Arithmetic and Language DeveloPment Learning the Language of Elementary Arithmetic By Rob Madell, Ph.D., and Jane R. Madell, Ph.D., CCC A/SLP, LSLS Cert. AVT olta Voices would like to intro- duce a 5-part column focusing on how arithmetic word prob- lems can aid in the development ofV listening and spoken language for children who are deaf and hard of hearing. Professionals may not recognize that learning to solve arithmetic word prob- lems involves language learning. Some children learn such language informally, uey Photography H either from parents or peers, but many raig do not. In the case of children who are C deaf or hard of hearing, parents, teach- redit: ers, speech-language pathologists and C listening and spoken language specialists Photo (LSLSs) should be prepared to help. The unique language of arithmetic word problems provide unique opportunities for developing listening and spoken language. Through this series, we will show that there is a connection between learning Jessica: (She takes a quick look at the candy corns with 4 of yours. (Rob helps arithmetic and learning language. This small pile of candy and, without count- her to do that.) article will show you that, in general, ing, answers.) I’ve got 4. Rob: (Points to Lynn’s additional candies.) word problems incorporate important Rob: Lynn, it looks like you have more These are the extra candies that you have. language distinctions. In the articles to than Jessica has. How many pieces of How many extra ones do you have? follow we will examine more thoroughly candy corn do you have? the language of addition, subtraction, Lynn: 1, 2, 3, 4, 5. multiplication and division. Lynn: (Counting the candies) 1, 2, 3, 4, 5, Rob: Right, you have 5 more candy corns Arithmetic word problems offer a large 6, 7, 8, 9. than Jessica. Here are your 9 (pointing), array of language development oppor- Rob: Lynn, how many more pieces of here are Jessica’s 4 (pointing), here are tunities. Here is an example involving a candy do you have than Jessica has? your 4 that match Jessica’s (pointing) child, Lynn, who has a profound hearing Lynn: (She seems to think that Rob is and here are your 5 extras (pointing). loss and uses bilateral cochlear implants. repeating himself. She looks at him like In trying to help, we show Lynn what to She is trying to solve an arithmetic word she doesn’t understand why and repeats count and use that language over and problem and it should be clear that her her previous answer.) I have 9. over again, always relating the words difficulty is as much about language as it Rob: Right, you do have 9, but I want back to the physical candies. We talk is about arithmetic. about “matching up” some of Jessica’s It is Halloween and Jessica and Lynn to know how many more you have than Jessica has. candies and about Lynn’s “extra ones.” At have collected way too much candy, the end of the conversation we use the including lots of candy corn. As it happens, Lynn: (No response.) words of the problem: “You have 5 more Jessica has collected 4 pieces of candy corn Rob: Well how about this, who has candies than Jessica has” and we essen- and Lynn has collected 9. more…you or Jessica? tially re-state the problem and the solu- Rob (Jessica and Lynn’s teacher): Jessica, Lynn: I have more. tion. We are trying to teach her what the how many pieces of candy corn do you words of the problem mean with respect have there? Rob: We need to figure out how many to how the candies must be manipulated more you have. Let’s match up Jessica’s 4 and what must be counted. 32 VOLTA voices • noveMber/deceMber 2010 Here is an example involving a multi- (essentially 9 2 4 5 ), we used the There are 3 shelves of books in your room. plication word problem. The child, Mike, actual candies that the words of the There are 4 books on each shelf. How many has a moderately severe hearing loss and problem referred to. In Mike’s outfit books do you have altogether? uses bilateral hearing aids. In this case, problem (essentially 3 3 4 5 ), he Mathematics educators disagree about it is not that Mike does not understand. represented his understanding of the number of different word problem Rather, he has misunderstood. Once the problem with the blocks that he models. But by our count and for the again his difficulty is as much about happened to be building with. purposes of this series, there are two language as it is about arithmetic. 2. For each word problem, the physical addition models, three subtraction mod- Jane (Mike’s teacher): I have a very hard model that represents the problem els, three multiplication models and two problem for you to try to solve. Suppose can be manipulated so that together division models. that your dog, Punch, has 3 collars. with appropriate counting, the Some children learn how to model problem can be solved. some word problems without direct, Mike: She only has 2. Mathematics educators agree that explicit instruction. But very few chil- Jane: OK. But just suppose that she had before children start to memorize dren (with or without typical hearing) 3 collars. And suppose also that she had “facts,” such as 9 2 4 5 5 and 3 learn to model them all in this way. 4 scarves. 3 4 5 12), they should first learn Parents, teachers, speech-language Mike: (He has been building with some to solve word problems by mak- pathologists and LSLSs should make wooden blocks and picks out 3 blocks to ing physical models of them and themselves familiar with all the dif- represent the collars and, in a separate counting. ferent models. They should systemati- pile, 4 blocks to represent the scarves. 3. If you pay careful attention to the cally introduce lots of different word Without even knowing what the question actual words of mathematical word problems and help children understand is, Mike knows that it will be helpful to problems, you see that problems, the language of those problems so that make a physical model of it.) which may seem similar, require they can model and solve them as well distinct models, manipulation and as develop better listening and spoken Jane: OK. Good. Now when Punch counting. As a consequence there language skills. gets dressed up for something special, are more types of word problems In the articles to follow we will she likes to wear a scarf and a collar. So than you might think. examine addition problems, subtrac- here is what I want to know. How many For example, we could have asked Lynn tion problems, multiplication problems outfits does Punch have? So if she wants the following: and division problems. Altogether, 10 to wear a scarf with a collar, how many You have 9 candy corns. Suppose that you different word problem models will different outfits can she make? gave 4 of them to Jessica. How many would be discussed. Collectively the articles Mike: (Matching 1 “collar” with 1 you have left? present all of the word problem models “scarf”), here is 1 outfit. Like Lynn’s original problem, this one that children are likely to see in school 2 5 Jane: OK. can also be represented by 9 4 . and will provide suggestions for helping But the words describe an entirely differ- children with hearing loss develop the Mike: (Matching a second “collar” with ent situation. The two problems require language of arithmetic. a second “scarf,” and then the third “col- very different models. lar” with a third “scarf.”) She can make 3 And here is a problem that, like Mike’s, Editor’s Note: Also available in Spanish at outfits. can also be represented by 3 3 4 5 . www.t-oigo.com / También disponible en Mike: (He pauses and then looks at the But it is much easier to model. español en la página web, www.t-oigo.com. fourth scarf.) But what should I do with this? Mike’s misunderstanding is not unrea- sonable. Punch can in fact only assemble 3 outfits at any one time. Although the words of the problem don’t make it explicitly clear, the intent is to ask for all possible combinations. You might want to think about how to help Mike represent those combinations so that he can count them. Examples like these illustrate three things about elementary arithmetic: 1. Every word problem can be repre- sented by a physical model of the problem. In Lynn’s candy problem VOLTA voices • noveMber/deceMber 2010 33.