Helping Children Succeed with Word ProblemsAuthor(s): Eula Ewing Monroe, Robert Panchyshyn and Damon L. BahrSource: Mathematics in School, Vol. 35, No. 1 (Jan., 2006), pp. 4-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215850 .

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HBLPING CHILDRBEN SUEBBD WITH WORD PROBLBMS

by Eula Ewing Monroe, Robert Panchyshyn and Damon L. Bahr

No other task inspires more math phobia in the hearts of adults, including a sizable number of elementary school teachers, than - WORD PROBLEMS! In a well-known comic panel from The Far Side series, Gary Larson depicted "Hell's library" filled with only such titles as Story Problems, Story Problems Galore, Story Problems Vol. 1, More Story Problems, and Big Book of Story Problems (Larson, 1988).

Nonetheless, the events of daily life are filled with the need for problem solving. These problems do not come prepackaged in ready-made algorithms (e.g. 7 x 42 = ?). They are often embedded in details from which we must sift out the information we need to answer our questions. Frequently the questions themselves are elusive; we spend much of our time in real-life problem solving trying to figure out exactly what it is we need to know. Thus we encounter real problems that must be described with words and numbers, i.e. word problems, and we must structure them and make meaning for ourselves. The ability to make sense of problem situations in real life is a key component in mathematical understanding (NCTM, 2000).

For students of mathematics, particularly the children we teach in elementary school, solving problems should be not only the goal of learning mathematics but also the means of facilitating that learning (NCTM, 2000). Children come to school with a great deal of informal mathematical knowledge couched in real-world contexts. Their experiences with solving real problems in their everyday lives (for example: "We have 4 biscuits and 2 people. How many for each person?") can serve to inform their problem solving in the classroom. Indeed, these contexts support children's mathematical thinking while abstract symbols often do not (Ambrose et al., 2004).

Extensive research is available to help us better understand children's problem solving. For example, we know that the way a child conceptualizes and thus solves a problem involving separating one quantity from another ('take away') can be much different from the way she or he might think about finding the difference between two numbers, even though both situations involve subtraction (Carpenter et al., 1999; Fuson et al., 1996). However, we also know that interference with problem solving may be caused, at least in part, by a problem with the words and contexts (Monroe and Panchyshyn, 1995-96), a 'word' problem, if you will! For students who speak English as a second language, this problem is typically exacerbated (Stoller and Grabe, 1993).

When word problems are carefully selected and sequenced for the mathematics they embody, and when they utilize words and contexts that are familiar to the children who are

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solving them, they need not be the nemesis they once were. The ideas below are provided to help you reduce the problems caused by words and contexts, thus enabling your students to direct their energies to the mathematical reasoning needed for solving problems successfully.

1. Remember the importance of teaching vocabulary in each mathematics lesson. Carefully teach unknown words and concepts just as you would in a literacy lesson.

2. To allow your students the time needed to explore the mathematics presented and come up with alternative solutions, assign only a few well-chosen problems. Oftentimes, two or three problems can provide sufficient contexts for focusing on the mathematical ideas at the heart of the lesson.

3. Allow additional time for those alternative solutions in small and large groups. Encourage students to compare, challenge, and defend their solutions and strategies.

4. Adapt the word problems provided in your curriculum materials. Use the names of people, places, and activities that are familiar to your students.

5. Utilize other curriculum areas as a source for word problems. A book you have read to the class can provide characters, settings and situations for meaningful problem solving, as in this example:

Little Red Riding Hood walked happily through the woods. In her basket she carried three sugar cookies and two chocolate chip cookies for her grandmother. How many cookies did she have?

6. Health and science also provide a wonderful supply of mathematical problem solving situations. For example:

On a cold, cloudy day, the temperature outside was 17' Celsius. Then the sun came out, and the temperature rose 8 degrees. What was the temperature then?

7. Use enough words to provide a meaningful context. A short word problem is not necessarily easier; rather, sufficient context needs to be provided for students to be able to 'see' the problem mentally.

8. Develop word problems from students' real-life experiences. For example, use the local holiday parade as the theme for a set of word problems:

Many people sat in the brightly decorated stands to watch the cricket match. The red section seated 500 people, the white section seated 400 people, and the blue section seated 450 people. How many people could sit in the stands?

Mathematics in School, January 2006 The MA web site www.m-a.org.uk

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9. Encourage children to write their own word problems or to rewrite textbook problems to reflect their interests and experiences. A problem written by 6-year-old Shawn:

I don't got 2 dogs. I don't got 3 cats. How many did I don't got?

10. Present problems in alternative ways (orally or in writing, with pictures, concrete objects, or symbols) and invite children to represent the same problem using other alternatives.

Word problems need not be a source of frustration for you or your students. Under appropriate instructional conditions, word problems not only can be negotiated successfully by your students, but they can also be the vehicle to promote greater mathematical reasoning and achievement. A well- developed mathematics curriculum can be a wonderful support in promoting the depth of mathematical thinking that the use of word problems can provide. However, because you know the students in your classroom, you are in the best position to make informed decisions about the words and contexts that are most likely to promote student success. I

Acknowledgement The authors wish to thank Robin J. Montgomery, research assistant at Brigham Young University, for her assistance in editing and proof reading this article, and Dr. Brad Wilcox, a valued colleague, for his review and recommendations.

References

Ambrose, R., Clement, L., Philipp, R. and Chauvot, J. 2004 'Assessing Prospective Elementary School Teachers' Beliefs about Mathematics and Mathematics Learning: Rationale and Development of a Constructed- response-format Beliefs Survey', School Science and Mathematics 104, 2, 56-69.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L. and Empson, S. B. 1999 Children's Mathematics: Cognitively Guided Instruction, Heinemann and National Council of Teachers of Mathematics, Portsmouth, NH.

Fuson, K. C., Carroll, W. M. and Landis, J. 1996 'Levels in Conceptualization and Solving Addition and Subtraction Compare Word Problems', Cognition and Instruction, 14, 3, 345-371.

Larson, G. 1988 Night of the Crash-test Dummies, Andrews McMeel, Kansas City, MO.

Monroe, E. E. and Panchyshyn, R. 1995-96 'Vocabulary Considerations for Teaching Mathematics', Childhood Education, 72, 2, 80-83.

National Council of Teachers of Mathematics 2000 Principles and Standards for School Mathematics, Author, Reston, VA.

Stoller, F. L. and Grabe, W. 1993 'Implications for L2 Vocabulary Acquisition and Instruction from L1 Vocabulary Research'. In Huckin, T., Haynes, M. and Coady, J. (Eds), Second Language Reading and Vocabulary Learning, (pp. 24-44), Ablex, Norwood, NJ.

Keywords: Word problems; Context.

Authors Professor Eula Ewing Monroe, Department of Teacher Education, Brigham Young University, Provo, UT 84602, USA. Professor Emeritus Robert Panchyshyn, Department of Teacher Education, Western Kentucky University. Damon L. Bahr, Associate Professor, Department of Elementary Education, Utah Valley State College. e-mail: Eula_Monroe@byu.edu

POWERS AND AGES: NUMBER THEORY AT THE BIRTHDAY PARTY

by Bonnie H. Litwiller and David R. Duncan

An acquaintance of the authors, Leonard, recently celebrated his 64th birthday. It was noted that this number is both a square and a cube (64 = 82 = 43). Besides the trivial case of 1, this was Leonard's first square/cube birthday age. The natural next question is when this square/cube situation will next occur for Leonard. To analyse it symbolically, we wish to find the next instance for which a2 = b3.

In our analysis, we first observe that the prime factors of a and b must be the same in order for a2 to equal b3. We will first consider the simplest case, in which a and b are both powers of 2 alone. Let us write a = 2" and b = 21. Then the required square/cube situation would yield (2")2 = (2')3 or 22u = 23v; implies that 2u = 3v or v = 2u. The simplest case for which this is true is: u = 3, v = 2. This yields the already noted solution pair of a = 23 = 8 and b = 22 = 4, for which a2 = b3 = 64.

The next larger (u, v) solution pair would be u = 6 and v = 4. This results in a = 26 = 64 and b = 24 = 16. The resulting square/cube birthday age is 642 = 16 = 4096, an unlikely possibility!

Mathematics in School, January 2006 The MA web site www.m-a.org.uk

Is there another square/cube 'birthday' age smaller than 4096? Suppose that a = 3u and b = 3v. Then a2 = b3, yielding 32u = 33v and, again, 2u = 3v. Using the smallest (u, v) pair for which this holds, u = 3 and v = 2:

a = 33 = 27 and b = 32 = 9

a2 = 272 = b3 = 93 = 729.

While 729 is smaller than 4096, Leonard's chances of reaching this age are indeed remote. He had better enjoy his 64th birthday, since this is the only square/cube age he is likely to see!

The reader and his/her students are encouraged to develop and solve other number theory curiosities. U

Keywords: Powers; Number theory.

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Authors Bonnie H. Litwiller and David R Duncan, Professors of Mathematics, University of Northern Iowa, Cedar Falls, Iowa 50614-0506, USA.

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