Mathematics+Problem+Solving



Magi is a very lively first grader she love to have fun and learning she enjoys as long as she is able to move about. She loved reading and was doing very well when I came to the school to tutor some children during my year off from being a regular classroom teacher. Her teacher ask me to work with her because she was not doing well in math. I thought she can not be that bad if she reads fine but the more I worked with this child the harder it was to get her to understand math concepts. We worked the whole 8 weeks on nothing but the concept of what happens when you add. This was my first time every working with a child that struggled so hard in math. I don’t know for sure what was going on with her but the following may help you and me out the next time we have a child like this. toc

= What is a Mathmatics Problem Solving Disability? =


 * Math realted disabilitys come in many forms and shapes and severity. There is Math Comprehension and Math Problem Solving disabilities they are both called Dyscalculia but the differences are that math comprehension deals with the understanding of number sense and Math problem solving has to do with more then just dealing with number it has more to do with the logical side of math. According to Paula's Special education website Dyscalculia is defined in the following way: **

Dyscalculia is a term used to refer to learning disabilities that involve arithmetic comprehension or computation. This difficulty in mastering concepts or computations is usually associated with neurological dysfunction or brain damage and is classified as developmental (occurring before birth from genetic or congenital problems) or acquired (occurring after birth usually from a traumatic brain injury). To be classified with dyscalculia, a child must have intellectual functioning that falls within or above the normal range and a significant discrepancy between his/her age and math skills (usually 2 years or more). Often children with dyscalculia show a spread of 20 points or more between their verbal and performance scores on WISC intelligence testing.
 * The Council for Learning Disabilities adds the following to the Dyscalculia deffinition. I think it is important to keep this in mind so that we don't over diagnoses children, when in reality it was simply a teaching problem. **

A learning disability in mathematics refers to unexpected learning problems after a classroom teacher or other trained professional (e.g., a tutor) has provided the student with appropriate research-based mathematics interventions for a sustained period of time. Appropriate mathematics interventions refer to the use of validated instructional practices and the monitoring of response to the mathematics interventions to determine the effect on mathematics performance. Typically, students with a mathematics disability have difficulty making sufficient progress in mathematics compared to others in their peer group despite the implementation and documentation of validated teaching practices over time. The term **dyscalculia,** “a severe or complete inability to calculate,” has been used for many years to refer to a math disability. Today, the terms **learning disabilities in mathematics** and **math disability** are more commonly used for individuals who exhibit continued difficulties learning and applying mathematical skills and concepts.

= Prevalence =


 * According to David C. Geary in the Journal of Learning Disabilities January/ February 2004 5-8 percent of children have some kind of math disability. This is often combined with some other disability like Dyslexia, ADD, ect. Children that are twins or have family members with some kind of disability are also more likely to have a math disability he goes on to state that this seems to be a pattern with all of the specific disabilities. **

= Describe Characteristics of Students =

** The National Council of Teacherslists the following as two of the major characteristics of a Math Disability. **

====//Slow or Inaccurate Retrieval of Basic Arithmetic Facts// ==== The one bedrock problem found in the literature about students with mathematics difficulties was their extremely slow retrieval of even the most rudimentary arithmetic facts (Hasselbring, Bransford, and Goin 1988). This finding was manifested repeatedly in a number of studies. In particular, one study showed that this problem seemed to persist even as these students progress through elementary school mathematics regardless of their increased proficiency with computation (Geary 1993).

In the early grades, this computation would include facts as simple as 3 + 6. For older students, it might entail simple multiplication and division facts, such as 4 × 2. Students who cannot retrieve these basic facts easily get lost and often cannot follow the logic of an explanation given by the teacher or a peer when the problems are embedded within more complex mathematical operations, such as simple algebra or long division. The teacher or the textbook assumes virtually automatic retrieval of these facts and bases explanations on this assumption. For example, during a lesson on equivalent fractions, teachers will assume that students “just know” that 2 × 5 is 10 and can use this information to follow an explanation of why five tenths is the same as one-half. If a student must use her fingers to count this out, the whole point of the lesson may well be lost.

There is no consensus on how best to assist students with this problem. Most efforts entail daily work on number families to see relationships between facts. The hope is that repeated practice or overlearning will improve retrieval speed. Unfortunately, no one has studied this issue systematically. Teachers need to recognize that some students will have difficulty with quick retrieval and that teachers will need to use alternative means so that students fully understand the concepts presented.

//Impulsivity//
A major problem found in the literature was impulsivity or a lack of inhibition. An example offered by Geary (2005) and Passolunghi and Siegel (2004) helped illustrate this problem. When asked what 4 + 8 is, a student might impulsively blurt out 5 or 9 because those numbers come next when counting. This example shows that the student has great difficulty with inhibiting irrelevant associations and with focusing on the problem at hand. This characteristic may explain why, as will be seen, instructional approaches that prompt students to think aloud or draw out a problem might be particularly helpful for students with a mathematics disability.

//Other Problems//
Three other characteristics of students who exhibit mathematics disability are the following:
 * Problems forming mental representations of mathematical concepts (e.g., a number line, a visual means to represent subtraction as a change process) (Geary 2004)
 * Weak ability to access numerical meaning from symbols (i.e., poorly developed number sense) (Gersten and Chard 1999; Rousselle and Noel 2006)
 * Problems keeping information in working memory (Passolunghi and Siegel 2004; Swanson and Beebe-Frankenberger 2004)



**The Council for Learning Disabilities adds the following of what might cause the problems they aren't so much characteristics as they are things to look out for that might be part of the Math learning problem.**

**// Memory //** Memory problems may affect students' math performance in several ways. Following are some examples. **// Cognitive Development //** Students with a math disability may have trouble because of delays in cognitive development, which hinders learning and processing information (Gersten et al., 2005). This might lead to problems with the following: **// Visual-Spatial Ability //** Visual-spatial problems may interfere with a child’s ability to perform math problems correctly. Examples of visual-spatial difficulties include the following: **// Language //** The language of mathematics involve recognition and comprehension of words and symbols. For students with language-based learning disabilities, the language of mathematics can present a challenge. Teaching the multiple meanings of words used in mathematics is essential for students with language-based problems. Words such as second, prime, and square have more than one meaning and can be easily misinterpreted by students struggling with language (Gurganus, 2007). For students whose primary language is not English (English Language Learners; ELL) who encounter problems with math, assessment must consider the linguistic background and experience of the student (Gutierrez, 2002). Some ELL students also have learning disabilities. For these students, the language of math and the vocabulary of mathematics can be problematic, especially as it relates to mathematical concepts and word problem solving. For such students, previewing and reviewing new mathematics vocabulary is critical to facilitate their understanding of math. In order for ELL students from diverse backgrounds to be successful with math, the teacher must make sure each student has the symbolic, semantic, and linguistic foundations for the mathematical concepts included in math lessons. Examples of strategies to assist ELL students with the acquisition of mathematics vocabulary include the following (Bryant, 2008):
 * Children might have memory problems that interfere with their ability to retrieve (remember) basic arithmetic facts quickly (Garnett, & Fleischner, 1983; Geary, 2004, Montani & Smith, 2005; Smith & Montani, in press).
 * In the upper grades, memory problems may influence students' ability to recall the steps needed to solve more difficult word problems (Geary, 2000, 2003, 2004), to recall the steps in solving algebraic equations, or to remember the meaning of specific symbols (e.g., å, s, π, ≥).
 * Students may exhibit inconsistent performance, leading teachers to say, ”He knew the math facts yesterday but can’t seem to remember them today.”
 * understanding relationships between numbers (e.g., fractions and decimals; addition and subtraction; multiplication and division).
 * solving word problems.
 * understanding number systems.
 * using effective counting strategies.
 * misaligning numerals in columns for calculation.
 * problems with place value that involves understanding the base ten system (aligning numbers correctly when setting up calculations).
 * trouble interpreting maps and understanding geometry (Jordan et al., 2003).
 * Include the math heritage of different cultural groups (different counting systems, e.g., abacus).
 * Provide assistance for students to learn the most essential math terms used in both English and the student’s primary language.
 * Provide pictures or have students generate pictures illustrating new vocabulary and math terms.

= How is this Disability Identified? =


 * It is very hard to diagnose a math learning disability do to so many factors that can play into the diagnosing it. According to the Education.com it is **//"Because we cannot yet identify dyscalculia based on brain function, we have to diagnose it based on its effect, i.e. difficulty with maths. This is difficult because "there are many reasons for being bad at maths!". Reasons other than dyscalculia include inadequate instruction, lack of motivation, attentional disorders, anxiety disorders, or across the board academic difficulties."//

**One test that is useful is the Brian Butterworth's Dyscalculia Screener.**

I//n general, however, what happens is that by the time parents and teachers realise that a child has a serious problem with maths and find out how to do something about it, the child is already 9 or 10 years old and is 3 years behind in school. It would be wonderful to be able to test children's behaviour and brain patterns in kindergarten, and pick out those at risk for dyscalculia so that they were already receiving extra monitoring and tuition. There is even the hope that dyscalculia could be "prevented" in this way.//

= How do these learners recive their education? =

** ﻿ **
 * What works the best is to teach them in a direct way while teaching them strategies to tack what ever kind of problems they may come across.[[image:http://school.discoveryeducation.com/clipart/images/blueawardribbon.gif width="117" height="200" align="right"]] **

= Strategies for Curriculum and Instruction =

Strategy training has been helpful to students with LD when learning mathematical concepts and procedures. Following are a few examples of strategies that are useful to teachers when instructing students with LD in problem solving.
 * <span style="font-family: Tahoma,Geneva,sans-serif;">All of the following was taken from the CLD website **

**Teaching Stragies**

**1. FAST DRAW (Mercer & Miller, 1992)** a. Find what you're solving for. b. Ask yourself, "What are the parts of the problem?" c. Set up the numbers. d. Tie down the sign.

a. Discover the sign. b. Read the problem. c. Answer, or draw and check. d. Write the answer.

**2. Questions and Actions (Rivera, 1994)** Step Questions Actions

a. Read the problem. Are there words I don't know? Underline words. Do I know what each word means? Find out definitions. Do I need to reread the problem? Reread. Are there number words? Underline.

b. Restate the problem. What information is important? Underline. What information isn't needed? Cross out. What is the question asking? Put in own words.

c. Develop a plan. What are the facts? Make a list. How can they be organized? Develop chart. How many steps are there? Use manipulatives. What operations will I use? Use smaller numbers. Select an operation.

d. Compute the problem. Did I get the correct answer? Estimate. Check with partner. Verify with calculator.

e. Examine the results. Have I answered the question? Reread question. Does my answer seem reasonable? Check question/answer Can I restate question/answer? Write a number sentence.

**3. TINS Strategy (Owen, 2003)** Different steps used to analyze and solve word problems are represented with this acronym. Thought: Think about what you need to do to solve this problem and circle the key words. Information: Circle and write the information needed to solve this problem; draw a picture; cross out unneeded information. Number Sentence: Write a number sentence to represent the problem. Solution Sentence: Write a solution sentence that explains your answer. Example: Kyle bought 6 baseball cards. The next day, he added 11 more cards to his collection. How many cards does he have in all? Thought: + Information: 6 baseball cards, 11 baseball cards Number Sentence: 6 + 11 = Solution Sentence: Kyle has 17 baseball cards in his collection.

**4. Problem Solving (Birsh, Lyon, Denckla, Adams, Moats, & Steeves, 1997)** Read the problem first. Highlight the question. Circle the important information. Develop a plan. Use manipulatives to represent the numbers. Implement the plan. Check your work.

** Techniques for solving calculation problems? **

beginning with the concrete level and moving to the abstract level, is an effective technique in helping struggling learners solve calculation problems. The Concrete-Representational-Abstract (CRA) Teaching sequence has been found to help students with LD learn procedures and concepts.
 * Multiple representations **
 * Concrete – use manipulatives (blocks, unifix cubes, counters) to represent numbers in the problem.
 * Representational – use tally marks, pictures.
 * Numbers alone.


 * Demonstrate - Guided Practice - Independent Practice sequence. **


 * 1) Verbalize steps while modeling calculations.
 * 2) Guide students while they solve problem and intervene as needed.
 * 3) Permit students to independently solve problems and then do error analysis as needed and provide immediate feedback.
 * 4) Provide lined paper for students who have difficulty with alignment of numbers.
 * 5) Draw a box around calculations to help students differentiate one problem from another.
 * 6) Use arrows to indicate starting points.
 * 7) Use green marks to indicate where to start and red marks to indicate where to end.

** Strategies to improve mathematics vocabulary **
 * Pre-teach vocabulary.
 * Use representations, both pictorial and concrete, to emphasize the meaning of math vocabulary (Sliva, 2004).
 * Pretest students' knowledge of glossary terms in their math textbook and teach vocabulary that is unknown or incorrect.
 * Teach mnemonic techniques to help remember word meanings.
 * Use the keyword approach (e.g., visualize a visor as the keyword for divisor; visualize quotation marks as the keyword for quotient (Mastropieri & Scruggs, 2002).

** Strategies to assist with teaching algebraic concepts ** Algebra is introduced in elementary school as students learn algebraic reasoning involving patterns, symbolism, and representations. Students experience difficulty with algebra for various reasons including difficulty understanding the vocabulary required for algebraic reasoning, difficulties with problem solving, and difficulties understanding patterns and functions necessary for algebraic reasoning.
 * Teach key vocabulary needed for algebra.
 * Provide models for identifying and extending patterns.
 * Model “think aloud” procedures for students to serve as examples for solving equations and word problems.
 * Incorporate the use of technology (e.g., graphing calculators) (Bryant, 2008).

In summary, students with learning disabilities in mathematics benefit from the use of direct instruction in the use of strategies to solve word problems and to perform calculations. For students whose memory weaknesses affect their ability to memorize math facts, research supports the use of strategies to quickly count up/down to determine the response to unknown facts. Other strategies that reinforce automaticity of basic facts help to increase the overall base of knowledge of basic facts. Following is a partial listing of websites that provide math software to address problem solving, calculations and math fluency, and instructional materials designed to address specific math concepts.

= Accommodations/Assistive Technology =

<span style="font-family: Tahoma,Geneva,sans-serif; font-size: small;">**The University of Washington recommends the follow accommodations for students with Math learning disabilities**.

**<span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin-bottom: 0pt;">Math specific accommodations **
 * The use of scratch paper to work out math problems during exams.
 * Talking calculators.
 * Fractional, decimal, and statistical scientific calculators.
 * Computer Assisted Instruction (CAI) software for math.
 * Computer Assisted Design (CAD) software for engineering.
 * Large display screens for calculators and adding machines.


 * <span style="color: #337407; font-family: Times New Roman; font-size: 110%; line-height: normal; margin-bottom: 0pt;">Other accommodations that often help these studemts **
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Notetakers.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Audiotaped or videotaped class sessions.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Extended exam time and a quiet testing location.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Visual, aural, and tactile demonstrations incorporated into instruction.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Concise course and lecture outlines.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Books on tape.
 * <span style="font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Alternative evaluation methods (e.g., portfolio, oral or video presentations).
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Providing projects or detailed instructions on audiotapes or print copies.
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Reinforcing directions verbally.
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Breaking large amounts of information or instructions into smaller segments.
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Computers equipped with speech output, which highlights and reads (via screen reading software and a speech synthesizer) text on the computer screen.
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Word processing software that includes electronic spelling and grammar checkers, software with highlighting capabilities, and word prediction software.
 * <span style="color: black; font-family: 'Times New Roman',Times,serif; line-height: normal; margin: 0in 0in 0pt;">Software to enlarge screen images.

=<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">Inclusive Practices =


 * Because this type of disability has now outward limitations or physical problems this child can be taught in a normal looking classroom the things that will need to change it the way the teacher runs the classroom. If the other children are allowed to make fun of the child or if the child does not receive the support he needs he will feel like a failure and then he will have more then just a Math Problem solving problem. The following are some examples of changes that need to be made to help the child be a success. They are all taken from //Teaching Students with Special Needs in Inclusive Settings// **
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Believe in a positive classroom community and the importance of empowered students: student will follow your lead.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Facilitate a sense of inclusion by helping students get to know and trust one another.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Have students create a map of the people and places in the school/classroom to help them understand their place in the community.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Have students interview one another and adults in the school.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Highlight commonalities like characteristics and favorite activities among students.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Play bingo, matching names with personal items.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Guide students in conducting classroom meetings to facilitate a sense of power in knowing what they say is important.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Give students choices of activities or assignments.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Reduce rewards for individuals and reward the group, especially for inclusive behaviors.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Assign classroom jobs and encourage community service.
 * <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;"> Use a “sharing chair” technique, where students validate others’ thoughts even if they disagree.
 * Display a celebration board highlighting group accomplishments and good citizenship. <span style="display: block; line-height: normal; margin-bottom: 0pt; margin-left: 0in; margin-right: 0in; margin-top: 0in;">


 * I found this You Tube video had some good ideas about inclusion some are math others are not.**

=<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">Special Challenges for General Education =


 * The biggest challenge facing a teacher who has a child with a Mathematics learning problems is giving affective differentiated instruction. This is were each child is working on the same objective but on his or her level.**

=<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">Provide a continuum of severity or extent possible in youngsters =



=<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">Impact: =

<span style="font-family: Tahoma,Geneva,sans-serif;">I am going to call this a school issue the child will grow up and live a normal life. He may not want to be a math teacher or an accountent but he will do just fine in life. He will need someone to check him at times but it manly a chanlge of school.
 * <span style="color: #337407; font-family: Tahoma,Geneva,sans-serif; font-size: 110%;">Impact: Is this a school issue or life issue **

<span style="color: #337407; font-family: Tahoma,Geneva,sans-serif; font-size: 110%;">**Impact: How does this affect home?** I don't think there will be much of an impact on the home life of this child. =<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">Name of someone who has achieved greatness =

<span style="color: #666666; font-family: Arial,Tahoma,Helvetica,FreeSans,sans-serif; line-height: 18px;">"Hilary Freeman is the British young adult author of ‘Loving Danny’ and ‘Don’t Ask’ released this month by Piccadilly Press. Hilary is a journalist and agony aunt, working for national newspapers, magazines and websites."She also just recently found out that she had Dyscalculia. A lot of people with a Math Learning disability will be very good at writing. To read her story click on the link below.

Hillery Freeman's story.

=<span style="color: #342e61; font-family: Tahoma,Geneva,sans-serif; font-size: 140%;">What Can You Do? =

<span style="font-family: Tahoma,Geneva,sans-serif;">**What can you do to make a difference for the child?** This child will need help with trying to come up with ways to remember math concepts. The may also be helped by have some to check them when math concepts are vital to life.

<span style="font-family: Tahoma,Geneva,sans-serif;">**What can you do to support parents?** The parents may never struggle with this child because it is manly and learn problem and after all a lot of people are no good at math. The parents my need to however be showen that their child has a problem.

<span style="font-family: 'Times New Roman',Times,serif;">