can turn most lessons into meaningful problem-solving and learning experiences. The teacher's approach is essential to making mathematics challenging in ways that are deeper than those found in doing drill-type lessons with progressively harder numbers.

Problems with a point Here is a great place to start fusing math skill with understanding. This searchable problem database classifies problems by topic, time required, suggested technology, required mathematical background, and the habits of mind that students develop or use as they work. Problems can be sequenced to build on student knowledge and to create individualized learning sequences. The list of favorite problems selected by teachers is a great place to start learning about the resources at this site.

Hot math! Hands-on tasks in the mathematics classroom These classroom activities link mathematics to real-world problems and situations. Each of the 13 activities takes less than 30 minutes and is designed to get students actively involved in mathematics. Teachers are encouraged to customize the activities for their students.

Word problems for kids Teachers can select grade-appropriate problems from this list to help students improve their problem-solving skills. Students are encouraged to carefully think about how they would solve a problem and to make sure that they understand all parts of the solution. Each problem contains a link to helpful hints and the solution.

Pre-Algebra Problem of the Week This site is a good source for math problems keyed to holidays and seasons. The problems are designed to foster students learning algebraic reasoning, identifying and applying patterns, ratio and proportion, and geometric ideas such as similarity. The problems are non-routine and focus on communication with students putting their solutions into words. The detailed solution to the previous posted problem and information about resources offered by Math Forum are available for a fee.

### Interdisciplinary Problems: Connecting Math to Science and Life

We can look at mathematics instruction as question and answer or as an opportunity to examine, think, and draw conclusions. Is it strictly one or the other? No—students need to develop not only skills with straight-forward problems, but also understanding with problems where the approach is not stated and the answer is not a forgone conclusion.

If you want to nurture students who are problem solvers, you must make them comfortable with the practice of explaining their solutions and methods. Converting a typical mathematics problem into a problem solving activity can be challenging work for teachers. But any problem—big or small—that the student does not know exactly how to approach can be an opportunity to foster problem-solving strategies. A teacher who encourages students to explain their methods and solutions can turn most lessons into meaningful problem-solving and learning experiences. The teacher's approach is essential to making mathematics challenging in ways that are deeper than those found in doing drill-type lessons with progressively harder numbers.

The Gulf Stream Voyage This multidisciplinary project uses both real-time data and primary source materials as students learn about the science and history of the Gulf Stream. Students investigate ocean currents with activities in fields that include marine and Earth science, math, history, and language arts.

Lotto or Life: What Are the Chances? Here is a one-day lesson where astronomy and probability are used as students apply problem-solving, reasoning, and communication skills to compare the probability of winning the lottery with the likelihood of intelligent life existing elsewhere in the universe.

Math in daily life Examine how math principles can be helpful when deciding whether to buy or lease a car, following a recipe, or decorating a home. The site is organized into six themes that offer hands-on application activities.

Puzzling and perplexing problems Here is a mixture of open-ended problems, problems that incorporate social studies topics, and problems with a linked web site for data. The problem sets, often containing background historical information, are listed by month and highlight holidays, seasonal events, and sports. To solve problems, students may be asked to create graphs, read information from tables, and execute multistep computations.

### Interactive Online Activities

Please don't underestimate the power of computer software to illustrate mathematical concepts! The computer's instantaneous ability to demonstrate the effects of a number or variable change or provide feedback on a solution can make for a very dynamic mathematics learning experience.

Collections

Alive maths Students explore a problem scenario through online simulations and uncover its mathematics through class discussion. Encourage students to identify relationships and patterns and to reflect upon and share their mathematical work. Each activity includes teaching suggestions, extension questions, and classroom narratives.

Math TV problem solving videos Students are challenged to solve math word problems after viewing a video with a step-by-step example solution. The online videos start with a typical word problem in algebra, geometry, probability, or another area. The host, Infinity Quick, solves the problem as in a tutorial, demonstrating each step as she proceeds. Viewers answer questions to help solve intermediate steps in the problem. At the end, students are presented with another similar problem to solve. An online calculator and a notepad for making sketches or figuring out the problem are available. Answers are checked automatically.

Middle school (geometry) This site features almost 100 virtual manipulatives or applets designed to help students acquire an intuitive understanding of fundamental concepts in geometry. Many applets are problem-based. Students are presented with a problem and must use the applet to explore the problem. Topics addressed in the applets include congruent triangles, transformations, and the Pythagorean theorem.

Selected Activities

Pythagoras' theorem Six interactive puzzle questions are presented to help students visualize the Pythagorean theorem and understand why it works. The puzzles feature a large square and four congruent right triangles with sides of length a, b, and c. Students use the triangles to cover part of the large square and examine the area of the remaining space. They may rearrange the triangles, note that the area of the remaining space does not change, and use this information to construct a relationship between the three sides.

From the National Library of Virtual Manipulatives for Interactive Mathematics

Number line bounce (grades 6-8) In this number line game for summing numbers, students are challenged to find a sequence of operations with four numbers that results in a given target number. The numbers are illustrated as bouncing balls on a number line. Each bounce can be in either a positive or negative direction. The student can use a guess-and-check approach to solve the problem or a more sophisticated strategy. After finding a correct sequence and reaching the target number on the number line, the student forms the number sentence that illustrates the sequence of operations used to arrive at the target number.

Fill and pour (grades 6-8) The student's goal when using this manipulative is to fill and pour juice in any order between two containers so that the target volume of juice is found in one of the two containers. The computer keeps track of the amount of juice in each container. In the challenge option, an impossible target number is given for specific sizes of containers, and the student is asked to explain why the situation is impossible.

Mastermind (grades 6-8) Here is an interactive version of the classic logic game of the same name. The student plays against the computer and has eight chances to guess the colored pattern of four pegs. The student may select to play with a pattern consisting of two to six colors.

From the Figure This! collection of 80 math challenges emphasizing math in the real world

Gone fishing: my, my little fish, how you've grown! Opening with a cartoon showing the weights of three combinations of fish, this activity challenges students to determine the weight of each fish. The introduction discusses algebraic reasoning and notes its importance to scientists, engineers, and psychologists. Students are encouraged to begin by adding the weights on all three scales. The answer page describes three strategies for solving the problem.

Map coloring: how many colors of states are on a map of the U. S? Students explore the classic mathematics map-coloring question known as the four-color problem. They must determine the minimum number of colors needed to color a map so that entities sharing a border have different colors. Initially, students investigate the minimum number of colors necessary to color a map of states west of the Mississippi River.

Rose Bowl: can a football team score 11 points in a game? Here's an activity for sports fans. The history of the Rose Bowl football tournament is used to examine numerical combinations. Students determine how many different ways a football team can end a game with 11 points, something that has never happened in Rose Bowl history. The activity suggests that students make tables to organize information as they examine possibilities. The usefulness of making a table in problem solving and prioritizing is noted.

Three squares: when do two squares make a new square? This activity challenges students to find a way to construct a third square from given diagrams of two unequal squares. It introduces the Pythagorean theorem and explains its importance in construction and engineering professions. Students are encouraged to model the problem using squares of paper.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related math-focused virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

### NCTM Problem Solving Standard

Marilyn Burns, internationally known mathematics teacher, tied problem solving and the other four NCTM process standards as the foundation for teaching when she said:

...the question that I always ask myself, is: "What experiences can I provide the children that would give them a way to start to make sense of this [mathematics] for themselves?" This is where I look at the process standards because they really address what children need to do to learn math— Problem solving: What kinds of problems can I present to children that would give them a chance to grapple with important ideas and skills? Reasoning and proof: What kinds of situations can I pose to children so that their reasoning is engaged and they have experience giving convincing arguments? Communication: How do I involve children in talking and writing to help them communicate what they are studying and learning, and hear the ideas of others? Connections: How do I help children see the connections among mathematical ideas rather than seeing concepts as isolated and separate from one another? Representation: How do I help children use the symbolism of mathematics to describe their thinking? (Herrera, 2001, p. 16)

In order to understand how the problem solving standard relates to the Standards for grades 6-8, go to the Standards document overview.

Herrera, Terese. (2001). An Interview with Marilyn Burns, Meeting the Standards—Don't Try to Do It All By Yourself. ENC Focus, 8(2), 16-19.