Word Problems in Math





Readers are encouraged to communicate their suggestions and ideas to David Moursund at the IAE address below. The original authors of this document are Robert Albrecht and David Moursund.

Introduction
Word problems or "story" problems are a well-established component of the math education curriculum. Indeed, some students and math teachers tend to equate "word problem" with "problem" when talking about problem solving in math. Moreover, many people seem to think that math is the only part of the school curriculum that teaches problem solving. Of course, these are both incorrect beliefs.

This document focuses on word problems in math. However, problems and problem solving are part of every academic discipline. Each discipline develops some special vocabulary, symbols, and methods for representing and communicating its problems. Each discipline develops methods for accumulating and storing information. Each discipline develops methodologies—educators call it content pedagogical knowledge—useful in helping students to learn the discipline.

Thus, the intended audience here is not only teachers of math, but also teachers in other disciplines. Moreover, there is considerable emphasis on transfer of learning across disciplines.


 * Food for thought: From time to time this document contains a short, indented section such as this one. If the materials in this document are being used in a workshop or in a course that includes class discussions, these indented sections might be useful discussion topics. They can also be used for reflection and self-assessment.


 * You are familiar with the idea of "reading across the curriculum." This reading education goal can be stated as: "We want students to learn to read, and we want students to read to learn." Thus, we want students to learn to read in various curriculum areas (such as math and science) well enough so that they can read to learn in these curriculum areas.


 * It is the authors' observation that relatively few precollege students learn to read math well enough to be able to learn math by reading math books, math websites, and other similar resources. To a very large extent, math is taught by "oral tradition" methods.


 * With this in mind, think about possible roles that math word problems might play in helping a student learn to read math. One measure of the quality of a math word problem is its contribution to helping a student make progress in learning to read math well enough to be able to read to learn math.

Here are some other observations from the authors that motivated the writing of this document:


 * 1) Math textbook series typically include a major emphasis on word problems. This emphasis is part of the overall emphasis on math problem solving.
 * 2) On average, students dislike math word problems and are not very good at solving them. Many students grow to hate this part of the curriculum.
 * 3) Many teachers, and a variety of books and websites, specifically teach how to identify and solve different types of math word problems. Often this is done in a mode that might be called "teaching to the test" or "having students learn by rote memory with little understanding."
 * 4) A high percentage of word problems used in math teaching and learning are "contrived," are of little interest to students, and are not designed to help students learn to deal with "real world" problems. Students do not tend to be intrinsically motivated by such problems. Typically, word problems do not draw on student knowledge and understanding of the "real world" and do not require students to do research (for example, research using the Web) to obtain information relevant to solving the problem.
 * 5) Problem solving requires a combination of domain specific knowledge and skills, and domain independent knowledge and skills. Math word problems are often designed so they make very little use of domain specific knowledge and skills from domains outside of math. Thus, students solving such problems cannot learn to develop and make use of this non-math domain-specific knowledge and skill. This topic or issue falls into the category of transfer of learning. Many students are quite weak at transferring math knowledge and skills to the task of solving math-related problems in non-math disciplines.
 * 6) Solving math word problems requires developing math models of problem situations. Thus, word problems are an excellent environment for teaching math modeling and computational thinking. Currently this aspect of word problems in math education is very under-emphasized in the math curriculum.
 * 7) While word problems are a very good vehicle for helping students to increase in math maturity, they typically are not taught or used in such a manner.

This document explores the various issues in the above list. It seeks to identify characteristics of "good" math word problems and accompanying teaching practices that will contribute to improving our math education system.

There is special emphasis on improving math maturity, improving transfer of learning, and improving computational thinking involving math and computer modeling.

Measuring Progress in Improving Math Education
You are familiar with the quotation, "Beauty is in the eye of the beholder." There are widely varying opinions as to what constitutes a good math education system of instruction and learning.

For example, math is a broad, deep, and quite old field of study. Math is thoroughly ingrained into our language and culture. What should students be learning about the history of math and about math as part of the various human cultures?

Mathematicians talk about the beauty of math. A proof or a problem-solving technique might well be described as being beautiful. What should students be learning about the beauty of math?

Math is an important aid to representing and helping to solve problems in non-math disciplines. What should math teachers know and what should students learn about the roles of math in representing and helping to solve problems in non-math disciplines?

Math is a huge discipline, with great breadth and depth. It is a vibrant, growing discipline. The discipline of math is being strongly impacted today by the disciplines of Computer Science and Information and Communication Technology (ICT).

Thus a K-12 math curriculum—indeed, a math curriculum extending through a doctorate in math—can cover only a small percentage of the discipline of math. What math content knowledge and math pedagogical content knowledge should be built into the K-12 math curriculum?


 * Food for thought: You may wonder about the idea of math content pedagogical knowledge being built into the curriculum. Each of us is a teacher, helping ourselves to learn and helping others to learn. Parents and other caregivers of children play a major role in the math education of children. They most often teach the way they were taught. We can improve math education by helping children learn good ways to teach and learn math.

Learning Facts and Learning to Think and Solve Problems
Here is an overly simplistic summary of the goals of math education. A similar simplistic view can be taken for any other content area being taught in schools:


 * 1) Learn some math facts (data, information, knowledge).
 * 2) Learn to think and solve math problems, both in math and in disciplines that make use of math.

People developing the math curriculum have given careful thought to an appropriate scope and sequence for #1. Different textbook series present some differences in both scope and sequence. Sometimes changes are made based on research and on increased insights into what works and what the various stakeholders believe students should be learning. Standards (school district, state, national, professional society) play a major role in #1.

Once the scope and sequence is agreed on, books can be developed, curriculum can be developed, teachers can be trained, and assessment instruments can be developed. The whole "bundle" represented by #1 lends itself to a factory model of education. It lends itself to setting standards and to measuring how well individual students or groups of students do compared to these standards and/or compared to other students and groups of students.

Item #2 in the list is quite different. Students vary considerably in both the nature and nurture components of their math intelligence and overall intelligence. They vary considerably in their overall cognitive development and their math cognitive development. They also vary considerably in their interest in math and other areas (think in terms of intrinsic motivation).

Item #2 has led many people to think about student-centric and highly individualized education. While the factory model of math education in #1 lends itself to multiple-choice assessment questions, #2 requires a different approach to assessment. It requires the use of open-ended questions in which students attack novel problems and demonstrate their thinking and overall problem-solving knowledge and skills.

Math Maturity
Mathematicians use mathematical maturity to mean, loosely, a mixture of mathematical experience and insight that are not taught directly, but which grow and ripen from substantial exposure to challenging mathematical concepts and processes.

Here is a list of some components of math maturity that relate to and/or can be increased through the study and use of math word problems.


 * Learn to learn math by completing the significant shift from learning by (rote) memorization to learning through understanding. Learn to learn math from a variety of aids such as teacher instruction, communication with other people, print materials, the Web, videos, Computer-Assisted Instruction, Distance Education, introspection and reflection, and so on.


 * Learn to communicate mathematics and math ideas orally and in writing using standard notation, vocabulary, and acceptable style.


 * Learn to represent (model) verbal and written problem situations as mathematical problems.


 * Learn to transfer one’s math knowledge and skills to address novel (not previously encountered) problems, both math problems and math-related problems in other disciplines.


 * Develop and use mathematical "sense" and intuition that allows one to detect errors in math problem solving and in use of math.


 * Recognize a valid mathematical or logical argument, and detect "sloppy" thinking. Provide solid evidence (informal and formal arguments and proofs) of the correctness of one’s efforts in solving math problems.


 * Have persistence when faced by a challenging problem. Learn to make use of challenging problems as a vehicle for increasing your own math problem-solving skills. A key aspect of this is metacognition and reflection.

The term math maturity is useful in talking about goals of math education and a student's level of progress in learning math. People want a math education system that leads to increasing math maturity for students. We recognize that because students vary so much in intelligence, cognitive development, and interests, the students will gain in math maturity at different rates. Also, the final level of math maturity that various students reach will vary considerably.

A "good" math word problem serves as a good vehicle or environment contributing to a student's progress in gaining an increased level of math maturity.

Math Problems and Math Problem Solving
George Polya was a 20th century mathematician and math educator leader. He wrote wrote extensively about math problem solving. The following is quoted from a talk that he gave to a group of elementary school teachers:


 * To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. (George Polya; 1887–1985.)

What Is a Problem?
Each academic discipline includes an emphasis on representing and solving problems. It is a challenge to write a definition of "problem" that cuts across all disciplines. Here is a definition of problem that fits reasonably well in many different disciplines.

You (personally) have a problem if the following four conditions are satisfied:


 * 1) You have a clearly defined given initial situation.
 * 2) You have a clearly defined goal (a desired end situation). Some writers talk about having multiple goals in a problem. However, such a multiple-goal situation can be broken down into a number of single-goal problems.
 * 3) You have a clearly defined set of resources that may be applicable in helping you move from the given initial situation to the desired goal situation. These typically include some of your time, knowledge, skills, and brain power. Resources might include money, computers, and access to the Internet. There may be specified limitations on resources, such as rules, regulations, guidelines, and timelines for what you are allowed to do in attempting to solve a particular problem.
 * 4) You have some ownership—you are committed to using some of your own resources, such as your knowledge, skills, time, and energy, to achieve the desired final goal.

The "resources" in the third part of the definition do not tell you how to solve a problem. Rather, they are used as aids to problem solving. In many problem-solving situations, Information and Communication Technology (including the Internet, Web, calculators, and computers) and computerized tools are resources. These resources have grown more powerful over the years. That is one reason why it is becoming more and more common to integrate teaching the use of computers in problem solving very thoroughly into the basic fabric of academic courses.

The fourth part of the definition of a problem is particularly important. Unless you have ownership—through an appropriate combination of intrinsic and extrinsic motivation—you do not have a problem. Motivation, especially intrinsic motivation, is a huge topic in its own right. Edward Vockell maintains an online book, Educational Psychology: A Practical Workbook. The sixth chapter provides a nice discussion of motivation.


 * Food for thought: Many students find they have little or no personal interest—little intrinsic motivation, little ownership—in the types of math problems they encounter in school. Thus teachers, parents, grading systems, and other extrinsic motivation factors are used to motivate and/or coerce students into learning math.


 * This is not a good situation. Of course, other academic disciplines also have to deal with this challenge. However, perhaps the challenge is more severe in math education than in most other disciplines. Starting at about the fourth grade, students must deal with a great deal of delayed gratification. They are steered through a math curriculum that prepares them for the next year's curriculum that prepares them for the next year's curriculum. They eventually are steered through algebra, geometry, and more algebra. Most of the content they are learning is not immediately useful to them, nor is it used in their everyday lives or in the other courses they are taking.

What Is Problem Solving?
This document considers problem solving to include:


 * Question situations: recognizing, posing, clarifying, and answering questions.
 * Problem situations: recognizing, posing, clarifying, and solving problems.
 * Task situations: recognizing, posing, clarifying, and accomplishing tasks.
 * Decision situations: recognizing, posing, clarifying, and making good decisions.
 * Using higher-order critical, creative, wise, and foresightful thinking to do all of the above. Often the results are shared, demonstrated, or used as a product, performance, or presentation.

Polya's 6-Step Strategy for Problem Solving
A strategy is a general plan. A company may have a strategy for increasing sales and profits. A military commander may have a strategy for an upcoming battle. A school district may have a strategy for improving the state and national test scores of its students.

Here is a strategy useful in attacking a wide range of math and non-math problems: (1) Break the problem into two or more smaller, more manageable problems; and (2) Solve these smaller, more manageable problems and then assemble the results into a solution to the original problem. This strategy is often called, "Divide and conquer." Another useful strategy is called, "Guess and check" or "Trial and error." There are a great many different strategies that are useful in math problem solving and in solving problems in other disciplines.

David Moursund's free book, Learning Problem-solving Strategies by Using Games: A Guide for Educators and Parents contains a long list of strategies useful in problem solving. The book includes a major focus on transfer of learning-especially, high-road transfer of learning. High-road transfer of learning for improving problem solving is based on learning some general-purpose strategies and learning how to apply these strategies in a reflective manner.

Math education research suggests that a typical student has a very small repertoire of math problem-solving strategies. Thus, helping a student to learn a new strategy and to develop fluency in its use can be a quite worthy goal of math education.

Even more important is helping students learn to make use of their problem-solving strategies over a wide rage of problems from a wide range of disciplines. That is exactly what the High-road Theory of Transfer of Learning is about.

George Polya (1887-1985) was a great mathematician and teacher who wrote extensively about problem solving. Polya's 6-step problem-solving strategy is useful in math and in most other disciplines. The following version of this strategy has been modified to be applicable in many problem-solving domains. All students can benefit from learning and understanding this strategy and practicing its use for a wide range of problems.


 * 1. Understand the problem. Among other things, this includes working toward having a clearly defined problem. You need an initial understanding of the Givens, Resources, and Goal. This requires knowledge of the domain(s) of the problem, which could well be interdisciplinary.


 * 2. Determine a plan of action. This is a thinking activity. What strategies will you apply? What resources will you use, how will you use them, in what order will you use them? Are the resources adequate to the task?


 * 3. Think carefully about possible consequences of carrying out your plan of action. Place major emphasis on trying to anticipate undesirable outcomes. What new problems will be created? You may decide to stop working on the problem or return to step 1 because of this thinking.


 * 4. Carry out your plan of action in a reflective, thoughtful manner. This thinking may lead you to the conclusion that you need to return to one of the earlier steps. (Note that reflective thinking leads to increased expertise.) Remember, there are many tools, such as computers, that can help in carrying out a plan of action.]


 * 5. Check to see if the desired goal has been achieved by carrying out your plan of action. Then do one of the following:


 * a. If the problem has been solved, go to step 6.


 * b. If the problem has not been solved and you are willing to devote more time and energy to it, make use of the knowledge and experience you have gained as you return to step 1 or step 2.


 * c. Make a decision to stop working on the problem. This might be a temporary or a permanent decision. Keep in mind that the problem you are working on may not be solvable, or solving it may be beyond your current capabilities and resources.


 * 6. Do a careful analysis of the steps you have carried out and the results you have achieved to see if you have created new, additional problems that need to be addressed. Reflect on what you have learned by solving the problem. Think about how your increased knowledge and skills can be used in other problem-solving situations. Work to increase your reflective intelligence!

Polya's 6-step strategy can be used in attacking a word problem. Thus, this section provides us with two possible characteristics of a good word problem:


 * 1) It helps a student learn and gain practice in using Polya's 6-step strategy.
 * 2) It helps students learn and/or practice a strategy that lends itself to high-road transfer of learning.

Word Problems
In this document, the term "word problem" is taken to mean any problem that makes use of natural language and the special vocabulary of a discipline to communicate a problem in that discipline. The following definition produced by the Google search engine is somewhat typical (and rather poor) dictionary definition of a math word problem.


 * A mathematics exercise presented in the form of a hypothetical situation that requires an equation to be solved; for example, “if George earns a salary of $18,500 and 28% of it is deducted in taxes, how much take-home pay remains?”

There is no particular reason for assuming that a math word problem requires that an equation be solved.

Each academic discipline has developed aids to communicating the problems in the discipline and in representing and storing accumulated knowledge about how to "attack" and possibly solve the problems. Each academic discipline has some discipline-specific ways of teaching—some pedagogical content knowledge. In some sense, one goal of education in a particular discipline is to help students learn the culture of the discipline.

Natural Language
The natural languages that humans use to communicate with each other contain many words that are used to communicate math ideas. Thus, we have number words such as one, two, three, and we have "abstract" symbols for numbers such as 1, 2, and 3. We have words such as plus, minus, times, and equals, and we have abstract symbols +, -, x, and =.

A language word often has several meanings, while a math symbol typically has a very precisely given single meaning. Thus, when we say, "All men are created equal." the word "equal" does not have the same precise meaning as the math symbol =. One of the challenges in learning math is to learn to effectively deal with and communicate precisely using the words and symbols of math.

We use number words and numbers in representing and dealing with age, time, distance, length, area, volume, money, and so on. In all of these situations, the number words are accompanied with some unit of measure. A student is 14 years old. A grocery store item costs $5.28. Carpet is sold by the square yard—and for a certain amount of money per square yard. Distance is measured in units such as centimeter or inch, meter or yard, kilometer or mile, and so on.

In natural language, we often make use of somewhat confusing and quite approximate measures. When asked, "How far is it from Eugene, Oregon, to to Portland, Oregon?" a person might respond, "About two hours." The answer is an approximate driving time for traveling to Portland by car. However, if two bicyclists were talking, the answer given might be seven hours.

Problems Expressed in Natural Language</Center>
We make extensive use of math and math-related ideas and vocabulary as we represent and attempt to solve many different types of problems. Think of a math word problem as being a problem expressed in oral or written language that makes use of the types of math and math-related words in our natural languages. From that point of view, math word problems are a routine part of our everyday lives.

Math is a huge and steadily growing discipline with a very large accumulation of mathematical knowledge and skills. Much of this is represented/communicated in the "language" of mathematics. Communicating in the Language of Mathematics. From this point of view, the purpose of including word problems in the math curriculum is two-fold:


 * 1) To help students learn to translate problems that are expressed in oral or written natural language into problems that can draw on the accumulated math and skills knowledge of the problem solver and also on the accumulated math knowledge available from other people or in physical and virtual libraries.
 * 2) To help students gain in their ability to more precisely use natural language to represent math-related problems and the processes used and results from solving math problems.

An Example from Problem Solving in Art</Center>
Each discipline has its own accumulation of problems, ways of thinking about problems, and ways to solve problems. The University of Oregon has recently completed construction of a new College of Education building. This building contains a variety of works of art. One problem to be solved was the design and selection of appropriate artwork to enhance the surrounding area and the entrance to the new building. The following is quoted from an email message sent to the faculty on 10/9/09:


 * Unity and Harmony: An Environmental Artwork


 * This weekend, the final pieces of public art will be installed in the College of Education complex. Five sculptures will be installed: one in front of the clinical services building, one on the grass leading to the plaza, and three on Kendall Plaza. This work is site-specific, designed for the College of Education.


 * The artist is Yuki Nagase. Masayuki Nagase began his career at the Academy of Fine Arts in Tokyo. In 1995, he became a U.S. resident and established a studio in Berkeley, CA.


 * Comments from the artist on the concept prior to starting the project:


 * “Upon my site visit, it was clear that the vision for the new College of Education complex would physically and symbolically re-connect the existing buildings that were scattered around the campus. My design concept comes from that vision to unify and to harmonize the diverse separate facilities into an inter-related whole.


 * "Key issues I wanted to address in this concept:
 * 1) Work with natural materials and natural imagery
 * 2) Provide functional seating
 * 3) Provide interaction with children visiting the complex
 * 4) Offer a symbolic focal point for the complex


 * "The artwork is an environmental work that visually and physically travels from [the clinical services building] through the plaza. It represents the concept of “Unity and Harmony”.


 * "The free-standing sculptural composition placed on the stretch of lawn…is composed of 3 elements carved out of one massive granite boulder. This sculptural composition is a metaphorical expression of a cycle of life. The main image that I envision is an abstract form of the universal concept of Heaven and Earth, in the Asian traditions of Yin and Yang.


 * "The benches in the plaza would be made from large natural granite boulders with the top surfaces hand carved and partially polished. The relief carvings would reflect the images of the 4 major elements found in nature: Water, Fire, Wind and Mountain.


 * "As one enters the plaza for the [new College of Education] HEDCO building, the visitor will see the sculptural benches. They will provide a strong tactile and interactive experience as well as on a visual level. Beyond them on the lawn, the visitor will see the vertical freestanding sculpture from the plaza and as one travels on the walkway. The forms of the sculpture are visually and tactilely inviting to children. They can spontaneously interact with the natural boulders by climbing, sliding, and touching, providing a spontaneous play experience with natural materials.”

The artist uses words to describe the problem and goals. However, these words do not communicate the same thing to each reader. Indeed, it takes considerable knowledge of the discipline of art to understand the problem being addressed, the specific goals, and the resources (materials, the artist's time and skills) available.

Notice also that there are many possible solutions to the original problem faced by the artist. If presented with the same problem sometime in the future, the artist would quite likely produce a very different solution.

Finally, think about the types of math involved in solving the problem. There are spatial problems, size and shape of seating problems, and so on. The artist likely built a scale model and/or did a careful scale drawing. How does the math a student learns in school relate to the math the artist used in addressing this problem?

Six-Step Strategy for Solving a Math Word Problem</Center>
Here is another way of representing Polya's 6-step strategy. Think about the six steps illustrated in the following diagram:




 * 1) Most people are not very precise when they start to talk about a problem situation. Step 1 is a process of moving from poorly stated and poorly defined problem situation to a carefully stated and precisely defined problem. Word problems in math books are usually clearly defined. This means that students do not get practice in doing step 1.
 * 2) Step 2 is the process of representing the clearly defined natural language problem as a math problem. The result is expressed in math vocabulary and symbols as a "pure" math problem to be solved. For example, it might be computations to be performed, equations to be solved, data to be graphed, or so on.
 * 3) Step 3 is the process of solving the "pure" math problem. This might be done mentally, using pencil and paper, using a calculator, using a computer, or so on. The problem solver may well draw upon information stored in physical or virtual libraries. A computer used to access the Web and/or the computers used by the Web may well be used to carry out needed symbol manipulation and computational activities. The WolframAlpha system provides an excellent example of combining information retrieval with symbol manipulation and computation.
 * 4) Step 4 takes the results from solving the "pure" math problem and translates them back into natural language.
 * 5) Step 5 checks to see if the clearly defined math problem has been solved.
 * 6) Step 6 checks to see if the original problem situation has been resolved.

In very rough summary, steps 1, 2, 4, 5, and 6 require careful, thoughtful human thinking and understanding.

Step 1 is a process of clarifying and understanding a problem situation. In very simple terms, what is the given initial situation and what is the goal? Problem solving is a process of moving from a given initial situation to a desired final (goal) situation.

Step 2 is a process of math and computer modeling. The modeling process produces a math problem, and quite often the problem can be solved by computer.

Step 3 might be carried out by some combination of mental math, paper and pencil math, use of simple tools such as ruler, compass, protractor, and calculator, and use of more sophisticated tools such as computers and computerized instruments.

Students find word problems to be hard because of the careful, thoughtful, human thinking and understanding that is required and because of the challenges of step 3.

Step 4 is a process of math or computer unmodeling. The process of solving the math problem produces results that need to be translated back into natural language.

A great may of the math-related problem situations that a person encounters in their day-to-day life can be handled easily and relatively quickly. The six steps blur together and are carried out mentally with little conscious thought. For example, I am talking to a friend on the phone. My friend says, "Let's meet for lunch in the eatery located on the corner of 4th and Jefferson."

My mind processes the problem situation. I glance at my watch, think about what I need to do before I can leave for lunch, think about the time involved in bicycling to 4th and Jefferson, glance outside to see if the weather will slow me down, think about what time it is and what traffic is like at this time, and I say, "All right. I can be there in 30 minutes—does 12:15 work for you?" Notice the complexity of the problem I am solving, the various uses of technology, and the "putting it all together" type of thinking that is required. Think about the high level of intelligence all of this takes. Any person who can handle such a problem is demonstrating a quite high level of intelligence of the type Robert Sternberg calls "street smarts."

Now, back to the 6-step diagram. It is interesting to explore how much of the math education time in schools is spent helping students to get better at steps 1, 2, 4, 5, and 6 versus how much time is spent on helping students to get better at step 3. Precollege teachers of math tend to estimate that step 3 gets about 75% of the student learning time and effort. Such informal data suggests that perhaps only a quarter of math schooling time is spent on the other five steps that are all quite dependent on human thinking. '''The one step that calculators, computers, and computerized instruments can do best gets by far the most student schooling time and effort. And, this effort focuses mainly on non-computer approaches to accomplishing this step.'''

This analysis suggests that word problems are a very important component of math education because they give student instruction and practice in doing the human thinking parts of problem solving. It also suggests that one aspect of a successful word problem is that it creates a good environment providing instruction and practice in human thinking.

Teaching About and Using Word Problems
Word problems can be an important component of a good math curriculum. However, the "goodness" of a particular word problem or set of word problems must be judged in terms of the contribution made to achieving the overall math education goals that guide the curriculum, instruction, and assessment.


 * Food for thought: Think about a parallel between a Language Arts teacher assigning an essay to write and a Mathematics teacher assigning a set of one or more word problems. In both cases, students are expected to "do" something. In both cases, the student has some knowledge, skills, and experience that can be used for self-assessment of the quality of work being done. In both cases, it is expected that summative assessment feedback (and perhaps formative assessment feedback) will be provided by others, such as other students, paper graders, and teachers.


 * It requires considerable knowledge and skill to provide good feedback on student writing. Moreover, this is a demanding and time-consuming task. The reader must read for understanding, have good insight into the writing skills of students at a particular level, and know what constitutes good writing. The reader must have good writing pedagogical knowledge and skills. The reader must be able to effectively deal with the different approaches students take to a particular writing task, and the major differences in content that different students will produce.


 * Similar statements hold for providing feedback on student work in solving math word problems. It requires quite good content knowledge and content pedagogical knowledge to provide good feedback on the work students do when solving challenging word problems.


 * This observation lies at the heart of major controversies in the math education of elementary school teachers. Math Departments stress the need for teachers to have very good content knowledge, while Colleges of Education stress the need for teachers to have very good pedagogical content knowledge.

Challenges of Word Problems
The Polya 6-step strategies diagram given earlier illustrates the general steps involved in attempting to solve a word problem. The math teaching challenge is to help students gain the knowledge and skill needed to deal with both commonly occurring and novel word problems.

If there were only a modest number of types of commonly occurring math word problems, then one might propose that rote memory is the right approach. Beginning in first grade, students memorize a portion of such problems—how to recognize them and how to solve them. Year after year in school a student continues this process, with the goal of mastery of how to solve a very large number of commonly occurring math word problems.

If the assessment of word problem solving skill at the school district, state, and national levels is based just on the list of the commonly occurring word problems being studied in the curriculum, then such a rote memory approach to this area of math teaching and learning will lead to students getting high scores on the assessments. Thus, it is not surprising that variations on this approach have been widely used in schooling in the past and are still widely used.

There are many serious flaws in this approach. In essence, the assessment process drives much of the curriculum content and pedagogy. Teaching becomes "teaching to the test." If the test is sufficiently authentic, than this is not a bad thing to do. But, typically math tests lack authenticity.

My 2/29/2016 Google search of the expression authentic assessment produced about 1.2 million results. Quoting from the website https://www.eduplace.com/rdg/res/litass/auth.html:


 * Authentic assessment refers to assessment tasks that resemble reading and writing in the real world and in school (Hiebert, Valencia & Afflerbach, 1994; Wiggins, 1993). Its aim is to assess many different kinds of literacy abilities in contexts that closely resemble actual situations in which those abilities are used. For example, authentic assessments ask students to read real texts, to write for authentic purposes about meaningful topics, and to participate in authentic literacy tasks such as discussing books, keeping journals, writing letters, and revising a piece of writing until it works for the reader. Both the material and the assessment tasks look as natural as possible. Furthermore, authentic assessment values the thinking behind work, the process, as much as the finished product (Pearson & Valencia, 1987; Wiggins, 1989; Wolf, 1989).


 * Working on authentic tasks is a useful, engaging activity in itself; it becomes an "episode of learning" for the student (Wolf, 1989). From the teacher's perspective, teaching to such tasks guarantees that we are concentrating on worthwhile skills and strategies (Wiggins, 1989). Students are learning and practicing how to apply important knowledge and skills for authentic purposes. They should not simply recall information or circle isolated vowel sounds in words; they should apply what they know to new tasks. For example, consider the difference between asking students to identify all the metaphors in a story and asking them to discuss why the author used particular metaphors and what effect they had on the story. In the latter case, students must put their knowledge and skills to work just as they might do naturally in or out of school.


 * Performance assessment is a term that is commonly used in place of, or with, authentic assessment. Performance assessment requires students to demonstrate their knowledge, skills, and strategies by creating a response or a product (Rudner & Boston, 1994; Wiggins, 1989). Rather than choosing from several multiple-choice options, students might demonstrate their literacy abilities by conducting research and writing a report, developing a character analysis, debating a character's motives, creating a mobile of important information they learned, dramatizing a favorite story, drawing and writing about a story, or reading aloud a personally meaningful section of a story. For example, after completing a first-grade theme on families in which students learned about being part of a family and about the structure and sequence of stories, students might illustrate and write their own flap stories with several parts, telling a story about how a family member or friend helped them when they were feeling sad.


 * The formats for performance assessments range from relatively short answers to long-term projects that require students to present or demonstrate their work. These performances often require students to engage in higher-order thinking and to integrate many language arts skills. Consequently, some performance assessments are longer and more complex than more traditional assessments. Within a complete assessment system, however, there should be a balance of longer performance assessments and shorter ones.

Synergy: Two for the Price of One
As noted earlier, each discipline has a focus on identifying, representing, learning about, and solving the types of problems that are consistent with the discipline. It has also been emphasized that math is often quite useful in representing and helping to solve problems in many different disciplines.

Thus, math educators face the following challenge. To what extent should math educators teach "pure" math (where there are few or no referents to disciplines other than math) and to what extent should they teach math with an emphasis on how it is embedded in and used in solving problems in other disciplines?

The "Two for the Price of One" title of this section is based on the idea that well-designed math word problems can be used to simultaneously helps studetns make progress in both pure and applied math. Indeed, a well designed math word problem can actually create "Three for the Price of One" learning situations, with studetns learning about and integrating Pure, Applied, and Computational mathematics.


 * Food for thought: College and University Math Departments often face this challenge by having some faculty and courses focusing on "pure" math, some faculty and courses focusing on "applied" math and statistics, and some faculty and courses focusing on "computational" math. That is, the overall discipline of mathematics is often divided into pure, applied, and computational components.


 * Such dividing up of the discipline and specialization of faculty seldom occurs below the high school level. At the current time there appears to be growing controversy as to whether all high school students should be required to take a relatively pure math collection of courses (algebra 1, geometry, algebra 2) or whether other options should be available.

Abstract Versus Concrete Word Problems</Center>
Here are two word problems:


 * 1. I am thinking of a number. If I double the number and then subtract 13, the result is 77. What is the number I am thinking of?


 * 2. If I double how old I am and subtract 13, I get my mother's age. My mother is 77. How old am I?

These are essentially the same problem. The second problem provides referents—people and ages of people. It also allows the problem solver to draw on the external information that a child is younger than his or her mother. This fact can be used as an aid to verifying the possible correctness of a proposed solution to the problem.

In this simple-minded example, the context of "real world" referents is of only modest use. For example, a negative age answer makes no sense, and everybody knows that a child is younger than his or her mother. When math is used to help solve a real world problem, the context of the problem often provides information about the reasonableness of a proposed answer.

Thus, one aspect of a "good" word problem is that it provides context and referents that the problem solver can combine with his or her real world knowledge as an aid to checking the reasonableness of a proposed answer.

It turns out that this is a very important idea in math education. Think about using a calculator, or a clerk in a store keying in the number and/or price for an item that is not in the scanner system. It is very easy to make keying errors. How does one detect keying errors?

An important part of an answer to this question lies in drawing on knowledge about the problem situation. For example, perhaps I am buying some fruit and I remember distinctly that I picked out the cheapest of the various varieties of apples from the various apple bins. I didn't want to spend more than about $4 or $5 for apples. I recall that some apples cost much more than others. I glance at the display being generated as each item is processed by the clerk, and I see that I have bought about $12 worth of apples. No! Surely I didn't buy that many. Something has gone wrong! Perhaps I took the apples from the wrong bin. Perhaps there was a keying error, a scanner error, or a wrong number entered into the computer that is hooked to the scanner. Perhaps an apple from a more expensive bin got into the bin I selected from.

A quite similar situation occurs when I use a calculator. Perhaps I am working on my family budget, on income tax, or on figuring out student grades at the end of a term. It is very easy to make a keying error. But, I have some knowledge about what constitutes reasonable answers. I often catch my own keying errors (including results from keys that stick, accidentally depressing two keys at once, accidentally depressing a key twice when I don't mean to, and so on). If I am mentally alert while doing the calculations, I will detect when an "A" student suddenly becomes a "C" students or when I unexpectedly seem to owe the IRS a lot of extra money. Of course, it is possible that my calculations are correct. But surely, I need to do them over again to help make sure I have not made a keying error.

Pure and Applied Mathematics</Center>
The "pure" versus "applied" math issue is evident in every precollege math curriculum. Some math books place much more emphasis on one side or the other. Some teachers are much more skilled at teaching a "blend" than others. Some students thoroughly enjoy learning math from a rather abstract, theoretical point of view, while others prefer that the math they learn is thoroughly grounded in practical, immediately useful ideas and applications.

Consider the math education challenge created by the second group of students. Each has their own individual knowledge, skills, experiences, and interests. If each student had a highly qualified personal tutor, the tutor could individualize the instruction and work one-on-one with the student to personalize the math topics being covered.

However, even the very best of the current Highly Interactive Intelligent Computer-Assisted Learning systems lack the breadth, depth, and knowledge to provide this quality of instruction.

Thus, individual students are left with the challenges of generalization and transfer of learning of the math they are being taught.

The pure versus applied, and transfer of learning, situations suggest that math education can be made more successful by a four-pronged approach:


 * 1) The precollege math curriculum becoming more balanced among pure, applied, and computational math.
 * 2) The precollege math curriculum placing greater emphasis on teaching for transfer of learning to problem solving in other disciplines.
 * 3) The precollege teachers in other disciplines placing a greater emphasis on problem solving and using math to help students learn to represent and solve these problems.
 * 4) Students being given considerable instruction about and practice in transfer of learning.

Math teachers at the precollege level can make use of appropriately chosen word problems and teaching methodology to help accomplish their part of this four-pronged approach.

Math Word Problems that Include Non-Math Content</Center>
Many of the math word problems one finds in typical math textbooks appear to be somewhat grounded in applications of math. Let's analyze the following math word problem:


 * Pat has an allowance of $8.50 a week and earns $3.75 a week doing extra chores. Pat spends an average of $.95 a day for candy, pop, and snacks, and saves all of the rest of her income. Pat wants to buy an electronic game that costs $49.95. How long will it take Pat to save that much money?

Well, you say. What's not "real world" about this? The problem deals with receiving, spending, and saving money. Now, try to imagine a student who has reached an age and educational level appropriate to this problem, who has not already learned about those aspects of money. Thinking about this may lead you to conclude that the math word problem will not help the student to learn anything about applications of math that he or she does not already know.

Now, think more deeply about the problem. Can you think of other somewhat similar problems? Here are a few examples:


 * 1) Consider a person eating to take in calories, using up calories through daily physical and mental activities, and experiencing a net increase or decrease in weight.
 * 2) Consider a highway surface that experiences substantial wear from traffic, is maintained via minor repairs from time to time, and gradually deteriorates.  One can describe similar situations for houses, office buildings, and other structures and infrastructures.
 * 3) Consider the world's known oil supply. It increases through discovery of new oil fields and decreases through use of oil. At current rates of discovery and use, how long will it be before the known oil supply is down to 1/4 of what it currently is?

Perhaps an "Aha!" is in order here. There are many different problems that are similar to the problem about Pat having income, spending, and savings. Some may involve negative savings, such as the known oil supply decreasing or a person's weight decreasing.

The income, outgo, and savings situation lends itself to developing a simple spreadsheet model. This provides an example of computational math from a computational thinking point of view. Spreadsheets making use of simple formulas and variables provide good concrete examples of some important aspects in algebra.

Value Laden Word Problems</Center>
Consider a word problem in which a child is saving money to buy a relatively expensive and fancy "outfit" versus a problem in which a child is saving money to contribute to providing food and clothes for homeless children.

The necessary thinking and problem-solving skills can easily be identical in the two problems. However, one of the problems might be seen as emphasizing a conspicuous and selfish type of consumption, while the other might be seen as emphasizing a type of providing help to others less fortunate than oneself.

Thus, while the math being learned is the same in the two cases, the values being emphasized are not. One can look for such synergy in every math word problem.

Think about some of the large problems in our world. You might think about global warming, sustainability, war and terrorism, medical problems, homelessness, hunger, irresponsible parents and children, crooked politicians and government officials, and so on. Now think about the context or "story" in a word problem. Can the story be designed to help teach values and points of view that you believe are important. Surely!

Are there values or points of view that we want our schools to help students to learn? Take sustainability as an example. Do you believe that it is important for students to grow up understanding some of the major issues of sustainability? Should students consider what they individually might do to help address these problem situations, and what they might accomplish in working with others?

If you say "yes," then one way to do this is through choice of context and referents in word problems. In essence, the story in a word problem can draw on real world problem situations and real world data, information, knowledge, and wisdom. The process of attacking such problems helps students to learn about these problem and to draw upon their own knowledge about the problems.

Reading and Writing Across the Curriculum</Center>
Reading and writing across the curriculum is a commonly accepted goal of education. Of course, we want students to learn to read well enough so that they can learn by reading. We want them to learn to write well enough so that they can write effectively in the various disciplines they study.

Reading and writing math are a major challenge to many students. We see some signs of this in students learning to deal with word problems. They tend to become bogged down in extracting the given information, adding to it information they already know, doing research to gain still more needed information, and then developing a clearly stated math problem. The whole task can be thought of in terms of reading and writing math. A good math word problem helps to increase the math reading and math writing knowledge and skills of students.

Along with learning to do the math reading and math writing necessary in attempting to solve math word problems, students can learn to create their own math word problems. Think of this in terms of creative writing in the discipline of math. A related writing topic is to keep a math journal that has some portfolio-like characteristics. In the journal, reflect on and explore one's math-learning and problem-solving activities.

Word Problems and Computational Thinking

 * Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination. (This quote is often mistakenly attributed to Albert Einstein; most likely the correct attribution is to Leo Cherne at the Discover America Meeting, Brussels, June 27, 1968.)

The statement quoted above captures the essence of computational thinking. Computational thinking involves using the capabilities of one's (human) brain and the capabilities of computer (brains) to represent and solve problems and accomplish tasks. Education for computational thinking involves learning to make effective use of these two types of brains.

Here is a more recent description of computational thinking:


 * Computational thinking is a way of solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computer science. Computational thinking is thinking in terms of abstractions, invariably multiple layers of abstraction at once. Computational thinking is about the automation of these abstractions. The automaton could be an algorithm, a Turing machine, a tangible device, a software system—or the human brain. (Carnegie Mellon, n.d.) [Bold added for emphasis.]

Human brains become better through informal and formal education and through regular use. Computer brains become better through the combined research and development of many thousands of people. Computer brains are getting better at a rapid pace. Thus, all students and all teachers need to learn and teach about the capabilities and limitations of the combination of human and computer brains.

Each type of brain has unique capabilities and limitations. Together they are incredibly powerful. Read more about this idea in the document Two Brains Are Better Than One.

What does this have to do with math word problems? Well, a simple answer is that a math word problem is a problem. Computational thinking is a type of thinking used in problem solving.

In problem solving, people draw on the thinking power of their own brains. However, they also draw on:


 * The "thinking power" of computer systems.
 * The knowledge and skills built into tools and other aids to problem solving that people have developed.
 * The accumulated knowledge of the human race that can be accessed from other people, from conventional print libraries, and from virtual libraries such as the Web.
 * Help from other people. Many problems are attacked by a team of people with differing specializations and problem-solving areas of knowledge and skills.

Thus, when faced by a word problem, a person needs to think about the specific resources he or she can bring to bear, as well as other types of resources such as those listed above.

Because of the steadily increasing power of computer systems and virtual libraries, computational thinking is becoming more and more important. A good problem solver has knowledge and skills to help determine what computer-based tools might prove useful—or even indispensable—in attacking a problem.

As indicated in earlier parts of this document, there are many reasons for using math word problems in the math curriculum. One very important reason is that solving math word problems helps students to gain knowledge and skills in problem solving.

Characteristics of Good Word Problems
Word problems can be an important component of a good math curriculum. However, the "goodness" of a particular word problem or set of word problems must be judged in terms of the contribution made to achieving the overall math education goals that guide the curriculum, instruction, and assessment.

Thus, "goodness" is dependent on the teacher's knowledge, interests, and skills as well as the students' knowledge, interest, and skills.

Similar statements hold for providing feedback on student work in solving math word problems.

Here is a summary list of possible features of a good math word problem that have been discussed in previous parts of this document.


 * 1) It helps students to learn and/or practice using strategies that are useful both in solving math problems and in solving problems in other disciplines. That is, it facilitates students learning and doing High Road transfer of problem-solving strategies.
 * 2) It helps students get better at math modeling and math unmodeling. (See the 6-step diagram version of Polya's six-step strategy.)
 * 3) It is fun. It has a good chance of being intrinsically motivating to students.
 * 4) It is cognitively challenging. It "pushes" the envelop for students.
 * 5) It serves as a good vehicle or environment contributing to a student's progress in gaining an increased level of math maturity.

Other considerations:
 * 1) Learning to draw on the accumulated knowledge of the human race. In real world problem solving, one of the most important ideas is drawing on this collected knowledge.
 * 2) Just in time learning. In real world problem solving, it is often necessary for the problem solver to learn new things in order to solve the problem.
 * 3) … Send your ideas of possible additions to this list to the authors of this IAE-pedia entry.

Analysis of Some Word Problems
Any math word problem can be analyzed and discussed from the point of view of ideas in this document. Here are some examples.

Counting Barnyard Animals</Center>
Here is a problem that might look somewhat familiar to you. There are many different variations on this type of problem.

While walking by a farmyard filled with chickens and rabbits, a farmer decided to count the feet of his animals. His count produces a total of 100 feet. How many chickens and how many rabbits does the farmer have?

My first thought is that this is not a very good word problem. What kind of person counts animals by counting their feet? What will I learn about farm animals and farming by solving this problem? Are these free range chickens and rabbits? If so, it seems to me that it would be very difficult to do an accurate count. Also, is not immediately obvious how what I learn by solving this problem will be applicable to other problems that might interest me.

My mind wanders a bit. My veterinarian has a three-legged cat and and I have seen one-legged birds. Perhaps some of the farmer's chickens and rabbits have more or less than the usual compliment of legs? If that is the case, this is a poorly defined problem. Thus, I will proceed by assuming all of the farmer's chickens each have exactly two legs and also that the farmer's rabbits each have exactly four legs.

Next, I think more about the counting task. I imagine it is quite difficult to produce an exact count of legs in a large group of animals. How do I know that 100 is a correct count? Perhaps there were actually 99 legs or 101 legs?

No—that cannot be. The total number of legs must be an even number? Aha! I am drawing on my math knowledge that 2 and 4 are even numbers, that the sum of a bunch of 2s is an even number, that the sum of a bunch of 4s is an even number, and that the sum of two even numbers is an even number. I have figured out that 100 is at least a plausible number for the total of the legs.

However, ignoring these types of questions, I proceed. I work to understand the problem. I create a picture in my mind's eye or perhaps on a piece of paper of a collection of chickens and a collection of rabbits. To get a "feel" for what is going on, I make a guess that perhaps there are 50 chickens and 50 rabbits. Hmm. That would produce a total of 300 legs—far more than the reported 100 legs.

On the other hand, suppose there were five chickens and five rabbits. That would produce a total of 30 legs—far less than the reported 100 legs. Okay, the problem is beginning to make sense to me. There is a challenge.

I am beginning to think that this problem is more interesting than I originally thought. I have convinced myself that the problem is poorly stated. I have convinced myself that I should assume that the farmer was able to make an exact count of legs—even though this seems somewhat unlikely. I have used some of my knowledge of even numbers to analyze the overall situation. I have used some mental "guess and check" to get a feeling for the sizes of numbers that are involved in the problem.

In doing these things, I am displaying some aspects of my math maturity. Mind wandering, thinking, and challenging the problem poser may be by far the most important learning activities going on as I address this problem. That is because these are all aspects of math problem-solving maturity that carry over to other problems.

Finally, I bring my mind back to the task at hand. How about another guess, to keep myself grounded. How about 25 rabbits, which gives me 100 legs. Sounds good to me. But, then there are no chickens. I look back at the original statement of the problem to see if it says that the farmer has both chickens and rabbits. If the farmer has zero chickens and 25 rabbits, would the problem poser say "chickens" and "rabbits?" Hmm. What assumption should I make about this possibly poorly stated aspect of the problem? My first thought was to assume that there are two or more chickens and two or more rabbits.

By now, of course, you have likely solved the problem. I hope that you got more than one correct answer. Many problems have more than one correct answer. It is good to provide students with examples of such problems.

Now, here is another question for you. Can you name any problem that you have ever encountered outside of a math class—either in school or outside of school—that is similar enough to the chickens and rabbits problem so that transfer of learning might occur between the problems? Let me give you a few hints of possibilities. Think about:


 * 1) Problems that were not clearly defined, and in which you had to make assumptions—perhaps based on your knowledge, perhaps based on guesses of the intended meaning—in order to proceed in attempting to solve the problem.
 * 2) Problems in which indirect measurements are used. Instead of counting or measuring exactly what you want to know, you gather the data needed to be able to figure out what you want to know.
 * 3) Problems with more than one solution.
 * 4) Problems that are easily solved by guess and check.

Hamburgers as a Unit of Measure</Center>
Consider the following math word problems.

Problem 1. How many hamburgers from HFFS (the Hamburger Fast Food Store) would it take to circle the earth at its equator if they are laid down flat?

Problem 2. How many hamburgers from HFFS would it take to reach from the earth of the moon if they were stacked on top of each other?

My first reaction is that these are silly problems. There is a lot of water along the equator, so one would have to deal with soggy, sinking hamburgers. Hamburgers stacked on top of each other will mush downward, flattening the ones nearer the bottom. Moreover, the earth and moon are both moving, so the task seems impossible.

On the other hand, think about a middle school student who comes home from school and is asked, "What did you do in math class today?" The student might well have found the hamburger problem fun and interesting, even though it seems silly and impractical. Aha! A good story to share with parents. Perhaps these problems are more likely to be intrinsically interesting than the chicken and rabbits problem.

Many middle school students do not know the circumference of the earth at its equator, and many do not know the distance from the earth to the moon. Moreover, the distance from the earth to the moon varies, as the moon does not have an exactly circular orbit about the earth.

Thus, the problem requires some research and some assumptions. Good! Also, the problem requires making an assumption about how to deal with the fact that the distance from the earth to the moon varies.

Notice also The moons orbit is not an exact circle and that the statement of the problem does not give data on the diameter or thickness of the HFFS hamburgers. It does not raise the issue that the HFFS likely sells a variety of sizes of hamburgers. Good! More data to be gathered or assumptions to be stated.

One of the things this problem illustrates is that to measure length, one needs a unit of measure. A foot is an interesting unit of measure. Is it better than a hamburger as a unit of measure? If the problems are being solved by small teams of students, perhaps some team will want to have a team member doing research on the history of units of measure. Such historical information could liven up the team's report to the class.

How do we know the diameter of the earth at the equator? Note that the equator passes over a lot of ocean, but it also passes over mountains. Does the length of the equator take into consideration going up and down these mountains?

What is the distance from the earth to the moon? Does it vary with the choice of a specific spot on the earth and a spot on the moon? How does one measure the distance from the earth to the moon? Aha. Sounds like indirect measurements to me.

One way we know the distance from the earth to the moon is by timing how long it takes a laser light beam to go from the earth to a reflector on the moon and bounce back to the earth. We measure the time, and make use of our knowledge of the speed of light. Is the speed of light in the earth's atmosphere the same as it is in the vacuum of "outer" space?

As you can see, the problem has a "fun feeling" but is very challenging. The problem can be solved by individual students working alone or by teams of students.

Carbon Footprint</Center>
Quoting from the Wikipedia:


 * A carbon footprint is "the total set of greenhouse gas (GHG) emissions caused directly and indirectly by an individual, organization, event or product". For simplicity of reporting, it is often expressed in terms of the amount of carbon dioxide, or its equivalent of other GHGs, emitted.

There are a variety of Carbon Footprint Calculators available on the Web. For example, The Nature Conservancy provides one to do a personal (individual) calculation and one for doing a household calculation.

The following examples might well be used in a math and science problem-based unit of study.


 * Problem 1: Determine your personal carbon footprint and develop a feasible plan of how to reduce it by 10 to 15 percent.


 * Problem 2: Determine your household carbon footprint and develop a feasible plan of how it could be reduced by 10 to 15 percent.


 * Problem 3: Determine your school's carbon footprint and develop a feasible plan of how it could be reduced by a significant amount. One approach to this problem would be to divide a class into teams of three to four students, and have each team work on the problem. After the team projects are completed, a whole class plan could be developed.

The overall idea of Carbon Footprint involves a lot of science and mathematics. Much of the input to a Carbon Footprint Calculator will consist of estimates. Moreover, the various methods for estimating a person's, household's or organization's carbon footprint will not produce identical answers.

This certainly is a real-world problem. The problem can be considered at a personal (individual) level, and each person is apt to come up with different approaches to decreasing his or her carbon footprint. Problem 3 can have individual groups producing quite different solutions, and may well generate competition among the teams.

The problem involves data gathering, where some of the data gathered or estimated may be much more accurate than other data. This introduces the idea of an estimate or measurement falling in an interval. It introduces the idea of calculating a range of estimates for a carbon footprint, rather than a specific number.

The problem lends itself to developing and using a spreadsheet model to do the needed calculations. Thus, students can learn about math modeling of an important science problem.

Improving Math Education
Many people are not satisfied with the quality of math education students are achieving. Many of them recommend that this "problem" in math education can be attacked by doing more of the same. Thus, we work to develop better text books and teacher materials, educate teachers to be better in teaching via such books and materials, require students to take more math, make the school day and the school year longer, and so on.

It is very difficult to substantially modify this approach. Our math education system is huge, well established, and highly resistant to change. For many years, Michael Fullan has been a world leader in educational reform. Over the years his thinking has evolved from focusing on the teacher as a unit of change, to the school being a unit of change, to the school district being a unit of change, to the state or province being a unit of change, and finally to an entire country being a unit of change.

Fullan's work can be considered as being supportive of current top down approaches being proposed and implemented in the United States. But, where does this leave the individual teacher? What can you, personally, be doing to improve the quality of math education your students are obtaining?

What You, Individually, Can Do</Center>
This document argues that a teacher of math can improve the math education of his or her students by making more effective use of math word problems. Part of the improvement will come through changes in the methods and approaches used by the individual teacher, and part will come through changes in the math word problems being assigned, and in their surrounding curriculum. The changes suggested in this document include:


 * 1) Place increased emphasis on math word problems—and, especially, on "good" math word problems. Learn to distinguish the good from the not so good, and significantly decrease use of the "not so good" word problems that are so prevalent in math textbooks and from other sources such as the Web.
 * 2) Use math word problems as a vehicle (as an environment and climate) in which to help students gain in math maturity. Make math maturity one of the ideas you hold in mind as you prepare math lesson plans and teach math classes.
 * 3) Reduce the use of math word problems in drill and practice, rote memorization, and teaching to the test modes.
 * 4) Use math word problems to increase the emphasis on math modeling, computer modeling, and other aspects of computational thinking in the math curriculum you teach.

References and Resources
The Web contains a large number of sites that provide free math word problems. Several are listed below. They may provide a useful start in finding math problems that you deem appropriate for use in your teaching.

The sites contain a number of sample problems. Please do not assume that the problems presented are all good examples of good math word problems. Albrecht, R. & Moursund, D. (2016), Math problems divorced from reality. Retrieved 3/18/2016 from http://iae-pedia.org/Math_word_problems_divorced_from_reality.

This site explores a variety of math word problems that are so unrealistic that they made the authors laugh. We believe they are examples of how not to teach math. Quoting from this site:


 * One of the key ideas in math problem solving is to check one's work and final results to "see" if they make sense. Well-designed word problems that are rooted in reality have the characteristic that one can use the reality-based roots to help check the correctness of one's results.

IXL Learning (n.d.). Word problems. Retrieved 3/18/2016 from https://www.ixl.com/math/word-problems. Quoting from the website:


 * Here is a list of all of the [K through precalculus] skills that cover word problems! These skills are organized by grade, and you can move your mouse over any skill name to view a sample question. To start practicing, just click on any link. IXL will track your score, and the questions will automatically increase in difficulty as you improve!

majortests.com (2016). SAT math problem solving. Retrieved 3/18/2016 from http://www.majortests.com/sat/problem-solving.php.

This site contains 12 problem-solving practice tests, each containing 10 multiple-choice problems. Some of them are word problems. Quoting from the site:


 * Practice your math problem solving skills with our tests. Use a calculator only where necessary. You shouldn't need more than three lines of working for any problem. Redraw geometry figures to include the information in the question.


 * Each test has ten questions and should take 12 minutes.


 * Initially don't worry too much about the time until you have a feel for the type of questions. But, by the time you have done two or three tests you should start getting tough about the time you take.

Math-Aids.com (n.d.). Word problem worksheets. Retrieved 3/18/2016 from http://www.math-aids.com/Word_Problems/. Quoting from the site:


 * Here is a graphic preview for all of the word problems worksheets. You can select different variables to customize these word problems worksheets for your needs. The word problems worksheets are randomly created and will never repeat so you have an endless supply of quality word problems worksheets to use in the classroom or at home. Our word problems worksheets are free to download, easy to use, and very flexible.


 * These word problems worksheets are a great resource for children in 3rd Grade, 4th Grade, and 5th Grade.

Montgomery, C. (11/21/2015). SAT/ACT Prep Guide to SAT Math word Problems. Retrieved 3/18/2016 from http://blog.prepscholar.com/sat-math-word-problems. Quoting from the website:


 * A word problem is any problem is based mostly or entirely on written description. You will not be provided with an equation, diagram, or graph on a word problem and must instead use your reading skills to translate the words of the question into a workable math problem. Once you’ve done so, you can then solve for your information (if necessary).


 * About 25% of your total SAT math section will be word problems, meaning you will have to create your own visuals and equations to solve for your answer. Though the math topics may vary, SAT word problems share a few commonalities, and we’re here to walk you through how to best solve them.

Syvum (n.d.). Kids math word problems. Retrieved 3/18/2016 from http://www.syvum.com/math/wordproblems/level1.html. Quoting from the website:


 * Free online quizzes on Math Word Problems and Math Exercises including Addition, Subtraction, Multiplication, Division and Arithmetic problems.


 * Recommended Age: 6-7 years. Knowledge Required of 1-2 digit numbers.


 * See also, Kids Math Word Problems II for kids 8-9 years of age.

IAE Math Resources Mentioned in This Document
Communicating in the language of mathematics.

Computational thinking.

Critical thinking.

Education for increasing expertise.

Improving our math education system.

Learning problem-solving strategies by using games: A guide for educators and parents.

Math maturity.

Math problems divorced from reality.

Problem solving.

Transfer of learning.

Two brains are better than one.

Materials Available Free from Bob Albrecht

 * A half-bakery of math, science, and other ideas by Bob & George.


 * Albrecht games.


 * Albrecht number.


 * Free Books by Bob Albrecht.


 * Hurkle.

Author or Authors
Initial versions of this page were developed by Bob Albrecht and David Moursund. Others are encouraged to contribute via email to moursund@uoregon.edu.