# “a problem is something you don’t want to have”: with learning the nursery rhyme “baa, baa,

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Although Adrian, when asked by his teacherwhat a problem is, was adamant that prob-lems are to be avoided, educators believeproblem solving is central to mathematics teachingand learning (NCTM 2000). Problem solving sup-ports students as they apply their skills and theirknowledge of mathematical concepts andprocesses to a range of different contexts and asthey construct knowledge by reflecting on theirown physical and mental actions. When childrensolve problems together, learning is a socialprocess in which they learn not only from theteacher but also by discussing, arguing, and nego-tiating with their peers.

In his book How to Solve It, first published in1945, Plya proposed four fundamental steps forsolving problems: (1) understand the problem, (2)devise a plan, (3) carry out the plan, and (4) lookback and check the solution. Although headdressed complex problems, these steps areapplicable to even simple word problems. Under-standing the problem requires children

to be able to read the problem; to comprehend the quantities and relationships

in the problem; to translate this information into a mathematical

form; and to check whether their answer is reasonable.

In a study of the errors children make when solv-ing word problems, Newman (1983) found thatalmost half the errors made by a large sample ofsixth-grade students involved their inability tocomprehend the problem or translate it into a math-ematical form rather than to carry out the mathe-matics.

The recent emphasis on the importance of stu-dents social interactions in their constructions ofmathematics, combined with Plyas fundamentalsteps, would seem to have implications for teach-ing problem solving to young children. Childrenneed to discuss different interpretations of prob-lems and the specific mathematical terminologythey encounter if they are to understand the prob-lem. To devise and then carry out a plan, children,individually or in small groups, need to identifynumerical quantities and relationships, as well asdiscuss how to represent them in a mathematicalform. Children can then attempt to solve the prob-lem, and perhaps refine and modify their plan.Finally, young children need to be encouraged toreflect on their solution; they might be asked toexplain to the teacher or their classmates why theirsolution makes sense or how they have representedthe problem.

146 Teaching Children Mathematics / October 2005

Lynne Outhred, Lynne.outhred@mq.edu.au, teaches in the Teacher Education Program atMacquarie University in Australia. Her research interests are childrens representations ofproblem situations and their development of measurement concepts, particularly the conceptof a unit of measurement. Sarah Sardelich, sezpost@ihug.com.au, believes teaching is aboutempowering children to engage in the discovery process so that each learning experience isexciting and fulfilling. She ensures that she has high but reasonable expectations of everychild and that she is as passionate about their discoveries as they are. Sarah currently teachesat Narraweena Public School in Sydney, Australia.

By Lynne Outhred and Sarah Sardelich

Problem Solving by Kindergartners

A Problem Is Something You Dont Want to Have:

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Copyright 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

Giving children opportunities to pose as well assolve problems allows them to engage with signif-icant mathematical ideas (English 1997). Problemposing is an open-ended activity that can be a pow-erful assessment tool because it reveals childrensunderstanding of problem structure; teachers mayfind that some children devise problems thatinvolve significantly more difficult mathematicsthan is being taught. The contexts within whichproblems are situated may also affect the types ofproblems that young children pose; familiar con-texts are easier and, therefore, may elicit moremeaningful problems (Lowrie 2002).

Both the problems that children pose and theirrepresentations of problem situations provide theteacher with a window into childrens thinking.The step of representing the problem is fundamen-tal because representations indicate childrensstructuring of fundamental ideas (Lopez-Real andVeloo 1993). Young children may represent numer-ical quantities and the relationships among thesequantities in a word problem by physically model-ing the situation with concrete materials, by draw-ing their solution strategy, or by writing a symbolicexpression. At present, however, we know little

about the ways in which childrens internal repre-sentations of problem situations generate externalrepresentations, such as drawings or models, dur-ing problem solving, especially for very youngchildren. We also do not know what factors influ-ence the development of childrens representationsor what the teachers role is in supporting thisprocess.

This article describes the results of an actionresearch project in which young children in oneclassroom were encouraged to solve and pose theirown problems in their first year at school.

The Kindergarten ClassThe children in this class were all 5 years of ageand were in kindergarten in a school in Sydney,Australia. The kindergarten class was separatedinto four mathematics groups on the basis of abil-ity at the beginning of term 2schooling is basedon a four-term yearalthough these groupingswere not fixed and some children later changedgroups. The groups rotated through different activ-ities over the week; the teacher worked with theproblem-solving group while volunteer parents

Teaching Children Mathematics / October 2005 147

worked with the other three groups. The childrenparticipated in a problem-solving session once aweek for sixteen school weeks, and during mathe-matics activity time they solved and later posedproblems (see fig. 1). The problems involved repe-tition of a number pattern and included addition,subtraction, multiplication, division, and simplefractions (one-half).

The contexts of the problems (see fig. 1) werebased on either literature read in class or familiarsituations, such as occurrences on the school play-ground. The initial impetus for solving problemsbegan with learning the nursery rhyme Baa, Baa,Black Sheep. Accordingly, some of the first prob-lems the children solved involved sharing situa-tions similar to those encountered in that poem.The teacher modeled the situation of three people

and three bags of wool by using different coloredcubes. She asked, We have three bags of wool;how many would each person get? The childrenreplied one, so she repeated the question for six,then nine, bags. Because the children had little dif-ficulty with this concept, she asked them, What ifwe have ten bags? Common responses were togive the extra bag back to the sheep or to divide thewool into equal shares.

In the problem-solving sessions, the teacherbegan by reading the problem to the children andshowing how the quantities could be modeled withcubes. Next the children modeled the situationindividually. This step was crucial because the chil-dren were not yet able to read, so the colored cubesacted as a memory aid. Then children were askedto draw a picture to show how they solved the

148 Teaching Children Mathematics / October 2005

Figure 1The problems used in the problem-solving sessions

Problems

Two monsters were playing, two monsters were flying, three monsters were dancing. Howmany monsters were at the monsters party?

Four monsters are at a party. There are eight little cakes with cherries on top. How manycakes will each monster get?

Grandma has baked 6 cookies. Tom, Lizzie, and David love her cookies. How many will theyget each?

There were 18 cookies on the plate. Tom was very hungry and ate half of them. How manycookies were left on the plate?

Three dinosaurs each had two mittens. How many mittens altogether?Derek the Dinosaur knitted 4 pairs of socks. How many dinosaur feet were kept nice

and warm?

In the story The Shopping Basket, Jeffrey had to buy lots of things. How many things did hehave to buy?

Make up your own shopping list using the same number pattern.

In the story of Stellaluna, the baby fruit bat makes friends with some baby birds called Pip,Flitter, and Flap. Stellaluna found 12 bugs. She shared them with Pip. How many bugs dideach get?

Flitter and Flap came to join the party. Stellaluna came to join the party. Stellaluna and Piphad to share their bugs. How many bugs did each get?

In Alexs Bed, Alex rearranges his bed to make more space. Think about your bedroom andthe things that are in it. Draw a plan of your bedroom.

Imagine that you could have the bedroom of your dreams. Draw a plan of how it wouldlook.

In Amys Place, Amy discovers some possums investigating her new tree house. If therewere 5 mother possums, and each had 3 babies, how many would there be altogether?

In the book Bear and Bunny Grow Tomatoes, Bear grew lots of large, juicy tomatoes andshared them with his friend Bunny. Bear gave Bunny two boxes of tomatoes. Each box had 8tomatoes inside. How many tomatoes did Bear give Bunny?

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problem. Finally, the children explained to theteacher how they had represented the quantitiesand relationships in their drawings. Considerableemphasis was given to showing their thinking intheir drawings, and aspects of childrens pictorialrepresentations were discussed in their group ses-sion. Groups generally attempted the same prob-lems, but some children attempted additionalproblems and some children required far moresupport than other

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