Michigan State University > College of Education : JDSDE-Author-Corner : The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students - messages

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : Singapore Math Model Drawing for Problem Solving

SueClark — Thu, 14 Feb 2008 04:24:56 GMT

Hi Claudia,
I enjoyed this forum! Unfortunately, I'm joining late but am hoping you'll get this and reply.

I've been reading more about the Singapore's approach to teaching math, especially since they are a world leader in mathematic education. I'm wondering if you are familiar with the Singapore Math problem solving approach and if you know of anyone who is trying this method with deaf and hard-of-hearing students. I strongly believe in instruction that follows the sequence of a concrete level, followed by a representational/pictorial level, and them moving to the abstract level. This sequence will help students become successful and confident problem solvers.

Please share any views you may have on the Singapore Math 8-Step Model Drawing. And thank you so much for this opportunity to share in the discussion.

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : Singapore Math Model Drawing for Problem Solving

SueClark — Thu, 14 Feb 2008 04:24:37 GMT

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : Singapore Math Model Drawing for Problem Solving

SueClark — Thu, 14 Feb 2008 04:24:30 GMT

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Paragraph #6: Suggestions ...? re needed data

reneet — Wed, 13 Feb 2008 15:56:37 GMT

Hi! That comment grabbed my attention also. Terms such as "altogether", "difference", "how many more/less than", etc. are typically difficult for my students to learn. Like you, I teach them the rote definition of math terms. I explain to my students that these words and phrases are "clues" within the word problem to help them know which operation (add/subtract) to perform. But, this only allows the students to get the problem right. By substituting a temporal word such as "now", it just seems to make more sense to the student. I think that it fits with the way that many hearing impaired students think about the world. I often rephrase as a strategy to support students in their mainstream classes and for test modifications. A simple change such as this can make a big difference in the student's understanding.

I don't believe that you have arrested her math development, however. It is important that your child understand the meaning of all words so that as she reads them independently, she will be able to do her work. Perhaps, in your explanations to her, you could add that when she sees "altogether" it means "how many now?". It is wonderful that you are working so closely with your child!

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Paragraph #6: Suggestions ...? re needed data

SK17 — Tue, 12 Feb 2008 06:50:39 GMT

I'm interested in the comment about utilizing the vocabulary of math in every day usage but not focusing on keywords such as in the example of "altogether" referring to "addition". I have focused on those words until my daughter learned them, and it did seem to assist her in solving story problems. Perhaps I have arrested her math development.
I'd be curious to hear an elaboration of this idea.
(And what an exciting opportunity for parents to participate in! Thank you!)
Sara

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: ASL numbers

niwri2 — Sun, 10 Feb 2008 23:51:44 GMT

The information about ASL numbers came from a college-level ASL class (most of the students were aspiring interpreters) taught by a native signer (CODA) who is also one of the top interpreters in my state. Numbers are like fingerspelling--except with fingerspelling you can catch some of the letters, the overall shape of the word, and use sentence context to figure it out. You don't get the same help with numbers. Since she pointed this out to the class, I've just been observing in a casual kind of way--the examples I gave are just things I've noticed and may not have any broader meaning. Another example I could give is the difference I noticed between the way numbers for a gift exchange were presented by a deaf signer to a deaf group, and the way numbers are called out for a drawing in a hearing environment. Much more repetition of the sign--and this is when everyone is watching for his or her number.

If I sign "let's meet at six," not only do I sign time-six, I will hold the six a little longer and give it a little bit of an up-and-down quiver. "Six dollars" has an inward twist. If I sign "78" I will move my hand slightly up for the "8" to help distinguish it from "87", for which my hand would move down. I can't even remember everything we covered in class--this was quite a few years ago--but the take-home message was that signing numbers is more complex than many people realize.

My daughter was in a school for the deaf, in a class with only about 4 students. They were in middle school by this time.

This is just something that I've been curious about. It's not so much a matter of whether numbers are being signed "properly" by a teacher. It's what ASL as used by native signers tells us about how ASL is perceived--signs have evolved to be readily understandable and easy to execute, and if numbers are given special emphasis by being modified in various ways, it suggests that there is a practical reason for this.

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: ASL numbers

Pagliaro — Sun, 10 Feb 2008 16:16:14 GMT

I've read this post several times since it was entered. In order to respond, however, I need clarification. First, what do you mean that ASL numbers are "small and easily confused"? Do you mean, for example, the difference between the sign for 7 and the sign for 8 is a minimal move of handshape? Second, I am not clear and/or maybe do not agree that the numbers signs in ASL are held, shaken, twisted, and moved to make them more visible. But again, I ask for clarification. Finally, could you also clarify your daughter's school situation as well as her and the teacher's signing abilities. There are a lot of factors that may be involved in children asking for repeated page numbers. I've been in many hearing classrooms where similarly the teacher had to repeat saying the numbers. So, I don't think that we can make any statements here about number processing.

Again, clarification of the above will help me to better address your final question.

Thanks! Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Fact Families

Pagliaro — Tue, 05 Feb 2008 16:26:10 GMT

First, we think that this will "work" with any child - deaf, hard of hearing, or hearing IF they first have an understanding of "number" and of cardinality (that is, that a number can represent a group of elements). Children need to first understand that a group of elements can be regrouped into smaller sets still representing the whole. Without this, the numerals in the number families that you mentioned don't really mean anything. Students will then try to memorize those families. The problem is that memorizing the abstract facts without having an understanding of what those facts mean or represent will limit the child in their application or use of those facts in problem solving.

There are two further issues that need to be addressed here. One is whether there is a particular order to best introduce the four arithmetic operations (addition/subtraction; multiplication/division). The second is how best to help children come to know arithmetic number combinations (facts and fact families). How one answers these two questions may be related.

By solving arithmetic problems in context (story problems), operations can be linked to an action or relation that children understand. The “operation” may be a movement made while modeling a problem’s solution (e.g. subtraction as separating one set of cubes from another and addition as putting one set together with another) or a signed counting string (e.g. counting up as addition, counting down as subtraction). Helping children find ways to use math symbols to record what they are doing, will provide a link between the operation symbols and their meaningful solutions.

If children have a chance to solve a variety of problems (different operations) frequently and talk about their solutions, their “math talk” will be about putting numbers together and taking them apart. If, while explaining their strategies children’s ideas are recorded using symbols, discussion of the different ways will lead to links between the operations and thus, in a meaningful way, to fact families.

The Authors

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Paragraph 4 and 5

Pagliaro — Tue, 05 Feb 2008 16:24:31 GMT

It is helpful to recognize that if children are solving a problem that is a multiplication situation such as the one you describe (number of groups is given and the value of each group is given or known but the idea that the group value needs to be repeated is implicit rather than explicit), they are doing multiplication, even if they are modeling the situation and counting by ones, as you indicate.
Given that this is the way these students chose to solve this problem, we would guess that they are not ready for the multiplication symbols. we would suggest that you continue to work on problems such as this one, allowing students to use and discuss the various ways in which they solve the problems with each other. As children have a chance to frequently solve multiplication situations and explain their thinking and listen to others describe their thinking, their strategies will develop. From modeling and counting all the legs one by one, they may begin to add 4’s (4+4 is 8 + 4 is 12, 13 14 15 16, 17 18 19 20, 24) or skip count 4, 8, 12, 13 14 15 16, 17 18 19 20, 24 (or some variations), and then use doubling strategies where appropriate (4+4=8, 8+8=16, 20, 24).

An important mediator for children solving and discussing this work is the recording of these strategies. As students begin to articulate their thinking, we can provide a way to record their thinking. Through recording, mathematical symbols can be introduced and there will be a record of work to compare strategies (e.g. discuss the similarities and differences between).

Another way to encourage the development of various strategies is to pay close attention to the problems selected and the numbers in the problems. The problem you gave above, for example, could be symbolized as 4x4 + 2x4. Perhaps it would be helpful for your students to solve problems that involve only one kind of animal (even if they have the same number of legs, this is a complex construction that I ignored in the solutions I gave above). Consider a sequence of problems that might encourage certain strategies. For example, if students tend to skip-count by 5’s, give a problem about starfish. You might give a problem about 2 starfish, then another problem that involves 6 starfish and see if they make use of their first solution in their solution to the second problem. You might also consider having students work together who bring different ideas (strategies) to the table.

In some instances, you, yourself, may want to model a good checking strategy by going back to the story and specifically the question after an answer is given, asking if the answer (even your own answer) makes sense?” You may think “aloud” saying, “That seems kind of low to me since I know that 1 lion as 4 legs. Perhaps we should draw this out.” Then, do as you did (applaud) and allow the students to draw the animals and count the legs.

Important >>>It needs to be okay (and valued) – to you and to your students - that at any given time students are likely to be at different places in their thinking.

The Authors

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Word Problems

Pagliaro — Tue, 05 Feb 2008 16:21:38 GMT

Our suggestion – one step at a time. Once students have opportunities to solve different one-step story problems, you will get indications that they are ready for more complex problems. By focusing on the story/situation, and not just performing an algorithm, students will figure out logically what it is they need to do to solve the problem. They will think about the story as a whole with the question and know what information is necessary in attaining the answer. Along the way, having students record their thinking (and assisting them with ways to record their thinking with mathematical symbols) will provide a vehicle for them to discuss their ideas and whether the order that they applied the “operations” matters. Comparing strategies is a great way to do that. The students then will be able to actually understand the logic behind the “order of operations.” It will actually make sense to them! How many of us can say that??!! 

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Paragraph 4 and 5

suekie — Tue, 05 Feb 2008 12:24:59 GMT

Sorry I have not gotten on in a while. I thank you for the suggestions and would love to discuss this. I meet every other month with other teachers of the deaf in our region of our state. We do a round table and all of us are itinerant. I am giving them the original information about this site at our next meeting. All of us are frustrated with the constraints put on us as itinerants as well as others misinterpretation of our jobs. I would love to sit down and discuss this and other topics. I would love however to get some of the other TOD's in my group in on this discussion. They have taught far longer than I (I stayed home for ten years to raise my kids), they are highly intelligent, well-experienced teachers and can contribute a lot to the discussion. I am going to print out the pages with this discussion on it and give it to them as well so that maybe they can join right in on it.

Thanks for your concern, it's nice to be listened to.

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Paragraph 4 and 5

JMerinar — Mon, 04 Feb 2008 01:37:59 GMT

In response to the 4 lions and 2 tigers problem...

Instead of skipping from counting each leg to pure multiplication, you could have the students count by 4's to get to the correct answer (4,8,12,16,20,24).

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Math-Problem Solving

Pagliaro — Sun, 03 Feb 2008 22:58:13 GMT

I'm not sure if I am responding to your question, but there are various areas of research with deaf/hh students that focus on cognitive processing from language to memory to specific tasks. In all areas, however, I think that we have just scratched the surface in our understanding. As a researcher, and a professional in Deaf Education for more than 20 years, it is true - the more I learn, the more I realize how much more I need to learn. Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: written English

Pagliaro — Sun, 03 Feb 2008 22:37:28 GMT

Using recipes are definitely a natural way to include English in a natural math activity. That's GREAT!

And, yes, I do believe that parents of deaf/hard of hearing children would be knowledgeable enough to create such opportunities primarily because you really don't have to create them - you're already doing them! Again, these activities stem from daily activities in the home. If the child goes to bed at 8pm, just make it a point to say to the child at 6pm that it is now 6:00 and s/he has 2 more hours to play before bedtime at 8:00. Then at 7:00, make a note again to tell the child that there is one more hour before bedtime at 8:00. You may want to mediate a bit here by just saying something like, "Let's see. It is now 6:00. You go to bed at 8:00, so 6, 7 (put up a finger), 8 (put up a finger). That's 2. You have two more hours before bedtime." Eventually, your child will be asking how many more hours there are until bedtime and soon, will be telling you! :)

Point is, the activities are already there. Just take advantage of them. If this is scary to some of you, or if you'd prefer some activity ideas, I gave some resources in another post that are very helpful (Ex. Family Math for Preschoolers; Family Math; Family Math for the Middle School come to mind - these are also in Spanish). There are free activities as well all over the Internet - particularly at the federal department of education website.)

I would NOT suggest that you sit and formally teach your child to solve addition or multiplication equations, however. Keep it natural and useful. If the child is frustrated, perhaps you could do the activity and think aloud to model for your child. Let your child see/hear what you are thinking and how you solve the problem. Start simple - even with the recipe. Make PB&J with 2 slices of bread and cut it in 1/2 (I am emphasizing 2 and 1/2 here). Or if you and your child are going to eat a sandwich each, how many pieces of bread do you need? etc. There are lots of recipe books for kids as well. Check them out!

If you are concerned about the signs for something, as your child's teacher or a Deaf adult. I'm sure they'll be more than happy to help.
Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Math phobic

Pagliaro — Sun, 03 Feb 2008 21:53:56 GMT

Thanks.
Perhaps with this type of natural approach to setting a good, meaningful foundation for mathematics, we won't have persons who feel uptight about math as you mention of yourself... BTW, I bet if you look around your environment and note the tasks that you do throughout your day, you'll find that you have no problem with math and use it quite well! :)

Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Bob's worms

Pagliaro — Sun, 03 Feb 2008 21:47:59 GMT

For our study, we included K-3rd graders from 9 schools for the deaf across the nation. We had an original sample of ~230 of these students (which is a large number in Deaf Education research!). Our goal was to get at the children's solutions to the story problems without language (ASL or English) barriers. Thus, as part of their tasks, each student was given an ASL receptivity assessment. No English assessment was necessary since the problems were not given in English (any form). We then took the scores on this task of all of the children at each age (5, 6, 7, 8, & 9) who had deaf parents, making the assumption that these children were then native signers on a typical trajectory for ASL development, and found the mean (average) score. Going back to the larger sample of ~230, we then took to analyze the data from only those children who met the mean score of native signers for their age group. Thus, we could be quite confident with this high minimum score, that these students could understand the story problems signed to them by the Deaf adult on the videotape. [Unfortunately, and quite telling in some respect, this reduced the number of children in our sample from ~230 to 59!]
BTW, in case anyone is wondering, each age group consisted of children whose parents were deaf and children whose parents were hearing. The hearing status of the parent was not significant in our sample or in any of our results.

Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Reverse operations/Bob's problem with worms

Pagliaro — Sun, 03 Feb 2008 21:28:52 GMT

We thought about this too. In fact, in our data, we looked for ways in which the children made use of signing in either their understanding of the story problems or in their solutions. In some instances, we could see where the signing of the story problems had an influence on the children's understanding of the problem - for better or worse. For example, in the signing of a "division" problem that told a story of a person having 15 caterpillars and putting three of them in each of how many jars, some children seemed to confuse the sign for the unspecified number of jars (JAR++) with a specified number (counting the repetitions of the sign's movement). [It could be the case, here, however, of the children not being developmentally ready for this linguistic aspect.] For the better, some children made use of the spatial set-up that occurred in the signing of the problem. For example, in the "multiplication" problem where candy was put into 3 bags each of which was signed in a different location on the signing shelf, the children made use of the spaces/locations as markers helping them to keep track and solve the problem.
In other instances however, it appeared that the signing of the problem did not have any impact on the children at all, as mentioned above.

Again, I think the notion of the signing of the problems possibly having an influence on the children's perception of the problems and/or their solutions is definitely worth further study. The research that we describe here, however, was not specifically designed for this. Future???

Claudia :)

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : ASL numbers

niwri2 — Sun, 03 Feb 2008 00:55:41 GMT

I wish I'd had your suggestions long ago! As the mother of a deaf adult, I remember working on the dreaded story problems. I can vaguely remember devising some kind of aid that involved circles of paper, big arrows and beans (as counters) to map out story problems. I really like what you said about not focusing on key words and phrases--I can remember needing to steer clear of those (as this technique had been introduced in school, and was not helping her THINK about the problem.)

My question concerns ASL numbers and how they are processed. In a second-year ASL class I took, the instructor pointed out that numbers in ASL are small and easily confused with each other. We spent a couple of days delving into the ways in which ASL numbers are held, shaken, twisted, moved up and down, etc. to make them more visible and add other information.

After that, I noticed a couple of things. Teachers at my daughter's school for the deaf complained that their students must not be paying attention when told to turn to page ###, because they'd ask for the information to be repeated multiple times. Attention or processing? Also, my daughter tested out quite high on most of the non-verbal portion of an IQ test, but below average on the subtest that required that she remember and repeat a series of numbers.

Is it possible that some deaf students are focusing on the harder-to-process numbers to the extent that they are missing the story, and falling back on addition as the first operation they learned, and therefore the most familiar?

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : re: Math-Problem Solving

LGS — Fri, 01 Feb 2008 20:47:02 GMT

What kind of research explores the cognitive processing distinction for DHH kids that you're describing? FASCINATING!

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students : Word Problems

AVanderwal — Thu, 31 Jan 2008 21:18:08 GMT

As a future educator to deaf/hh students I think this website is amazing! I am learning so many new and interesting teaching strategies. I have to admit though, mathematics is intimidating to me. In this article, it is stated that the kids with more flexible strategies did the best on the story situations but what would happen when these stories become more complicated? Would they work just as well when you have to take into account the order of operations or would the story become so complex to be overwhelming? I know that I struggle with word problems because I get tripped up when there is more information than what you really need. What are your suggestions?