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The Educators Room
Empowering Teachers as the Experts
The Difference Between Calculation and Mathematics
This piece originally ran on Bluffcityed.com on July 29, 2014;
Lets be clear; the shift to Common core is hard. In some courses like geometry, it turns the entire structure of the way we’ve taught it on its head and requires both kids and teachers to make new connections. Common Core was also a struggle at first because these standards are much more rigorous than what we’ve been demanding from our children for years. They require, in the words of Jordan Ellenberg , prioritizing actual mathematics over calculation. Let me explain what I mean by this. For years we’ve been requiring math students to primarily calculate, that is, find a numerical solution, and do a small level of thinking. Common core flips this on its head and puts the emphasis on mathematical thinking by requiring students to explain their strategy, their work and their answer.
After a year of wresting with the Common Core state standards I have three takeaways that I think might be useful to teachers embarking on their own journey with common core for the first time:
Don’t Panic – there’s a learning curve because Common Core requires us to teach predominantly in ways and strategies that we aren’t use to utilizing on a daily basis; exploration and manipulation. For example, the first few times I tried using exploration based learning, I fell flat on my face simply because I hadn’t taught using that strategy before. It took a few tries before I figured out how to do it well, but once it’s become one of the more rewarding teaching techniques I employ!
Enjoy the freedom – the new geometry standards have given me freedom in what and how to teach. It eliminates some standards and expounds on others. Consider trigonometry. Under the common core state standards I’m to teach trigonometry in any contextual situation that I want, as long as students understand that they are ratios. Contrast this with the old standards, which required me to do so using only surface area and volume.
Dive deeply – the common core state standards reduced what I needed to teach, freeing me up to dive deeply into certain contents that I would have had to skim previously. For example,previously compasses and protractors were fun tools, but not all that essential to the way geometry was taught. Now they’ve become essential tools that allow us to explore concepts in a much more direct way, through our own tactile senses.
Like I said, last year was a huge learning experience, as it will be for most math teachers adjusting to the common core. Now that I’ve had a chance to grapple with the standards, I have a much better understanding that I can use to shift my focus for this year to three key things – concepts, application and proofs.
Concepts – I’ve figured out even more so where the deadwood is and can focus much more deeply on the most important Common Core concepts. I’ve cut out a few units
Applications – with fewer concepts I can start thinking through applications for each unit. For example, my first unit with lines and angles will be using the theme of minigolf, with the culminating project being to create a minigolf course where a hole in one is possible using segments and angles.
Proofs – I’ve completely retooled the way I have students display rational thinking through proofs. Old geometry required 2 column proofs. Common core doesn’t require this, so my goal this year is to have my kids focus on the paragraph proof as the method of displaying their thinking because this mirrors more what they will be required to do in real life.
Common core is new and it has a learning curve. But it’s the law of the land and if we can get past the initial struggle, I feel that we’ll find as a profession that the standards are much more liberating than we at first might have believed. It will be tough, but it doesn’t have to be a negative experience. Learn from my struggles and my mistakes and shorten your own learning curve and before you know it you’ll be doing Common Core in your classroom like a pro!
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20 Effective Math Strategies To Approach ProblemSolving
Katie Keeton
Math strategies for problemsolving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.
Problemsolving skills are essential to math in the general classroom and reallife. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.
This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in reallife situations.
What are problemsolving strategies?
Problemsolving strategies in math are methods students can use to figure out solutions to math problems. Some problemsolving strategies:
 Draw a model
 Use different approaches
 Check the inverse to make sure the answer is correct
Students need to have a toolkit of math problemsolving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better.
Strategies can help guide students to the solution when it is difficult ot know when to start.
The ultimate guide to problem solving techniques
Download these readytogo problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.
20 Math Strategies For ProblemSolving
Different problemsolving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem.
Here are 20 strategies to help students develop their problemsolving skills.
Strategies to understand the problem
Strategies that help students understand the problem before solving it helps ensure they understand:
 The context
 What the key information is
 How to form a plan to solve it
Following these steps leads students to the correct solution and makes the math word problem easier .
Here are five strategies to help students understand the content of the problem and identify key information.
1. Read the problem aloud
Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.
2. Highlight keywords
When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed. For example, if the word problem asks how many are left, the problem likely requires subtraction. Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.
3. Summarize the information
Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary. Summaries should include only the important information and be in simple terms that help contextualize the problem.
4. Determine the unknown
A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer. Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.
5. Make a plan
Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it. The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer. Encourage students to make a list of each step they need to take to solve the problem before getting started.
Strategies for solving the problem
1. draw a model or diagram.
Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process. It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.
Similarly, you could draw a model to represent the objects in the problem:
2. Act it out
This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives . When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts. The examples below show how 1stgrade students could “act out” an addition and subtraction problem:
The problem  How to act out the problem 
Gia has 6 apples. Jordan has 3 apples. How many apples do they have altogether?  Two students use counters to represent the apples. One student has 6 counters and the other student takes 3. Then, they can combine their “apples” and count the total. 
Michael has 7 pencils. He gives 2 pencils to Sarah. How many pencils does Michael have now?  One student (“Michael”) holds 7 pencils, the other (“Sarah”) holds 2 pencils. The student playing Michael gives 2 pencils to the student playing Sarah. Then the students count how many pencils Michael is left holding. 
3. Work backwards
Working backwards is a popular problemsolving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution. This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.
For example,
To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71. Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.
4. Write a number sentence
When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved. It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.
5. Use a formula
Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve. Make sure students know the important mathematical formulas they will need in tests and reallife. It can help to display these around the classroom or, for those who need more support, on students’ desks.
Strategies for checking the solution
Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense.
There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.
Here are five strategies to help students check their solutions.
1. Use the Inverse Operation
For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work.
2. Estimate to check for reasonableness
Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable. Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten. For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10. When the estimate is clear the two numbers are close. This means your answer is reasonable.
3. PlugIn Method
This method is particularly useful for algebraic equations. Specifically when working with variables. To use the plugin method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.
If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓
4. Peer Review
Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly. Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixedability partners or similarability partners. In mixedability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills. If partners have comparable ability levels and problemsolving skills, they may find that they approach problems differently or have unique insights to offer each other about the problemsolving process.
5. Use a Calculator
A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to realworld situations.
Stepbystep problemsolving processes for your classroom
In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4step process to solve problems.
Polya’s 4 steps include:
 Understand the problem
 Devise a plan
 Carry out the plan
Today, in the style of George Polya, many problemsolving strategies use various acronyms and steps to help students recall.
Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom.
Here are 5 problemsolving strategies to introduce to students and use in the classroom.
How Third Space Learning improves problemsolving
Resources .
Third Space Learning offers a free resource library is filled with hundreds of highquality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking.
Explore the range of problem solving resources for 2nd to 8th grade students.
Oneonone tutoring
Third Space Learning offers oneonone math tutoring to help students improve their math skills. Highly qualified tutors deliver highquality lessons aligned to state standards.
Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problemsolving to independent practice.
Throughout each lesson, tutors ask higherlevel thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problemsolving skills.
Problemsolving
Educators can use many different strategies to teach problemsolving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra.
Teaching students how to choose and implement problemsolving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to reallife problemsolving.
READ MORE :
 8 Common Core math examples
 Tier 3 Interventions: A School Leaders Guide
 Tier 2 Interventions: A School Leaders Guide
 Tier 1 Interventions: A School Leaders Guide
There are many different strategies for problemsolving; Here are 5 problemsolving strategies: • draw a model • act it out • work backwards • write a number sentence • use a formula
Here are 10 strategies for problemsolving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model • Act it out • Work backwards • Write a number sentence • Use a formula
1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back
Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.
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Ultimate Guide to Metacognition [FREE]
Looking for a summary on metacognition in relation to math teaching and learning?
Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.
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LDs in Mathematics: EvidenceBased Interventions, Strategies, and Resources
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By Hanna A. Kubas and James B. Hale
Mathematics. Some love it, some loathe it, but there are many myths about math achievement and math learning disabilities (LDs). The old belief – boys are naturally better at math than girls – may be more a consequence of teacher differences or societal expectations than individual differences in math skill (Lindberg, Hyde, Petersen, & Linn, 2010).
Similarly, the old belief that reading is a left brain task, and math is a right brain task, is not a useful dichotomy as clearly multiple shared and distinct brain regions explain these academic domains (e.g., Ashkenazi, Black, Abrams, Hoeft, & Menon, 2013).
Math is a language with symbols that represent quantity facts instead of language facts (i.e., vocabulary), so rules (syntax) are important for both (Maruyama, Pallier, Jobert, Sigman, & Dehaene, 2012). You might be surprised to learn that approximately 7% of schoolaged children have a LD in mathematics (Geary, Hoard, Nugent, & Bailey 2012).
Let’s first explore the fundamental skills needed for math achievement.
Number Sense / Numerical Knowledge
Children develop knowledge of quantity even before math instruction in schools, and kindergarten number sense is predictive of math computation and problem solving skills in elementary school (Jordan et al., 2010). These basic math skills include understanding of number magnitudes, relations, and operations (e.g., adding). Children link basic number sense to symbolic representations of quantity (numbers); the math “language”. Poor early number sense predicts math LDs in later grades (Mazzocco & Thompson, 2005).
Math Computation vs. Math Fluency
Children often rely on various strategies when solving simple calculation problems, but math computation requires caring out a sequence of steps on paper or in your mind (working memory) to arrive at an answer. Math fluency refers to how quickly and accurately students can answer simple math problems without having to compute an answer (i.e., from memory 6 x 6 = 36), with no “steps”, calculation, or number sense needed.
Children with fluency deficits often use immature counting strategies and often do not shift from computation to storing and retrieving math facts from memory, taking more time to provide an answer. Difficulty with retrieval of math facts is a weakness/deficit associated with math LDs (Geary et al., 2007; Gersten, Jordan, & Flojo, 2005). Without math fact automaticity, working memory may be taxed when doing computation, and the child “loses his place” in the problem while computing each part to arrive at a final answer.
Developmental Sequence of Math Skills
1. Finger Counting Strategies : Students first display both addends/numbers with their fingers; this is the most immature strategy.
2. Verbal Counting Strategies: Next, students begin to develop basic adding skills and typically go through three phases.
 Sum: counting both addends/numbers starting from 1, this is a beginning math counting skill;
 Max: counting from the smaller number; and finally
 Min: counting from the larger number (most efficient strategy).
3. Decomposition (Splitting) Strategies: Students learn that a whole can be decomposed into parts in different ways, a good problem solving strategy for unknown math facts
4. Automatic Retrieval from LongTerm Memory: Students become faster and more efficient at pairing problems they see with correct answers stored in longterm memory (as is the case with sight word reading), no computation is required
The Role of VisualSpatial Skills
Basic arithmetic skills are factual, detailed “left hemisphere” functions (similar to basic reading), but Byron Rourke (2001) discovered many students with nonverbal or “right hemisphere” LDs had math calculation problems, suggesting left was verbal and right nonverbal.
Students need “right hemisphere” visual/spatial skills to align numbers when setting up multistep math problems , they need to need to be able to understand and spatially represent relationships and magnitude between numbers, and they need to be able to interpret spatially represented information (Geary, 2013).
Neuropsychology has also taught us that children with visual/spatial problems may neglect the left side of stimuli (the left visual field is contralateral to the right hemisphere) (Hale & Fiorello, 2004; Rourke, 2000).
Math Reasoning and ProblemSolving
Word problems require both receptive and expressive language skills , unlike simple calculation, so students with languagebased LDs may struggle even if math skills are good. Students must translate math problem sentences/words into numbers and equations , so they must identify what the sentences are asking them to do in terms of calculation, and then perform the calculation
Students with LDs are typically poor strategic learners and problem solvers , and often manifest strategy deficits that hinder performance, particularly on tasks that require higher level processing (Montague, 2008). So there is a strong relationship between fluid reasoning, executive functioning, and quantitative reasoning (Hale et al., 2008). Students with LDs often benefit from explicit instruction in selecting, applying, monitoring, evaluating use of appropriate strategies to solve word problems.
The Brain, Math, and LDs
Click here to access a printable PDF version of LD@school's diagram of brain areas and math skills .
Strategies for Promoting Math Computation and Fluency
Note : Your understanding of foundational mathematical concepts and skills is critical for targeted interventions that are developed, implemented, monitored, evaluated, and modified until treatment efficacy is obtained!
Remember: early identification and intervention are key!
Click here to access a printable PDF version of the Strategies for Promoting Math Computation and Fluency explained below .
Strategic Number Counting
Fuchs et al. 2009
Goal : Improve counting strategies (e.g., MIN; decomposition) to efficiently pair problem stems and answers
Skills Targeted: Explicitly teach math counting strategies when number sense or algorithm adherence is limited
Target Age Group: Elementary students struggling with basic computation and quantitynumber association
Description:
 Direct instruction of efficient counting (g., MIN for addition), followed by guided practice.
 For twonumber addition, students start with larger number and count for smaller number/addend.
 For twonumber subtraction, students start at ‘minus number’ and count up to ‘starting number,’ tallying numbers
 Flashcards used to math fact encoding, storage, and/or retrieval deficits. Optional number line can enhance method.
E mpirical Support:
 Fuchs et al. (2009) found strategic counting led to better math fact fluency compared to control groups, even better if combined with intensive drill and practice
 Strategic counting with and without deliberate practice better math fluency, with deliberate practice better than controls (Fuchs et al., 2010)
Additional Resources:
 Click here to access a number line and a strategic number counting instruction score sheet available on the Intervention Central .
Drill and Practice
Fuchs et al. 2008
Goal: Drill and practice interventions help children quickly and accurately recall simple math facts
Skills Targeted: Practice and repetition of math fact calculations
Target Age Group : Students struggling with basic math facts, especially with limited automaticity
 May be paperandpencil and/or computerized drill and practice in either a game or drill format, typically includes modeling, practice, frequent administration, and brief, timed practice, selfmanagement, and reinforcement
 Drill and practice with math problem solving strategies may be more effective
 Software to ensure correct student response; math facts appear for 13 seconds, and students reproduce the whole equation and answer from shortterm memory
 Students visually encode both the number question and answer for longterm memory storage
 Connection between math fact rehearsal and increased fact retention and generalization (Burns, 2005; Codding et al., 2010; Duhon et al., 2012)
 Promotes efficient paring of problems and the correct answers (Fuchs et al., 2008)
 Computer versions improve math fact retrieval fluency (Burns et al., 2010; Slavin & Lake, 2008)
 Click here to access a website for free math fact flashcards .
 Click here to access free math computation worksheets and answer keys for addition, subtraction, multiplication, and division .
CoverCopyCompare
Skinner et al. 1997
Goal: Improve accuracy and speed in basic math facts
Skills Targeted: Students taught selfmanagement through modeling, guided practice, and corrective feedback
Target Age Group : Students learning basic math facts, those with executive, sequential, or integration problems
 Students learn 5step strategy to solve simple math equations and selfevaluating correct responses
 Students look at math problem, cover it, copy it, and evaluate response to compare to original
 For errors, brief error correction procedure undertaken before next item introduced
 Strategy requires little teaching time or student training
 CCC procedures enhance math accuracy and fluency across general education (Codding et al., 2009; Grafman & Cates, 2010) and special education (Poncy et al., 2007; Skinner et al., 1997)
 Metaanalysis of many studies shows CCC improves math performance, especially when coupled with other evidencebased methods (e.g., token economies, goal setting, correct digits, increased response opportunity; Joseph et al., 2012)
 Click here to access a CCC intervention description including worksheet and performance log at Intervention Central .
DetectPracticeRepair
Poncy, Skinner & O’Mara, 2006
Goal: Promote efficient basic math fact practice targeting problems not completed accurately and/or fluently
Skills Targeted: Encoding and retrieval of math facts from longterm memory
Target Age Group : Students developing basic math facts, may be useful for executive memory difficulties
 DPR is a 3phase testteachtest procedure for individualizing math fact instruction for basic fact groups (e.g., addition)
 (1) Detect phase  metronome determined rate to determine automatic (< 2 seconds) vs. slow (>2 second) math fact responding
 (2) Practice phase using CoverCopyCompare (CCC; see description above)
 (3) Repair phase using 1minute math sprint with items requiring practice embedded in automatic ones
 DPR validated across grades, skills, and research designs (Poncy et al., 2013)
 Improves subtraction, multiplication, and division fluency (Axtell et al., 2009; Poncy et al., 2006; 2010; Parkhurst et al., 2010)
 Differentiation possible because DPR targets specific difficulties (Poncy et al., 2013)
Reciprocal Peer Tutoring
Fuchs et al., 2008
Goal: Peer tutoring procedure includes explicit timing, immediate response feedback, and overcorrection
Skills Targeted : Basic math fact retrieval and automaticity through constant engagement in dyads
Target Age Group : All students, but especially useful for students with poor attention or persistence
 Students are paired up and take turns serving as the “tutor”
 Flashcards with problem on one side (e.g., 2 x 3 = ___) and answer on other side (e.g., 6)
 Student tutors shows flashcard, tutee responds verbally
 Tutor states either “correct,” (and puts in correct stack) or “incorrect” (and puts in incorrect stack)
 If incorrect, tutee writes problem and correct answer 3 times on paper
 Roles change after 2 minutes; then students complete 1 minute math probes and grade each other
 Multicomponent approach + other evidencebased efforts improve math fact rates (Rhymer et al., 2000)
 Improves math achievement, engagement, and prosocial interactions (Rohrbeck et al., 2003)
 Improves achievement, selfconcept, and attitudes (BowmanPerrot et al., 2013; Tsuei, 2012)
Strategies for Promoting Math ProblemSolving
Click here to access a printable PDF version of the Strategies for Promoting Math ProblemSolving explained below .
Schema Theory Instruction
Jitendra et al. 2002
Goal : Teaches mathematical problem structures, strategies to solve, and transfer to solve novel problems
Skills Targeted: Expanding student math problem solving schemas
Target Age Group: Students in any grade learning math problem solving skills, helps conceptual “gestalt”
 Encourages math problem solving schemas for word problems, identifying new, unfamiliar, or unnecessary information, and grouping novel problem features into broad schema for strategy use
 Explicit instruction in recognizing, understanding, and solving problems based on mathematical structures; can be used with schemabroadening instruction for generalization (e.g., Fuchs et al., 2008)
 Randomized controlled trials show improved math word problem solving (Fuchs et al., 2008, 2009)
 Schemabased approach generalizes into better math word problem solving (Jitendra et al., 2002; 2007; Xin, Jitendra, & DeatlineBuchman, 2005)
Mercer & Miller, 1992
Goal : Selfregulated strategy instruction method for increasing math problem solving skills
Skills Targeted: Targets selfteaching, selfmonitoring, and selfsupport strategies for identifying salient math words in sentences, determining and completing operation, and checking accuracy
Target Age Group: Students struggling with executive monitoring and evaluation skills
 Teaches 8step math word problems strategy and selfregulation
 The mnemonic FAST DRAW cues students, can use as checklist
 Increases math achievement and improves math attitude (Tok & Keskin, 2012)
 Increases math achievement in math LD (Miller & Mercer, 1997; Cassel & Reid, 1996)
Click here to access LD@school’s template for the FAST DRAW mnemonic .
Cognitive Strategy Instruction
Montague & Dietz, 2009
Goal : Teach multiple cognitive strategies to enhance math problem solving skills
Skills Targeted: Focuses on cognitive processes, including executive functions (selfregulation/metacognition)
Target Age Group: Useful for differentiating instruction based on processing weaknesses
 Teaches 7step cognitive strategy for solving math word problems, with 3step metacognitive selfcoaching routine for each step
 Direct instruction includes structured lesson plans, cognitive modeling, guided practice cues and prompts, distributed practice, frequent teacherstudent interaction, immediate corrective feedback, positive reinforcement, overlearning, and mastery
 Read the problem for understanding
 Paraphrase the problem in your own words
 Visualize a picture or a diagram to accompany the written problem
 Hypothesize a plan to solve the problem
 Estimate /predict the answer
 Compute the answer
 Check your answer to make sure everything is right Say, Ask, Check metacognitive routine in each of the 7step cognitive processes
 Say requires selftalk to identify and direct self when solving problem
 Ask requires selfquestioning, promoting selftalk internal dialogue
 Check requires selfmonitoring strategy for checking understanding and accuracy
 Selfregulation strategies foster math problem solving in metaanalyses (Kroesbergen & van Luit, 2003)
 Cognitive strategy instruction increases math problem solving skills in general education (Mercer & Miller, 1992, Montague et al. 2011) and ADHD and LD (Iseman & Naglieri, 2011)
 Click here to access the SayAskCheck handout for student selfcoaching .
Related Resources on the LD@school Website
Click here to access the article Math Heuristics .
Click here to access the article Helping Students with LDs Learn to Diagram Math Problems .
Click here to access the answer to the question: There is a lot of information about identifying learning disabilities in mathematics. However, information about strategies and ideas for working with these disabilities is limited. What strategies work? .
Click here to access the video Using Collaborative Teacher Inquiry to Support Students with LDs in Math .
Click here to access the recording of the webinar Understanding Developmental Dyscalculia: A Math Learning Disability .
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 Transl Pediatr
 v.7(1); 2018 Jan
Specific learning disability in mathematics: a comprehensive review
Math skills are necessary for success in the childhood educational and future adult work environment. This article reviews the changing terminology for specific learning disabilities (SLD) in math and describes the emerging genetics and neuroimaging studies that relate to individuals with math disability (MD). It is important to maintain a developmental perspective on MD, as presentation changes with age, instruction, and the different models (educational and medical) of identification. Intervention requires a systematic approach to screening and remediation that has evolved with more evidencebased literature. Newer directions in behavioral, educational and novel interventions are described.
Introduction
Mathematics (math) is not only the science of numbers, but also is used in everyday life from calculating time and distance, to handling money and analyzing data to make decisions in financial planning and insurance purchasing, and is essential in the STEM (science, technology, engineering and math) fields. Infants have an innate capacity for “numerosity” or the number of things. In early childhood, counting is learned to bridge this innate capacity to more advanced math abilities like arithmetic facts and concepts ( 1 ). Math is a core subject taught in primary and higher education, which builds a foundation of math skills for real life situations. Numeracy is the knowledge and skills required to effectively manage and respond to the mathematical demands of diverse situations ( 2 ), and is a necessary skill in both bluecollar and professional work places. Analyses in both the United Kingdom (UK) and United States (US) revealed that poor numeracy skills impacted employment opportunities and wages, even in the presence of adequate literacy skills ( 3 , 4 ). Hence, it is concerning to note that the Nation’s Report Card from the National Assessment of Educational Progress (NAEP) yielded poorer results for 4 th , 8 th and 12 th graders in the US in 2015 compared to 2013 ( 5 ), and only 9% of US adults scored in the highest numeracy levels in a 23 country comparative study ( 6 ). This has led to recommendations in the US aimed at improving math education, around curricular content, learning processes, instructional practices and materials, assessment, and teacher education ( 7 ). New standards like the Common Core State Standards (CCSS) (currently adopted in 42 states, District of Columbia and four territories) are designed to be relevant to the real world, while preparing students for postsecondary education and careers ( 8 ).
Terminology utilized to describe children who have difficulties with math varies in the literature depending upon how the study populations are defined and what instruments are used. Generally speaking, the term mathematical difficulties refers to children whose poor mathematics achievement is caused by a variety of factors from poor instruction to environmental factors, and represents a broader construct than does the term math disability (MD). Children with mathematical difficulties have low average performance or poor performance in math, but not all children with mathematical difficulties will have MD, which is hypothesized to be due to an inherent weakness in mathematical cognition not attributable to sociocultural or environmental causes ( 9 ). The continuously changing diagnostic criteria and varying definitions between the educational and medical/mental health realms add an additional confounder between the two terms. Terms like dyscalculia and poor math achievement add to the confusion as it is unclear if the terms are meant to be synonymous or overlapping ( 10 ). For the purposes of this review, the terms math learning disorder, math learning disability and dyscalculia will be considered as synonymous and represented by MD.
In many international clinical settings, criteria for MD is outlined in the World Health Organization International Statistical Classification of Diseases and Related Health Problems 10 th edition (ICD10). It is defined as a specific impairment in arithmetical skills not solely explicable on the basis of general mental retardation or of inadequate schooling, which involves mastery of basic computational skills rather than more abstract mathematical skills ( 11 ). For the US, the definition in the most recent revision of the Diagnostic and Statistical Manual of Mental Disorder s (DSM) 5 th edition, DSM5 ( 12 ) is in the single category of specific learning disorders with specifiers for the area of math (others being reading and written expression). The definition states that difficulties should have persisted for at least six months despite interventions, and skills should be substantially below those expected for age. Deficits should interfere with functioning, as confirmed by individually administered standardized achievement measures and comprehensive clinical assessment. It includes possible deficits in number sense, memorization of math facts, calculation, and math reasoning.
Across reviews of studies ( 13 ), and as a matter of clinical practicality, most MD is identified by the school systems (educational and school psychologists, special educators among others). As such, it is critical to understand the changing criteria for MD determinations in the education system. With passage of the original version of Individuals with Disabilities Education Act (IDEA) in 1975 when specific learning disabilities (SLD) was recognized as a disability category for special education services, the operational definition was based on abilityachievement discrepancy in 1977 ( 14 ). Over time, the discrepancy model was deemed flawed and lacking in validity, and in 2004 reauthorization changed the definition to be based upon inadequate response to research based interventions ( 15 ). Yet, reviews of research indicate that despite the abandonment of the discrepancy model, studies continue to use this to identify subjects. Furthermore, much of the research on MD appears to be focused on elementary school children and basic math skills due to an emphasis on early identification and doesn’t pay adequate attention to MD in older children ( 16 ) where math involves complex domains like algebra, and math reasoning.
There are almost 2.5 million students (5%) with SLD receiving special education services in US public schools, but this number has declined in the last decade, largely due to increased use of instructional strategies, shifts in SLD identification and attention to early childhood education ( 17 ). There is a wide range of prevalence data in the research, depending on criteria used for identification and cutoffs for determination, as well as the country of study. Prevalence data has varied from 3–7% ( 12 ). With regards to gender, results have varied depending upon the criteria used for identification, and overall gender differences were not identified when using absolute thresholds or relative discrepancy criteria in defining MD ( 18  21 ).
Neurobiological basis
MD is considered a neurodevelopmental disorder, involving dysfunction in specific brain regions that are implicated in math skills. Numerosity is considered the building block of math skills, and relies on visual and auditory association cortices and the parietal attention system, specifically the intraparietal sulcus (IPS) within the posterior parietal cortex (PPC) ( 22 ). There is a developmental sequence in childhood to math acquisition; children initially rely on procedurebased counting (which when repeated, results in associations for retrieval). This is reflected in greater activation in functional imaging studies of the dorsal basal ganglia, which is involved in working memory (WM) ( 23 ). Gradual development shift mostly occurs in second and third grades to retrieval from long term memory, laying the foundation for more complex math skills ( 24 ). Brain involvement for complex math is based in the medial temporal lobe, with connections to other brain areas, especially the hippocampus and the prefrontal cortex (PFC). Younger children exhibit greater involvement of the hippocampus, and parahippocampal gyrus ( 25 , 26 ) compared to adults. The greater activation seen in this region in younger subjects may reflect the greater recruitment of processing resources for memorization and may also reflect novelty effects ( 27 ). As children mature, there is greater activation in the left PPC and lateral occipital temporal cortex along with lower activation in multiple PFC areas evincing more specialization ( 27 ).
Individuals with MD have reduced activation during math tasks in functional imaging studies involving the IPS ( 28 , 29 ) and structural imaging also showed reduced gray matter in the IPS in those with MD ( 30 ) and reduced connectivity between parietal and occipitotemporal regions ( 31 ). A recent study postulated that deficient fiber projection in the superior longitudinal fasciculus (particularly adjacent to the IPS) which connects parietal, temporal and frontal regions in children with MD is related to MD via a “disconnection” or interruption of integration and control of distributed brain processes ( 32 ). Since research in MD is not as established as reading disorders (RD) and is ongoing, no single hypothesis accounts for all children with MD. Multiple theories exist as described below.
Core deficit hypothesis
This predicts using neuroimaging data that numeric concepts such as quantity, magnitude, numerosity are associated with the IPS. This is further distinguished as deficit in processing number magnitude ( 33 ) or deficit in number sense ( 34 ). However, developmental literature is inconclusive and inconsistent as MD has a complex and heterogenous clinical presentation ( 35 ).
Deficits in general domain hypothesis
This presumes there are subtypes of MD based on impairment in underlying cognitive processes. Three subtypes are thought to be associated, deficits in; verbal WM (necessary to acquire math procedures), longterm memory (necessary for storage and retrieval of math facts), and visuospatial processing ( 36 ).
Deficits in domainspecific math areas
The Triple Code model ( 37 ) of number processing hypothesizes there are three math domains: (I) numerical quantity representation (similar to number sense); (II) visuospatial numerical representation associated with attentional shifting; and (III) auditory verbally representation (associated with math facts and retrieval). This has some neuroanatomical support with IPS corresponding to first, posterior superior parietal lobule to second, and angular gyrus (AG) and perisylvian areas to the third domain respectively.
Procedural deficit hypothesis
This states that MD is a deficit of procedural memory involving a neural network of frontal, parietal, basal ganglia and cerebellar systems involved in storage and recall of skills. After practice, learned information is processed rapidly and automatically ( 38 ). However, while promising, it does not account for all cases of MD.
Genetic basis
There are thought to be a variety of intrinsic genetic contributors to MD, including genetic mutations and polymorphisms with hypothesized influences ranging from alterations in neural development to connectivity to epigenetic effects ( 39 ). Populationlevel studies take into account genetic variants on genotyping platforms and provide a lower bound estimate of heritability, while twin models capture all genetic factors and produce higher estimates due to the presence of dominance or interaction effects ( 40 ). Studies revealed that monozygotic and dizygotic twins of individuals with MD were twelve and eight times more likely to have MD, respectively ( 41 ) and >50% of siblings of individuals with MD also had MD ( 42 ). Math ability is thought to be influenced by many genes generating small effects across the entire spectrum of ability in genomewide association studies ( 43 ) and around half of the observed correlation between math and reading ability is due to shared genetic effects ( 40 ). In a twin study, 60% of the genetic factors that influenced math ability also influenced reading ability and about 95% of the phenotypic correlation between the two is explained by these shared genetic influences ( 44 ).
Certain genetic conditions increase the risk of MD. Children with 22q11.2 deletion syndrome have deficits in calculation, math word problem solving and numerical quantities despite intact math fact retrieval ( 45 ). Children with Turner syndrome have intact number comprehension and processing skills, but have slower and more errorprone functioning on complex math problems ( 46 ). Children with fragile X syndrome have both MD and RD ( 47 ), while children with myelomeningocele have deficits in simple math, counting and math word problem solving despite relative strengths of basic number knowledge and language skills ( 48 ). Children with neurofibromatosis1 have deficits in math calculation and math word problems and more variability in math skills than other genetic condition ( 49 ), and children with Williams syndrome have significant visuospatial deficits which may be responsible for their MD features ( 50 ).
Clinical presentation
While the theories described above are tested against cognitive neuroscience models, from a clinical perspective, it is helpful to classify presentations depending on the skills and deficits demonstrated. One such model from Karagiannakis et al. ( 51 ) has 4 subtypes of skills:
 Core number: which involves numerosity, estimating numbers and quantities, number line ability, managing symbols and basic counting;
 Memory: retrieving math facts, performing calculations, remembering rules and formulae;
 Reasoning: grasping math concepts, complex math procedures, logical problems and problem solving;
 Visuospatial: geometry, written calculations, graphs and tables.
Typically developing children may have an experiential knowledge of math even prior to formal school math education, where they get exposed to math facts and calculation procedures. Clinical features of a child presenting with MD ( Table 1 ) largely depends on three factors: the age/developmental age of the child, the existing math instruction and curriculum exposure, and the presence of comorbid conditions.
Age  Examples of expected math skills  Symptoms of MD 

Toddlers and preschoolers  Symbolic representation  Difficulty learning to count 
Recognize and label small numbers  Difficulty sorting  
Simple magnitude awareness  Difficulty corresponding numbers to objects  
Simple subitizing  Difficulty with auditory memory of numbers (e.g., phone number)  
Matching and naming simple shapes  
Cardinal rule  
One to one correspondence  
Rote counting followed by meaningful counting  
Kindergarten  Rapid recognition of small quantities  Difficulty counting 
Mastery of counting (individually and in classes)  Difficulty subitizing  
Recognition of numbers  Trouble with number recognition  
Solve simple math word problems  
Able to recognize and name parts of shapes  
1 –3 grade (early elementary)  Naming and writing large whole numbers  Difficulty with magnitude comparison 
Counting forwards, backwards, skip count  Trouble learning math facts  
Identifying place value of digits  Difficulty with math problemsolving skills  
Simple addition and subtraction  Over reliance on ﬁnger counting for more than basic sums  
Compare collections even if disparate in size and quantity  Anxiety during math tasks  
Number conservation  
Understand complementarity of addition and subtraction  
Retrieve some arithmetical facts from memory  
Classifying and sorting shapes based on attributes  
4 –8 grade (late elementary through middle)  Numbers up to 100,000, comparing and ordering numbers  Difficulties with precision during math work 
Place value of larger numbers  Difficulty remembering previously encountered patterns Difficulty sequencing multiple steps of math problem  
Understanding of fractions  Difficulty understanding realworld representation of math formulae  
Multiplication and division and their inverse relationship  Anxiety during math tasks  
Basic algebraic concepts to solving linear equations  
Reading graphs, tables, charts  
Understanding ratio, proportions, percent, unit pricing  
Learning to draw, compare and classify two and threedimensional shapes  
Learning to measure objects by length, area, weight, volume using standard units  
High school  Equations and inequalities  Struggle to apply math concepts to everyday life, including money matters, estimating speed and distance 
Logic and geometry  Trouble with measurements  
Polynomials  Difficulty grasping information from graphs or charts  
Exponential and logarithmic functions  Difficulty arriving at different approaches to same math problem  
Trigonometry  Anxiety during math tasks  
Sequences, series, and probability  
Statistics: sampling, designing experiments, measures of center and spread  
Statistics: normal distributions, confidence intervals, tests of significance, graphical displays of association 
Not an inclusive list, dependent on instruction and local curriculum. MD, math disability.
The coexistence of another condition along with the primary condition under study is considered “comorbidity” ( 52 ). For MD, the most prevalent comorbidity is RD, with rates as high as 70%, and correspondingly for children with RD, rates of MD can be as high as 56% ( 18 , 53 , 54 ). The rates tend to be lower when more stringent cutoffs are applied to the definitions of the various disorders ( 55 ), and when population samples are studied compared to identified SLD samples. Distinguishing MD from comorbid MD with RD has focused on performance on nonverbal and verbal tests with studies showing that students with RD (and MD + RD) experienced more difficulty with phonology; students with MD (and MD + RD) more difficulty with processing speed, nonverbal reasoning, and most mathematical performances ( 56 ). While there have been suggestions that inattention and poor planning associated with attention deficit hyperactivity disorder (ADHD) may be responsible, MD and ADHD are thought to be comorbid separate disorders that are independently transmitted in families ( 57 ). A Spanish study ( 58 ), which attempted to distinguish the cognitive profiles between ADHD and MD children, revealed that simultaneous processing was more predictive in the MD group while executive processes was predictive in ADHD group.
Math anxiety is specific to math and is a negative emotional reaction or state of discomfort involving math tasks ( 59 ). It is not a rare phenomenon and has been found in 4% of high school students in the UK ( 60 ). Children with MD can develop a negative attitude towards math in general, and avoidance of math activities associated with anxious feelings. Negative experiences with math teachers compounds the situation. Poor math achievement is strongly related to math anxiety, especially when children are expected to work rapidly towards a single correct response ( 61 ). There is debate whether math anxiety is a distinct entity from generalized anxiety, as some studies have shown correlation with measures of general anxiety ( 62 ) while other studies show measures of math anxiety correlate more with one another than with test anxiety or general anxiety ( 63 ). Math anxiety is thought to develop as young as first grade ( 64 ), and brain activity shows that math anxious children show increased amygdala activity (emotional regulation) with reduced activity in dorsolateral PFC and PPC (WM and numerical processing) ( 65 ).
This is the first step in a diagnostic process, and is usually conducted with the general student population to identify “atrisk” children (as early as kindergarten) who need to proceed to a formal diagnostic process. Gersten et al. ( 66 ) outlines common components included in screening batteries such as: magnitude comparison, strategic counting, retrieval of basic arithmetic facts, and (more recently) word problems and numeral recognition. Single proficiency screening measures, which are easy to administer quickly with large numbers of students, have comparable predictive validities to multiple proficiency screening measures that cover a wider range of mathematics proficiencies and skills. No single test score is found to be predictive, though performance on number line estimation ( 67 ) and reading numerals, number constancy, magnitude judgments of onedigit numbers, and mental addition of onedigit numbers ( 68 ) are most correlated with math achievement. Additionally, testing WM is recommended along with mathspecific items ( 68 ).
In the UK, a computerized screener has been used ( 69 ). In the US, the Response to Intervention (RTI) model ( 70 ) recommends screenings on a schoolwide basis at least twice yearly, using an objective tool and focused on critical math objectives for each grade level that are prescribed at the state level (or CCSS as applicable). Curriculumbased measurement (CBM) probes are empirically supported for screening ( 71 ) and for math generally is a single proficiency measure (digit computation). Health professionals’ role in screening generally involves providing anticipatory guidance on development and behavior and surveillance for “early warning signs” of MD (especially in preK children) of difficulty counting loud and struggling with number recognition and rhyming ( 72 ). Other roles include investigating and treating potential medical problems that can affect the patient’s ability to learn ( 73 ) while ruling out medical differential diagnoses for learning problems like seizures, anemia, thyroid disease, sensory deficits and elevated lead levels.
The process of diagnosis of MD depends on the discipline of the clinician first encountered, what criteria are used, as well as local and regional regulations. In early grades (kindergarten and first grade), diagnostics probe for functioning on foundational skills like core number processing. This is reliant on exposure to, and interaction with, symbolic processing skills as well as language and spatial processing. As children get older, increasing abstract concepts are relied on for math processing, and deficits in these are used to make the diagnosis ( 74 ). Differences observed in young children may result from exposure to mathematics before formal schooling or from student performance on more formal mathematics in school ( 66 ). Diagnostics can occur either in a medical interdisciplinary or educational model depending on local and regional availability and access to such clinicians.
Specialists in interdisciplinary clinics (like Child Neurologists, DevelopmentalBehavioral Pediatricians, Pediatric Psychologists) can be involved, often at the request of caregivers with referrals from primary care providers, or for independent consultations in event of questions or concerns about the adequacy of school services. This may engender tension between educators and medical professionals despite the acknowledged importance of collaboration; particularly around the perception that medical clinicians foster dissension between parents and schools, don’t acknowledge the competence of school personnel and make recommendations not reflecting the individual needs of the child or impact on the cost or structure of the child’s education ( 75 ). In the medical model, history is usually elicited around the child’s symptoms and current functioning in all academic areas, history of interventions or grade retention, family history of MD, presence of other developmental delays (gross, fine motor, visuomotor, language, adaptive) and behavioral symptoms (including anxiety, somatic symptoms and attentional issues). Apart from a physical and neurological examination, screening tests for learning disabilities are conducted. In the interdisciplinary team, clinical psychologists are invaluable in conducting diagnostic assessments including cognitive and academic batteries.
Neuropsychological testing (NPT) has been presumed to be broader than psychoeducational assessment and suggested as an essential part of SLD identification, as it can provide information on strengths and weaknesses, particularly if other medical conditions exist ( 76 ). While studies ( 77 , 78 ) have shown that performance on neuropsychologicalpsychological batteries can predict academic achievement later, such testing is time intensive and many insurances (both public and private) only consider NPT medically necessary in the assessment of cognitive impairment due to medical or psychiatric conditions. They explicitly exclude coverage for educational reasons (unless a qualifying medical disorder is present like metabolic disorders, neurocutaneous disorders, traumatic brain injury) and suggest that testing be provided by school systems. NPT is of value in context of medical disorders or when function deterioration (due to neurological conditions) is not adequately explained by socialemotional or environmental factors ( 79 ). However, one limitation of NPT is that reports offer hypotheses about a child’s level of functioning that may not account for, or be relevant to, planning interventions within the school setting and is more of a snapshot than a progressively developing picture that school personnel are privy to ( 79 ). There is emerging interest that technology and biopsychosocial data, such as eyetracking data in combination with number line estimation tasks, might be a promising tool in diagnosing MD in children ( 80 ).
In an educational model, prior to the 2004 reauthorization of IDEA, eligibility for determination as SLD (including math) rested on the ability—achievement discrepancy model which required assessment of cognitive and academic functioning and relied on formulae that determined cutoffs. However, since 2004, local education agencies were permitted to adopt criteria that could either be the discrepancy model or an alternate based in evidencebased science. One of the latter, the RTI model ( 70 ) changed eligibility to students who fail to respond to increasing hierarchy of datadriven interventions based on individual student problems. However, despite the law and subsequent regulations, there is no directive regarding magnitude of achievement and progress targets for struggling learners which continues to create varying identification practices ( 81 ).
Another approach emphasizes the role of strengths and weaknesses in cognitive processing measured by individually administered standardized tests. This patterns of strengths and weaknesses (PSW) approach needs additional empirical evidence to determine the robustness of this model as an alternate to existing procedures ( 82 ). There continues to be considerable variability in the state practices using identification techniques, with 34 states continuing to use the discrepancy model, and 10 states explicitly prohibiting its use. While 45 states provide guidance on RTI implementation, only 8 states exclusively use RTI models in LD identification, but with variability regarding how to implement models. Less than half of the states allow use of PSW models, but with little information regarding identification practices. Prior to 2004, only half of the states included math reasoning as an area for LD identification, and this has increased to all presently ( 83 ). School eligibility evaluations are conducted by credentialed special education teachers, licensed educational psychologists, or school psychologists.
Generally, an evaluation consists of history and review of records, followed by psychometric testing for academic skills, intellectual abilities, sometimes executive function (EF), socioemotional and behavioral assessment including qualitative information, classroom observations and questionnaires from caregivers and teachers. For MD, the assessment includes whether the student has mastery of math skills compared to the state’s academic content standards despite appropriate instruction and if the impact of the disability entitles the student to specialized instruction to benefit. The school evaluators prioritize educational goals and place them in the context of the school, which is the child’s natural environment ( 79 ).
Educational Interventions
Teaching individuals with MD creates unique challenges in regular education environments, as teachers often struggle to provide individualized attention due to large class sizes, limited resources and learners with different styles. The overall goal is building knowledge and skills to develop automaticity. With students with MD, many math processes never become “automatic,” and they need extra time to improve WM and extra work on cognitive functions ( 84 ). The coteaching model, in which children with MD are taught in the general education classroom, involves a general education teacher delivering overall instructional content, and a second special education teacher designing and delivering more intensive interventions and learning strategies as needed. This model is built around inclusion, to foster positive interaction and behavior among students, and to foster professional development among educators, but it has been criticized for not detecting or differentiating students with MD early in their education ( 85 ).
While the adoption of CCSS standardizes the expected knowledge and abilities of students at a certain grade level, and allows an easier transition if students move between districts or states, some suggest that it also may cause children to fall further behind their peers, and leave gaps in important skills and general math understanding. It can create hardship for students with MD as well; since the time constraints and the expected amount of material that students are expected to master may lead to students with MD not getting enough procedural practice, and repetition of the basic concepts in order to be successful ( 86 ). Since CCSS is reading intensive, students with comorbid LD and MD struggle to a greater extent, and there are additional challenges transitioning from an older curriculum, and the lack of professional development educators receive during this transition ( 87 ).
The RTI model is a three tiered system is used for early identification (and intervention) of children who may be at risk of future educational failure. The typical representation is a pyramid ( Figure 1 ) of larger numbers in a generalized education environment with progressively smaller groups receiving additional instructional supports and finally students needing intensive instruction similar to specialized education. The model has the benefit of creating accountability by encouraging and guiding practitioners to intervene earlier with the great number of children at risk of failure, and by introducing a more valid method to identify student with MD with progress monitoring and motivational strategies, particularly for students in Tier 2 and Tier 3 instruction ( 88 ). The impetus for RTI has been primarily for reading, with slower adoption for math, though programs have been slowly expanding over the past five to ten years. There has been debate about the value of investment in smallgroup intervention, whether interventions should be aligned with core curriculum, and which students are likely to benefit from intervention. Small group interventions have consistently improved students’ math performance on proximal measures, that is, difficult grade level content and sophisticated topics. However, the effect on distal measures, that is building a general capacity in math, is less clear ( 89 ).
Organizing the school for tiered instruction. Used with permission from the RTI Action Network: http://www.rtinetwork.org/essential/tieredinstruction/tieredinstructionandinterventionrtimodel .
The National Center for Educational Evaluation and Regional Assistance recommends that instructional material should focus intensely on whole numbers in kindergarten through grade five, and on rational numbers in grades four through eight. Instruction should include at least ten minutes devoted to building fluent retrieval of facts, should be explicit and systematic, including providing models, verbalization of thought process, guided practice, corrective feedback, and frequent cumulative review, and students should have the opportunity to work with visual representations of mathematical ideas. Interestingly, in classrooms with a high percentage of students with MD, teachers are more likely to instruct with the use of manipulatives/calculators, or movement/music, or ordering and number/quantity skills, though these activities have not been associated with math achievement gains by students with MD. Students without MD benefit from both studentcentered and teacherdirected activities. However, only increased classroom time doing teacherdirected activities is associated with positive gains in students with MD ( 90 ). One metaanalysis on teaching methods for children with MD revealed that students with lower math performance tend to respond better to instruction, and while interventions may help in many math domains, a smaller effect is seen in early numeracy and general math proficiency. Explicit teacherled instruction, and peerassisted learning have the largest affect, and providing instructional recommendations to teachers, and the use of technology may also have positive effects. Specific instructional components that may be beneficial for students with MD include controlling task difficulty, greater elaboration on topics, and working in a small group setting. Additionally, cognitive and metacognitive strategies computerbased interactive lessons, videos, and handson projects have been shown to aid math understanding and performance. Using concrete and visual representations along with teacherfacilitated instruction and virtual manipulatives have also been shown to be effective ( 91 ).
Three general strategies have been shown to be useful in accommodating children with MD including; improving reading skills, improving mathematical problem solving skills, and altering general instructional design ( Table 2 ) ( 92 ). There are several specific math intervention programs which have gained research support. For a detailed description of these programs, see: http://www.hanoverresearch.com/2015/04/06/bestpracticesinmathinterventions/ . A detailed list of math interventions describing the appropriate age of learners and the level of evidence is also available at the What Works Clearinghouse, an online database created by the U.S Department of Education’s Institute of Education Sciences ( 93 ).
Methods  Accommodations 

Improving reading skills  Break up the text into smaller sections 
Use a simple font  
Do not justify the text  
Use colored overlays to reduce glare  
Improving mathematical problem solving skills  Photocopy math books with the relevant sections placed in order 
Separate complicated problems into small steps  
Use markers to highlight, and guide attention  
Use color to delineate columns and rows in spreadsheets  
Simplify tables  
General instructional design  Supplement incomplete notes 
Use posters to remind students of various basic concepts  
Use flash cards  
Provide flow charts to clarify procedures  
Engage visual learners with manipulatives  
Encourage students to move at their own pace  
Teach organization, studying, and time management skills  
Focus on revision prior to an examination 
Not an exhaustive list. MD, math disability.
While emphasis is usually on implementing an intervention in the way it was researched and manualized, interventions are often slightly modified by special education teachers in response to their students’ notions of mathematics, and the particular need of individual students, or small groups of students. They do this through changing the pedagogy, materials, or tasks to bridge students’ prior and informal knowledge. They provide additional practice opportunities and connect procedural practices to larger concepts, provide additional time to practice in areas of difficulty, foster greater student interaction, and allow students more opportunity to think out loud and justify their thought processes ( 94 ). These practices should not be discouraged, because research has shown that training tailored to a child’s specific needs can create positive results ( 95 ).
Behavioral interventions
While educational interventions comprise the bulk of effective interventions for MD, there have been behavioral interventions that are effective in improving the math skills of children with MD, in the areas of EF, cognitive tutoring, and cognitivebehavior therapy (CBT) to improve performance, decrease math anxiety, and foster a positive attitude towards math.
EF broadly refers to the processes of attention, WM, long term planning, volition, and behavior inhibition ( 96 ) that allows students to organize and prioritize information, monitor progress, and adapt. Students who struggle with EF may have difficulty determining key information in math word problems, performing mental math, or starting a task, and they may make careless mistakes if they fail to check their work ( 97 ).
Strategies that can be implemented by school personnel include teaching positive selfregulatory skills through engagement, awareness of strengths and needs, goal setting, skill mastery, and generalization. A highly structured environment and schedule, limiting distractions, and providing interventional cues can foster selfregulation. For students who have difficulty with WM, teachers should limit the number of concepts presented at any one time, and may find it helpful to group information into chunks ( 97 ).
Improving attention in children with low arithmetical achievement may help to improve their abilities. Strategies include using instruction in improving reaction time, sensory (visual, auditory) selectivity, and attention shift ( 98 ). WM is specifically related to the ability of learn new math skills and is widely accepted to be impaired in individuals with MD. WM is correlated with math performance and is indicative of future math performance. WM training shows improvement in math skills especially when involving visuospatial WM ( 99 ). Cognitive tutoring has been found to be successful in improving math performance, effecting neuroplasticity, and changing the brain function of children with MD. Functional imaging studies have shown that initial differences in the prefrontal, parietal and ventral temporaloccipital cortices normalized after eight weeks of oneonone tutoring focused on strengthening conceptual and procedural knowledge to support attention, WM, visuospatial skills, and cognitive inhibition necessary for math fluency ( 100 ).
Behavioral interventions are also important in addressing math anxiety, and early intervention is important as math anxiety tends to increase with age. Interventions include systematic desensitization, and cognitive behavior therapy (CBT); for example, expressive writing before a test may cause the student to reevaluate the need for worry, and therefore increase the WM available to perform specific tasks ( 101 ). Studies have shown that cognitive behavioral intervention with targeted tutoring may be helpful in decreasing anxiety, fostering a positive attitude towards math, and improving performance ( 102 ).
Novel approaches
Computer games and tutorials have been used for the past 30 years in special education to aid students with MD to improve basic math knowledge and skills ( 103 ). Rapid advances in broadband connectivity and ubiquity of mobile computing has led to almost every student having access to a computer connected to the internet, and increased digital content ( 104 ). Technology helps to emphasize important concepts, engage multiple sensory modalities, divide complex material into smaller components, and provide immediate feedback about accuracy ( 104 ). In one study, the use of math apps allowed both students with and without MD to make gains in math assessments. The larger gain was made in the struggling group, which helped to close the achievement gap ( 105 ).
Other technologies that can aid students with MD include computers and tablets with touch screens that are easier to use than traditional mouse and keyboards, and children can use them with little instruction ( 105 ). For students with MD, this type of feature allows for accommodations such as “talking calculators,” and the easy transfer of numbers to graphs, tables, and charts to create visual aids. Now, programs containing an artificial intelligence element can interactively tailor the features and settings of these programs, to the needs and abilities of particular students. For additional information about specific programs, please see the paper by Campuzano et al. ( 106 ).
Recent research has shown that virtual manipulatives can be invaluable when integrated with general math concepts to ensure a meaningful learning experience to match the individual student’s needs and abilities. For classrooms already equipped with computers, the cost is negligible since many virtual manipulative websites are easy to access, free of charge and don’t require storage or cleaning which benefits busy teachers ( 103 ). Video prompting using tablet devices have the added benefit of giving educators more time to work with small groups, or oneonone, while others work in a selfdirected manner, and students benefit by gaining independence, and becoming more accountable for their own learning ( 107 ).
The importance of identification and intervention is underscored by the impact of MD on child and adult functioning. Shalev et al. ( 108 ) demonstrated in a longitudinal study over three and six years that MD is persistent, which is similar to other developmental disorders like ADHD and RD with educational interventions not being protective against persistence of MD. There is limited research on SLD outcomes in adults, much less with respect to MD, and it tends to focus on young adulthood with limited data points which are less helpful about trends. The National Longitudinal Transition Study – 2 only provides a snapshot of the immediate years after leaving school, lacks specific information about SLD in absence of other disabilities, and most information is about literacy and not numeracy. It also does not include persons who have dropped out of school, and describes attendance (but not completion) of twoyear postsecondary schools. It does not track the types of employment and job advancement ( 109 ). Educational achievement is hypothesized to translate to higher SES through higher attained qualifications, improving occupational status and career development opportunities ( 110 ) and numeracy is associated with successful financial decision making ( 111 ).
Conclusions
Changing diagnostic manuals, different hypotheses regarding the core precepts of the disorder, and different criteria adopted in medical and educational systems can lead to different perspectives on MD. Advances in neuroscience and genetics offer promise in etiological determinations and understanding the neural processes underpinning MD. Clinical presentation varies depending on age and developmental status of the child, presence of other comorbid conditions, and the degree and nature of instructional methods used. The core interventions for MD continue to be educational in nature, with a varying degree of evidencebase that continues to grow. Additional avenues include behavioral interventions, which are primarily for comorbid conditions and math anxiety. Exciting new directions utilizing informational technology can help supplement educational interventions.
Acknowledgements
Conflicts of Interest: The authors have no conflicts of interest to declare.
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Assessment plan: a guide to evaluating for dyscalculia.
Assessment Considerations for Dyscalculia:
Using the wj iv cog & ach.
According to the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM5; APA, 2013), Dyscalculia is characterized by academic achievement that is substantially below age expectations in the areas of understanding number concepts, number facts or calculation; and/or mathematical reasoning (e.g., applying math concepts or solving math problems). Complications may be present during formative school years, although some deficits may not present until later in life when demands increase. These mathematical deficits must cause problems in the individual’s daytoday functioning (e.g., school), and must not be due to any other condition (e.g., intellectual disability, lack of access to adequate instruction).
The WJ IV COG and ACH are powerful tools in the assessment of Dyscalculia¹, allowing the examiner to investigate various latent and applied abilities related to these aspects of mathematical achievement. Clusters and tests of interest from the respective instruments are listed below for your review.
Using the WJ IV COG for Dyscalculia
Administering the WJ IV COG allows an examiner to investigate underlying abilities which lend themselves to academic achievement.
Fluid and Quantitative Reasoning
Tests in the Fluid Reasoning cluster assess a host of cognitive functions (e.g., drawing inferences, identifying and forming concepts, and identifying relationships). Fluid Reasoning supports an individual’s ability to think flexibly (“cognitive flexibility”) and apply their knowledge across various domains. Quantitative Reasoning ² is assessed as an extension of Fluid Reasoning and is defined as the extent to which an individual can understand and reason using mathematical concepts. An examiner can obtain scores for these clusters by administering Number Series, Concept Formation, and AnalysisSynthesis.
 Number Series assesses quantitative reasoning and inductive reasoning (i.e., drawing general conclusions from specific details). This test requires the examinee to determine the missing number in a sequence.
 Concept Formation measures categorical and inductive reasoning, while incorporating an element of mental flexibility, as the examinee must remember what is learned while completing the task.
 AnalysisSynthesis is a controlledlearning task, which tests deductive reasoning (i.e., the ability to form conclusions based on general principles)
¹ Please refer to Schrank et al. (2017) for more comprehensive details on how to use the WJ IV CoreSelective Evaluation Process to identify a SLD. For more detailed information regarding cluster and test descriptions, please consult the WJ IV COG and ACH examiner’s manuals (Mather & Wendling, 2014) and the WJ IV Interpretation and Instructional Interventions Program Manual (WIIIP; Schrank & Wendling, 2015). For more information regarding the intraachievement variations, please refer to the WJ IV Technical Manual (McGrew et al., 2014).
² AnalysisSynthesis is needed to derive the extended Fluid Reasoning Cluster and the Quantitative Reasoning Cluster.
Number Facility
Number Facility can be defined as an individual’s speed and accuracy when working with numbers. It includes Numbers Reversed and NumberPattern Matching.
 Numbers Reversed measures an individual’s shortterm working memory and attentional capacity. Examinees are asked to hold strings of digits in their immediate awareness and then manipulate those digits by reversing the sequence.
 NumberPattern Matching is a measure of processing speed, which assesses how quickly and accurately an examinee can visually discriminate to locate identical numbers amongst distractors.
Scholastic Aptitude Cluster for Mathematics
Examiners can also elect to administer tests that are evidencebased to predict mathematical achievement with respect to calculation and problemsolving. Examiners can then use Riverside Score to analyze Scholastic Aptitude/Achievement Comparisons. These comparisons can be useful in discerning whether an examinee’s tested academic performance level aligns or is inconsistent with, their performance on tasks of related cognitive abilities. The WJ IV COG tests for the Mathematics Aptitude Cluster are listed below:
 Oral Vocabulary tests vocabulary knowledge via two subtasks. One subtask requires the examinee to provide a synonym for a target word, and the other requires antonyms.
 Visualization is another twopart task assessing distinct aspects of visual processing including the ability to mentally manipulate stimuli.
 Pair Cancellation is a measure of cognitive processing speed which provides information related to an examinee’s ability to sustain attention and maintain vigilance.
 Number Series (discussed above under Fluid and Quantitative Reasoning)
Using the WJ IV ACH for Dyscalculia
Examiners using the WJ IV ACH can derive several mathematics clusters of interest in the evaluation of Dyscalculia. These include Mathematics, Broad Mathematics, Math Calculation Skills, and Math Problem Solving. The breakdown of these clusters is listed within the WJIV Selective Testing Table (Mather & Wendling, 2014, p. 14). There are also several tests of mathematical achievement on the WJ IV ACH, including Applied Problems, Calculation, Math Facts Fluency, and Number Matrices. These tests are listed below in relation to the DSM5 criteria:
Understanding Number Concepts
 Applied Problems assesses logical and quantitative reasoning. The examinee is required to determine appropriate mathematical operations and the values needed for those operations. The calculations increase in difficulty from relatively simple to complex.
 Calculation assesses an examinee’s ability to execute mathematical computations. Items progress in difficulty from being asked to write individual numbers, to completing various mathematical operations (e.g., addition, subtraction, multiplication, division, geometry, trigonometry, etc.). Other math concepts are included in this task, as well (e.g., whole numbers, negative numbers, percentages, decimals, and fractions).
Number Facts
 Math Facts Fluency measures an examinee’s accuracy and speed with addition, subtraction, and multiplication. Qualitative item analysis can help the examiner determine whether the examinee has more difficulty with one operation over another and can provide insight into an examinee’s approach to the task (e.g., working quickly and inaccurately, working slowly, yet accurately).
 Calculation (described above)
Calculation and Mathematical Operations
Math Reasoning and Problem Solving
 Number Matrices tests quantitative reasoning, while also tapping an examinee’s fluid reasoning capacity. Examinees are shown number matrices and asked to identify the missing number.
 Applied Problems (described above)
IntraAchievement Variation Procedure
When reviewing an examinee’s performance, examiners can run an intraachievement variation procedure using Riverside Score . This procedure compares an examinee’s performance in one achievement area to their expected performance³. Examiners with access to the WJ IV Interpretation and Instructional Interventions Program Manual ( WIIIP ; Schrank & Wendling, 2015) can also run comprehensive reports for an indepth analysis of an examinee’s profile and generate personalized interventions and accommodations to support educational planning. There is a sample comprehensive report available with a WIIIP subscription.
With the selective testing model of the WJ IV and the robust scoring methods offered via Riverside Score and WIIIP, examiners can efficiently and effectively evaluate for Dyscalculia.
³ An examinee’s “expected” or predicted performance is based on their average performance on other achievement areas. Table 515 in the WJACH Examiner’s Manual offers an outline of tests required for the intraachievement procedure (Mather & Wendling, 2014, p. 102)
American Psychiatric Association (2013). Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM5). Washington, D.C.: American Psychiatric Association Publishing.
Mather, N., & Wendling, B.J. (2014). Examiner’s Manual. WoodcockJohnson IV Tests of Achievement. Rolling Meadows, IL: Riverside Publishing.
Mather, N., & Wendling, B.J. (2014). Examiner’s Manual. WoodcockJohnson IV Tests of Cognitive Abilities. Rolling Meadows, IL: Riverside Publishing.
McGrew, K. S., LaForte, E. M., & Schrank, F. A. (2014). Technical Manual. WoodcockJohnson IV. Rolling Meadows, IL: Riverside Publishing
Schrank, F. A., StephensPisecco, T. L., & Schultz, E. K. (2017). The WJ IV CoreSelective Evaluation Process Applied to Identification of a Specific Learning Disability (WoodcockJohnson IV Assessment Service Bulletin No. 8). Itasca, IL: Riverside Assessments, LLC
Schrank, F. A., & Wendling, B. J. (2015). Manual and Checklists. WJ IV Interpretation and
Instructional Interventions Program. Rolling Meadows, IL: Riverside Publishing
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Computation and ProblemSolving Strategies
One of the most difficult areas identified by afterschool staff is helping students with math. We include some tips on helping students who are being taught computation with some newer and more traditional strategies. We also provide a model for helping students think through word problems that has emerged from highachieving East Asian countries.
A. Strategies that use and build number sense
Addition and subtraction: Working with ten(s) Help students use numbers that are easy to work with. Get the very young to work with 10. It is easy to add numbers that end in zero in your head. Students should become very comfortable with all the combinations that make 10!
To make 9 into 10 you need 1 more so think of 6 as 1 & 5.  
To make 27 into 30 you need 3. Split 18 into 3 & 15.  
To subtract from 10, first take away 4 ones. See 6 as 4 & 2.  
It is easy to subtract numbers ending in zero. Subtract 40. But we subtracted 2 too many. We must put them back now. 
B. Multiplication and division
Children need to understand that multiplying is counting by groups of things, each group having the same quantity. Help them see how the tables are built one group at a time. Some solution strategies that help students understand the multiplication algorithm are:
1. Build an array to show the meaning of multiplication.
(4 rows of 12)


OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO  1 row has 12 2 rows have 24 3 rows have 36 4 rows have 48 
2. Ttables. These numbers can be recorded in a ttable that gives meaning to the multiplication tables.


1  12 
2  24 
3  36 
4  48 
Later, students can begin reasoning with such tables and not have to write every number.
 
1  18  Given 
2  36  Add another 18 
4  72  Double amount for 2 boxes 
6  108  Add amounts in 2 and 4 boxes 
3. Area models
These models are more abstract and can allow students to see partial products and can eventually be linked to the traditional algorithm.
Begin by building an array model with base10 blocks. Students must know how to break numbers apart by place value to do this. Here is an illustration for 24 x 45. (You would start with something simpler, such as 8 x 12.)
24 is split into 20 and 4. 45 is split into 40 and 5.
100  100  100  100  10  10  10  10  10  
100  100  100  100  10  10  10  10  10  
10  10  10  10  1  1  1  1  1  
10  10  10  10  1  1  1  1  1  
10  10  10  10  1  1  1  1  1  
10  10  10  10  1  1  1  1  1 
There are eight 100 blocks or 8 x 100 = 800. There are 26 blocks of 10 or 26 x 10 = 260. There are 20 blocks of 1 or 20 x 1 = 20.
800 260 + 20 1080
Eventually students chunk numbers and draw four large blocks. This mirrors the fundamental distributive property.
 20x40
 20x5

4x40
 4x5

800 + 160 = 960 100 + 20 = 120 960 + 120 = 1,080
Then you can connect the numbers from the area model to a multiplication problem. Keep the full quantities visible at first. It is 20 times 40 not 2 times 4. When this level is clear, do the same thing but show how you can start the calculation in ones place. Continue to record the full quantities. Finally, when that routine is comfortable, show how the traditional algorithm is another way of doing the same thing. This time not all the partial products are recorded. There is some remembering in the head that you have more tens. What is very tricky is that they cannot be added to the tens place number until that multiplication has been done. Students who have difficulty “moving over” or remembering when to add in the regrouped figures should be allowed to write the complete numbers as partial products.
4. Division
The language used for division can be an impediment to students understanding what division is all about. For example, to say “9 goes into 81, 9 times” is difficult to visualize. What does that mean? Why does it “go in” so many times? Ask instead, how many groups of 9 are there in 81?
Before students are forced to think abstractly, help them understand what these numbers stand for. Since division is the inverse of multiplication, we are still dealing with groups. In the problem 3,528 divided by 24, they are finding how many groups of 24 there are in 3,528 or how many groups of size 24 could be made. Could there be 100? Certainly. 100 x 24 is 2,400. That leaves 1,128. Then some students might recognize there are at least 20 more. Others might stick with thinking 10 more at a time.
As they work through the problem, help them understand what has been accounted for so far and how many more are still left to put into groups.
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Math disability in children: an overview
by: The GreatSchools Editorial Team  Updated: June 13, 2023
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Recently, increased attention has focused on students who demonstrate challenges learning mathematics skills and concepts that are taught in school across the grade levels. Beginning as early as preschool, parents, educators, and researchers are noticing that some students seem perplexed learning simple math skills that many take for granted. For example, some young children have difficulty learning number names, counting, and recognizing how many items are in a group. Some of these children continue to demonstrate problems learning math as they proceed through school. In fact, we know that that 5% to 8% of schoolage children are identified as having a math disability.
Research on understanding more completely what a math disability means and what we can do about it in school has lagged behind similar work being done in the area of reading disabilities. Compared to the research base in early reading difficulties, early difficulties in mathematics and the identification of math disability in later years are less researched and understood. Fortunately, attention is now being directed to helping students who struggle learning basic mathematics skills, mastering more advance mathematics (e.g., algebra), and solving math problems. This article will explain in detail what a math disability is, the sources that cause such a disability, and how a math disability impacts students at different grade levels.
What is a math disability?
A learning disability in mathematics is characterized by an unexpected learning problem after a classroom teacher or other trained professional (e.g., a tutor) has provided a child with appropriate learning experiences over a period of time. Appropriate learning experiences refer to practices that are supported by sound research and that are implemented in the way in which they were designed to be used. The time period refers to the duration of time that is needed to help the child learn the skills and concepts, which are challenging for the child to learn. Typically, the child with a math disability has difficulty making sufficient school progress in mathematics similar to that of her peer group despite the implementation of effective teaching practices over time. Studies have shown that some students with a math disability also have a reading disability or AttentionDeficit/Hyperactivity Disorder (AD/HD). Other studies have identified a group of children who have only a math disability.
Several sources of math disability
When a child is identified as having a math disability, his difficulty may stem from problems in one or more of the following areas: memory, cognitive development, and visualspatial ability.
Memory problems may affect a child’s math performance in several ways. Here are some examples:
 A child might have memory problems that interfere with his ability to retrieve (remember) basic arithmetic facts quickly.
 In the upper grades, memory problems may influence a child’s ability to recall the steps needed to solve more difficult word problems,to recall the steps in solving algebraic equations, or to remember what specific symbols (e.g., å, s, ?, ?) mean.
 Your child’s teacher may say, “He knew the math facts yesterday but can’t seem to remember them today.”
 While helping your child with math homework, you may be baffled by her difficulty remembering how to perform a problem that was taught at school that day.
Cognitive development
Students with a math disability may have trouble because of delays in cognitive development, which hinders learning and processing information. This might lead to problems with:
 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
Visualspatial
Visualspatial problems may interfere with a child’s ability to perform math problems correctly. Examples of visualspatial difficulties include:
 misaligning numerals in columns for calculation
 problems with place value that involves understanding the base ten system
 trouble interpreting maps and understanding geometry.
What math skills are affected?
According to the Individuals with Disabilities Education Act of 2004 (IDEA), a learning disability in mathematics can be identified in the area of mathematics calculation (arithmetic) and/or mathematics problem solving. Research confirms this definition of a math disability.
Math calculations
A child with a learning disability in math calculations may often struggle learning the basic skills in early math instruction where the problem is rooted in memory or cognitive difficulties. For example, research studies have shown that students who struggle to master arithmetic combinations (basic facts) compared to students who demonstrated mastery of arithmetic combinations showed little progress over a twoyear period in remembering basic fact combinations when they were expected to perform under timed conditions. According to Geary (2004), this problem appears to be persistent and characteristic of memory or cognitive difficulties. Students with math calculations difficulties have problems with some or most of the following skills:
 Identifying signs and their meaning (e.g., +, , x, <, =, >, %, ?) Automatically remembering answers to basic arithmetic facts (combinations) such as 3 + 4 =?, 9 x 9 = ?, 15 – 8 = ?.
 Moving from using basic (less mature) counting strategies to more sophisticated (mature) strategies to calculate the answer to arithmetic problems. For example, a student using a basic “counting all” strategy would add two objects with four objects by starting at 1 and counting all of the objects to arrive at the answer 6. A student using a more sophisticated “counting on” strategy would add two with four by starting with 4 and counting on 2 more to arrive at 6.
 Understanding the commutative property (e.g., 3 + 4 = 7 and 4 + 3 = 7)
 Solving multidigit calculations that require “borrowing” (subtraction) and “carrying” (addition)
 Misaligning numbers when copying problems from a chalkboard or textbook
 Ignoring decimal points that appear in math problems
 Forgetting the steps involved in solving various calculations
Math word problems
A learning disability in solving math word problems taps into other types of skills or processes. Difficulties with any of these skills can interfere with a child’s ability to figure out how to effectively solve the problem.Your child may exhibit difficulty with some or most of the processes involved in solving math word problems such as:
 Reading the word problem
 Understanding the language or meaning of the sentences and what the problem is asking
 Sorting out important information from extraneous information that is not essential for solving the problem
 Implementing a plan for solving the problem
 Working through multiple steps in more advanced word problems
 Knowing the correct calculations to use to solve problems
Math rules and procedures
Students with a math disability demonstrate developmental delay in learning the rules and procedures for solving calculations or word problems. An example of a math rule includes “any number × 0 = 0.” A procedure includes the steps for solving arithmetic problems such as addition, subtraction, multiplication, and division. A delay means the child may learn the rules and procedures at a slower rate than his peer group and will need assistance in mastering those rules and procedures.
Math language
Some children have trouble understanding the meaning of the language or vocabulary of mathematics (e.g., greater than, less than, equal, equation). Unfortunately, unlike reading, the meaning of a math word or symbol cannot be inferred from the context. One has to know what each word or symbol means in order to understand the math problem. For instance, to solve the following problems, a child must understand the meaning of the symbols they contain: (3 + 4) x (6 + 8) =? or 72 < 108 True or False?
Math disability at different grade levels
As the curriculum becomes more demanding, a math disability is manifested in different ways across the grade levels. For example, the specialized language of mathematics — including terms and symbols — must be mastered in more advanced mathematics curriculum. Problems with counting strategies, retrieving basic facts quickly, and solving word problems seem to persist across grade levels and require extra instruction to reinforce learning.
Ongoing research in math disabilities
We do not fully understand how a math disability affects a child’s ability to learn mathematics in all of the different areas because of the limited research base on math disability. To date, the majority of research has focused mostly on the skills associated with mathematics calculations including number, counting, and arithmetic (e.g., arithmetic combinations or basic facts) and on solving word problems. Much less is known about development and difficulties in areas such as algebra, geometry, measurement, and data analysis and probability.
We know that a group of students exhibit problems learning mathematics skills and concepts that persist across their school years and even into adulthood. We understand that specific problems in the areas of memory, cognitive development, and visualspatial ability contribute to difficulties learning mathematics. Fortunately, researchers and educators are focusing efforts on better understanding the issues these students face as they encounter the math curriculum across the grade levels. In my next article, I will explore methods for identifying a math disability and offer parents ideas for working with their children and teachers to address such difficulties.
Get more information on math disabilities — also known as dyscalculia — at Understood.org , a comprehensive free resource for parents of kids with learning and attention issues .
 Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 415.
 Robinson, C., Menchetti, B., and Torgesen, J. (2002). Toward a twofactor theory of one type of mathematics disabilities. Learning Disabilities Research and Practice, 17(2), 8189.
 Hallahan, D. P., Lloyd, J. W. Kauffman, J. M., Weiss, M. & Martinez, E. A. (2005). Learning disabilities: Foundations, characteristics, and effective teaching. Boston : Allyn and Bacon.
 Bryant, D. P., Bryant, B. R., & Hammill, D. D. (1990). Characteristic behaviors of students with LD who have teacheridentified math weaknesses. Journal of Learning Disabilities, 33, 168177.
 Geary, D. C. (2000). Mathematical disorders: An overview for educators. Perspectives, 26, 69.
 Geary, D. C. (2003). Learning disabilities in arithmetic. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook of learning disabilities (pp. 199212). New York: Guilford.
 Jordan, N., Hanich, L., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74(3), 834850.
 Garnett, K., & Fleischner, J. E. (1983). Automatization and basic fact performance of normal and learning disabled children. Learning Disability Quarterly, 6, 223231.
 Bryant, D. P., Bryant, B. R., & Hammill, D.D. (1990). Characteristic behaviors of students with LD who have teacheridentified math weaknesses. Journal of Learning Disabilities, 33, 168177.
 Hallahan, D. P., Lloyd, J. W. Kauffman, J. M., Weiss, M. & Martinez, E. A. (2005). Learning disabilities: Foundations, characteristics, and effective teaching. Boston: Allyn and Bacon.
 Gersten, R., Jordan, N., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293304.
 Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and mathematical problemsolving performance of students with learning disabilities. Learning Disabilities Research and Practice, 29, 251261.
 Rivera, D. P. (1997). Mathematics education and students with learning disabilities: Introduction to the special series. Journal of Learning Disabilities, 30, 219, 68.
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What Is Dyscalculia? Math Learning Disability Overview
Dyscalculia is a learning disability that makes math challenging to process and understand. symptoms range from difficulty with counting and basic mental math to trouble with telling time and direction. learn more about this math learning disability, including potential causes and treatments here., dyscalculia definition.
Dyscalculia is a math learning disability that impairs an individual’s ability to learn numberrelated concepts, perform accurate math calculations, reason and problem solve, and perform other basic math skills. 1 Dyscalculia is sometimes called “number dyslexia” or “math dyslexia.”
Dyscalculia is present in about 11 percent of children with attention deficit hyperactivity disorder (ADHD or ADD). 2 Other learning disorders, including dyslexia and dysgraphia, are also common – up to 45 percent of children with ADHD have a learning disorder. 3
Dyscalculia Overview
Individuals with dyscalculia have difficulties with all areas of mathematics — problems not explained by a lack of proper education, intellectual disabilities, or other conditions. The learning disorder complicates and derails everyday aspects of life involving mathematical concepts – like telling time, counting money, and performing mental calculations.
“Students and adults with dyscalculia find math puzzling, frustrating, and difficult to learn,” says Glynis Hannell, a family psychologist and author of Dyscalculia: Action Plans for Successful Learning in Mathematics (#CommissionsEarned) . “Their brains need more teaching, more targeted learning experiences, and more practice to develop these networks.”
Dyscalculia frequently cooccurs with dyslexia , a learning disability in reading; about half of children with dyscalculia also have dyslexia. 4 While figures vary, the estimated prevalence of dyscalculia in school populations is 3 to 6 percent. 5
[ Take the Dyscalculia Symptom Test for Children ] [ Think You Have Dyscalculia? Take This Screener for Dyscalculia in Adults ]
Dyscalculia Symptoms
What are the signs of dyscalculia? Symptoms and indicators include 6 7 :
 Connecting a number to the quantity it represents (the number 2 to two apples)
 Counting, backwards and forwards
 Comparing two amounts
 Trouble with subitizing (recognize quantities without counting)
 Trouble recalling basic math facts (like multiplication tables)
 Difficulty linking numbers and symbols to amounts
 Trouble with mental math and problemsolving
 Difficulty making sense of money and estimating quantities
 Difficulty with telling time on an analog clock
 Poor visual and spatial orientation
 Difficulty immediately sorting out direction (right from left)
 Troubles with recognizing patterns and sequencing numbers
Fingercounting is typically linked to dyscalculia, but it is not an indicator of the condition outright. Persistent fingercounting, especially for easy, frequently repeated calculations, may indicate a problem.
Calculating errors alone are also not indicative of dyscalculia – variety, persistence, and frequency are key in determining if dyscalculia is present.
[ Watch: Early Warning Signs of Dyscalculia ]
Dyscalculia Causes
When considering dyscalculia, most people are actually thinking of developmental dyscalculia – difficulties in acquiring and performing basic math skills. Exact causes for this type of dyscalculia are unknown, though research points to issues in brain development and genetics (as the disability tends to run in families) as possible causes. 8
Acquired dyscalculia, sometimes called acalculia, is the loss of skill in mathematical skills and concepts due to disturbances like brain injury and other cognitive impairments. 9
Dyscalculia Diagnosis
Dyscalculia appears under the “specific learning disorder” (SLD) section in the Diagnostic and Statistical Manual of Mental Disorders 5th Edition (DSM5). 10 For an SLD diagnosis, an individual must meet these four criteria:
 Individuals with dyscalculia exhibit at least one of six outlined symptoms related to difficulties with learning and using academic skills. Difficulties with mastering number sense and mathematical reasoning are included in the list.
 The affected academic skills are below what is expected for the individual’s age, which also cause trouble with school, work, or daily life.
 The learning difficulties began in school, even if problems only became acute in adulthood.
 Other conditions and factors are ruled out, including intellectual disabilities and neurological disorder, psychosocial adversity, and lack of instruction.
Individuals whose learning difficulties are mostly mathbased may be diagnosed with “SLD with impairment in mathematics,” an SLD subtype equivalent to dyscalculia.
Diagnostic evaluations for dyscalculia are typically carried out by school psychologists and neuropsychologists, though child psychiatrists and school health services and staff may play a role in evaluation. Adults who suspect they have dyscalculia may be referred to a neuropsychologist by their primary care provider.
There is no single test for dyscalculia. Clinicians evaluate for the disorder by reviewing academic records and performance in standardized tests, asking about family history, and learning more about how the patient’s difficulties manifest in school, work, and everyday life. They may also administer diagnostic assessments that test strengths and weaknesses in foundational mathematical skills. Tools like the PALII Diagnostic Assessment (DA), the KeyMath3 DA, and the WIATTIII are commonly used when evaluating for dyscalculia.
Dyscalculia Treatment and Accommodations
Like other learning disabilities, dyscalculia has no cure and cannot be treated with medication. By the time most individuals are diagnosed, they have a shaky math foundation. The goals of treatment, therefore, are to fill in as many gaps as possible and to develop coping mechanisms that can be used throughout life. This is typically done through special instruction, accommodations, and other interventions.
Under the Individuals with Disabilities Education Act ( IDEA ), students with dyscalculia are eligible for special services in the classroom. Dyscalculia accommodations in the classroom may include 11 :
 allowing more time on assignments and tests
 allowing the use of calculators
 adjusting the difficulty of the task
 separating complicated problems into smaller steps
 using posters to remind students to basic math concepts
 tutoring to target core, foundational skills
 computerbased interactive lessons
 handson projects
If left untreated, dyscalculia persists into adulthood, leaving many at a disadvantage when it comes to higher education and workplace success. 12 Adults with dyscalculia , however, may be entitled to reasonable accommodations in their workplace under the Americans with Disabilities Act ( ADA ). They can also commit to brushing up on math skills on their own or with the help of a trained educational psychologist. Even the most basic improvements in math skills can have longlasting impacts on dayto day life.
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Dyscalculia At a Glance
· Dyscalculia is present in about 11 percent of children with attention deficit hyperactivity disorder (ADHD or ADD).  
· Slow to develop counting and math problemsolving skills · Trouble understanding positive versus negative value · Difficult recalling number sequences · Difficulty computing problems · Problems with time concepts · Poor sense of direction · Difficulty completing mental math  
Evaluation should be conducted by a school psychologist or special education professional. School supports may be provided by special education professionals and/or your child’s classroom teacher.  
· There is no medication to treat learning disabilities · Your child may qualify for an IEP to receive specialeducation services including math supports  
· · · · · by Daniel Ansari, Ph.D. 
Dyscalculia: Next Steps
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View Article Sources
1 American Psychiatric Association. (2018, November). What is Specific Learning Disorder? https://www.psychiatry.org/patientsfamilies/specificlearningdisorder/whatisspecificlearningdisorder
2 Soares, N., & Patel, D. R. (2015). Dyscalculia. International Journal of Child and Adolescent Health. https://psycnet.apa.org/record/201529454003
3 DuPaul, G. J., Gormley, M. J., & Laracy, S. D. (2013). Comorbidity of LD and ADHD: implications of DSM5 for assessment and treatment. Journal of learning disabilities, 46(1), 43–51. https://doi.org/10.1177/0022219412464351
4 Morsanyi, K., van Bers, B., McCormack, T., & McGourty, J. (2018). The prevalence of specific learning disorder in mathematics and comorbidity with other developmental disorders in primary schoolage children. British journal of psychology (London, England : 1953), 109(4), 917–940. https://doi.org/10.1111/bjop.12322
5 Shalev, R.S., Auerbach, J., Manor, O. et al. Developmental dyscalculia: prevalence and prognosis. European Child & Adolescent Psychiatry 9, S58–S64 (2000). https://doi.org/10.1007/s007870070009
6 Haberstroh, S., & SchulteKörne, G. (2019). The Diagnosis and Treatment of Dyscalculia. Deutsches Arzteblatt international, 116(7), 107–114. https://doi.org/10.3238/arztebl.2019.0107
7 Bird, Ronit. (2017). The Dyscalculia Toolkit. Sage Publications.
8 Szűcs, D., Goswami, U. (2013). Developmental dyscalculia: Fresh perspectives. Trends in Neuroscience and Education, 2(2),3337. https://doi.org/10.1016/j.tine.2013.06.004
9 Ardila, A., & Rosselli, M. (2019). Cognitive Rehabilitation of Acquired Calculation Disturbances. Behavioural neurology, 2019, 3151092. https://doi.org/10.1155/2019/3151092
10 American Psychiatric Association (2014). Diagnostic and Statistical Manual of Mental Disorders. DSMV. Washington, DC: American Psychiatric Publishing
11 N, Soares., Evans, T., & Patel, D. R. (2018). Specific learning disability in mathematics: a comprehensive review. Translational pediatrics, 7(1), 48–62. https://doi.org/10.21037/tp.2017.08.03
12 Kaufmann, L., & von Aster, M. (2012). The diagnosis and management of dyscalculia. Deutsches Arzteblatt international, 109(45), 767–778. https://doi.org/10.3238/arztebl.2012.0767
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What is a specific mathematics disability?
By Sheldon H. Horowitz, EdD
Q. What is a specific mathematics disability?
A. You may hear the terms specific math disability , specific learning disability in math , or dyscalculia . These terms all refer to a type of disorder that significantly impacts a person’s ability to learn and perform in math.
There is no single profile of this disability. The signs of dyscalculia will vary from person to person. And they will affect people differently at different times in their lives.
Some people with dyscalculia have no trouble memorizing basic math facts. It’s performing calculations and solving problems that cause trouble. Others struggle with calculation and basic math operations like multiplication and division. But they can grasp the big concepts and easily understand how a problem can be solved.
Disabilities in math are often missed in the early years because kids are learning many basic skills through memorization. Young kids with dyslexia can often memorize their ABCs. But they might not understand the complex relationship between letters and sounds. Similarly, kids with dyscalculia may be able to memorize and recite their 123s. But they may not be building the number sense that is essential to future math learning.
IMAGES
COMMENTS
Math areas of deficit include math calculation and math problem solving. Math calculation is the knowledge and retrieval of facts and the application of procedural knowledge in calculation. Math problem solving involves using mathematical computation skills, language, reasoning, reading, and visualspatial skills in solving problems; and
Math problemsolving is a crucial skill that helps people understand and deal with the complexities of the world. It's about more than just doing calculations; it involves interpreting problems, creating strategies, and using logical thinking to find solutions. Many influential educators and mathematicians have established the foundations of ...
Reply Fri 4 Nov, 2011 12:04 pm. This is in line with the other answers given, but I would say "math computation" is the ability to do straight up math problems: addition, subtraction, multiplication, division, etc. Math reasoning is the ability to build a math equation given a problem. "If Mary is twice Sue's age and Sue is twelve...". 1 Reply.
They require, in the words of Jordan Ellenberg, prioritizing actual mathematics over calculation. Let me explain what I mean by this. For years we've been requiring math students to primarily calculate, that is, find a numerical solution, and do a small level of thinking. Common core flips this on its head and puts the emphasis on ...
Problem solving is not necessarily just about answering word problems in math. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in math is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.
Math calculation is the knowledge and retrieval of facts and the application of procedural knowledge in calculation. This is associated with deficits in number sense and operations, onetoone correspondence, and learning basic facts. Math problem solving involves using mathematical computation skills, language, reasoning, reading, and visual ...
either math calculation or math problem solving4. It also involves including the child in a scientific, research based intervention designed to address targeted area of concern (math calculation and/or problem solving), along with "repeated assessments of achievement at
The purpose of this study was to explore patterns of difficulty in 2 domains of mathematical cognition: computation and problem solving. Third graders (n = 924; 47.3% male) were representatively sampled from 89 classrooms; assessed on computation and problem solving; classified as having difficulty with computation, problem solving, both domains, or neither domain; and measured on 9 cognitive ...
The purpose of mental computation is to make the calculation mentally without using tools such as paper, pencil or a calculator (Reys, 1984 ). Mathematical reasoning that is higher level of thinking aims to reach a reasonable result by considering all aspects of a problem or case (Erdem, 2011 ). 1.1. Mental computation.
Here are five strategies to help students check their solutions. 1. Use the Inverse Operation. For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7.
Math Computation vs. Math Fluency. Children often rely on various strategies when solving simple calculation problems, but math computation requires caring out a sequence of steps on paper or in your mind (working memory) to arrive at an answer.
Children with 22q11.2 deletion syndrome have deficits in calculation, math word problem solving and numerical quantities despite intact math fact retrieval . Children with Turner syndrome have intact number comprehension and processing skills, but have slower and more errorprone functioning on complex math problems .
Calculation (described above) Calculation and Mathematical Operations. Calculation (described above) Math Reasoning and Problem Solving. Number Matrices tests quantitative reasoning, while also tapping an examinee's fluid reasoning capacity. Examinees are shown number matrices and asked to identify the missing number. Applied Problems ...
Children need to understand that multiplying is counting by groups of things, each group having the same quantity. Help them see how the tables are built one group at a time. Some solution strategies that help students understand the multiplication algorithm are: 1. Build an array to show the meaning of multiplication. (4 rows of 12) 4 x 12.
The Mathematics Subtests on the WIAT4 (WIATIV) are designed to measure a student's mathematics proficiency, as well as their fluency or how quickly and accurately they can solve simple math facts. They assess a student's math skills in a variety of areas, including basic computation, math fluency, math reasoning, and math problemsolving.
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Students with a math disability demonstrate developmental delay in learning the rules and procedures for solving calculations or word problems. An example of a math rule includes "any number × 0 = 0.". A procedure includes the steps for solving arithmetic problems such as addition, subtraction, multiplication, and division.
Dyscalculia is a math learning disability that impairs an individual's ability to learn numberrelated concepts, perform accurate math calculations, reason and problem solve, and perform other basic math skills. 1 Dyscalculia is sometimes called "number dyslexia" or "math dyslexia.". Dyscalculia is present in about 11 percent of ...
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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by highschool and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and ...
It's performing calculations and solving problems that cause trouble. Others struggle with calculation and basic math operations like multiplication and division. But they can grasp the big concepts and easily understand how a problem can be solved. Disabilities in math are often missed in the early years because kids are learning many basic ...
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