How to Teach Number to Young Children

This video is about Introducing a Number using Visual Cluster Cards and Cuisenaire Rods. We discussed in another video (The Parallels between learning Literacy and Numeracy), knowing the letters of the alphabet is a prerequisite to (a) learning sight words, (b) phonemic awareness, (c) word attack skills, and, ultimately, acquiring reading skills. Similarly, knowing one’s mathematics alphabet (The ten numerals/numbers: 1, 2, 3, 4, 5, 6, 7, 9, 9, and 10) is a prerequisite to number concept, numbersense, and ultimately, learning numeracy skills.

In this video, we introduce the concept of number 5. Same lesson plan, as used in this video, can be used for other numbers. A sketch (essential steps, sequence, questioning, and the principles of pedagogy) of this lesson is available (free of cost) by sending an email to: mahesh@mathematicsforall.org.

For the full treatment of teaching number concept watch the video: How to Teach Number Concept. Future videos will deal with numbersense and numeracy. In my opinion, almost every elementary teacher is a better reading teacher than a numeracy teacher. However, using the same principles, same care, and same teaching skills can deliver an effective numeracy lesson. The only requirement is to understand number concept mastery in the same way we expect them to know the letter concept.

As we expect every child to read fluently with comprehension by the end of third grade, we should expect and work diligently toward the goal of for every child to have mastery of numeracy with understanding by the end of fourth grade so that they can learn mathematics easily, effectively, and efficiently and appreciate its reach, power, and beauty. It is about number concept, numbersense and building the brain for numeracy and beyond. it is all about how we make it easier for students to understand it and communicate it. Through a systematic way of learning, practicing, and applying knowledge about the number concept, numbersense, numeracy, and mathematical way of thinking children have access to higher mathematics..

The videos on this site are to help achieve this goal. Join me in reaching this goal, not just in our country, but the whole world. Numeracy is the new literacy. Mahesh For a list of publications from the Center and my ideas about learning and teaching mathematics, understanding learning problems in mathematics, please visit my website http://www.mathematicsforall.org and my blog: http://www.mathlanguage.wordpress.com.

I am so thankful to my friend and colleague, Ann Lyle, for her expertise in photography. She is an accomplished, award winning, and professionally recognized photographer.

How to Teach Number to Young Children

Dyscalculia and Other Mathematics Learning Difficulties

How to Create a Dyscalculia Friendly Classroom

Mahesh C. Sharm

One-day Workshop

October 11, 2019

Professional Development Series

at

Several professional national and international groups, the National Mathematics Advisory Panel and the Institute for Educational Sciences, in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings—number conceptualization, arithmetic facts, place value, fractions, and integers make new learning overwhelming for many students and teachers to teach. 

Some students have difficulty in mathematics because of their learner characteristics—neuropsychological and cognitive profile, poor linguistic skills, lack of prerequisite skills for learning mathematics, learning difficulties/ disabilities, such as, dyscalculia, dyslexia, and/or dysgraphia. While some others may have difficulty learning mathematics due to factors contributed by socio-cultural environmental conditions. One of the outcomes of these factors is poor number concept, numbersense, and numeracy.

Clearly, there exists a need for instruction and interventions that go beyond “typical” classroom instruction. These interventions should be effective, efficient, and elegant (that can be generalized and extrapolated). They must be based on sound principles of learning mathematics, reflecting the characteristics of the difficulty and focused on the practices that deliver outcomes envisioned. They cannot be based on just modifying the content by diluting the standards. The goal is not just being proficient in applying numeracy skills in routine situations (that can be delivered by mindless use of technology). 

Being proficient at arithmetic/numeracy skills is certainly a great asset (a necessary, but not a sufficient condition) when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for mastery, that Common Core State Standards in Mathematics (CCSS-M)— set for arithmetic for early elementary grades are specific.  To have mastery in a particular concept or procedure students should have its: 

(a) understanding (e.g., have appropriate language and possess efficient and effective strategies, based on authentic conceptual schemas), 

(b) fluency (at acceptable standards), and 

(c) applicability (can apply to other concepts, procedures, and in problem solving). 

Such a level of mastery ensures that students form strong, secure, and developmentally appropriate numeracy foundations for learning algebraic concepts and procedures so that these students can learn easier and go higher.  The development of those foundations is assured if we implement the Standards of Mathematics Practices (SMP) along with the CCSS-M content standards. 

As we expect every child to read fluently with comprehension by the end of third grade (approximately age 9), we should expect and work for every child to have mastery of numeracy by the end of fourth grade (approximately age 10) so that they can learn mathematics easily, effectively, and efficiently.  A fluent reader with comprehension can apply his/her reading skills to reading content in any discipline and in any context. Similarly, a student with fluent numeracy skills (language, understanding, fluency, and applicability) should be able to apply those skills for learning algebra and model problems in intra-mathematical, interdisciplinary, and extracurricular settings. 

A nation-wide discussion, verging on a political fight, is going on right now, concerning pro and con of the Common Core State Standards in Math (CCSS-M) and it involves every school in the country, whether they have adopted CCSS-M or not. This discussion has international and long-lasting implications for mathematics education.  Even those who have not adopted them, have given considerable thought to these standards. As we see it, the implementation of CCSS-M has considerably upped the ante in mastery, rigor, coherence, and developmental trajectories of mathematics ideas.  But, they have also opened the option of introducing ‘modeling,’ of realistic problems in the curriculum and teaching. That means the mathematics we teach should relate to other disciplines (i.e., STEM) and the real world (i.e., careers and professions). As a result, all school mathematics can be enlivened and made relevant, even exciting, to students by dipping into the vast array of applications (e.g., intra-mathematical, interdisciplinary, and extra-curricular) that mathematics has to real life. Our message, therefore, is: math, properly taught, need not turn away our students from “good” mathematics.

This year, in this series of workshops on mathematics education, we will cover content and strategies related to key developmental milestones in mathematics—topics that form the backbone of school curriculum.  The objective is to help you learn about the content and instructional practices for teaching mathematics to all students effectively, including, those who struggle with the critical concepts and skills necessary for mastering numeracy and success in algebra.

This session will provide specific strategies and recommendations for content, instruction, intervention, remediation and draw upon currently available research-based evidence for teaching mathematics.  For more information, one can go to the many posts on this blog relevant to this topic.

The topics, in this workshop, deal with understanding the issue of learning problems in mathematics, including dyscalculia and mathematics difficulties due to dyslexia, dysgraphia, and other language related difficulties from Kindergarten through high school. 

In the field of dyscalculia, we have arrived at a place, where we need to move on from just focusing on the characteristics and profiles of students with dyscalculia to         

  • what are the reasons for the incidence and conditions of dyscalculia, and
  • what to do to help students who show such characteristics. 

In other words, on one hand, we should focus on understanding the nature of mathematics instruction that responds to these students’ learning needs. On the other hand, we need to focus on providing the classroom instruction so that the emergence of conditions due to dyscalculia are reduced significantly (i.e., preventive teaching). We also need to create, the classroom instruction that is dyscalculia friendly environment that minimizes the impact of conditions that may exacerbate the risk factors for dyscalculia. 

In most research on dyscalculia suggestions given for instruction are too fragmented and compartmentalized. We need to present a conceptual framework that can help mathematics education professionals—classroom teachers, special educators, and interventionists better understand what their students are facing as they learn mathematics systems: language, concepts, procedures, and skills. In this area, we can learn a great deal from the progress made in the science of reading that provides suggestions for instruction and remediation for dyslexic students. 

Several national and international panels have recommended instructional components for improving mathematics outcomes but presented these instructional components as a list without explicitly addressing their interrelations with learning needs, either in terms of instruction or cognitive development. We need to explore the key cognitive capacities underlying learning and conceptualizing specific mathematics ideas that specify the relationships between them. The central objective of this presentation is to provide help to classroom teachers and intervention specialists achieve better outcomes. In other words, what cognitive capacities undergird learning mathematics skills, particularly, that provide the basis for developing numeracy skills in all children. The suggestions, here, are intended to help teach the content and also improve the underlying cognitive capacities, such as executive functions: working memory, inhibition control, organization, and flexibility of thought. 

“Whether you think you can, or you think you can’t, you’re right.” 

— Henry Ford

A.  Introduction

I have been working with people who demonstrate difficulty learning mathematics or are gifted and talented in mathematics. In this work, I meet people who have direct professional interest in the topic and others who are indirectly affected by it. People range from children age 3 to high school students to doctors and university professors and school administrators. These people raise many and diverse questions and issues to tackle problems related to mathematics learning, teaching, assessing, and organizing instruction, both classroom instruction and interventions. 

Many just want to know what is “dyscalculia?” Some of them are looking for specific definition of dyscalculia, its symptoms, its causes, and information about protocols for diagnosis and treatment of mathematics learning difficulties, particularly dyscalculia. School administrators seek procedural advice about—what are their responsibilities, what programs and resources can help students with difficulties in mathematics. Parents search for advice on school issues related to math learning, testing, enrichment, and remediation of these issues—meeting the needs of their children. 

Students, on the other hand, seek survival skills, relief from troubling math failures, and concessions and accommodations from instructors and institutions. Some of them want to learn mathematics that excites them and challenges their abilities. Many adults, who achieve success in other areas of their lives, wonder why they need to seek remedial and coping strategies to overcome baffling and frustrating conditions in learning and applying mathematics. Almost all dyscalculics seek vindication of their intelligence, and illumination and understanding of this disability.

Many students have difficulty in learning mathematics for a variety of reasons. Individuals having difficulties in learning mathematics manifest the symptoms in varying degrees and forms. One of these forms is known as dyscalculia. Not all students having difficulty in learning mathematics have dyscalculia. However, there are some basic areas of mathematical activity in everyday life that may indicate a dyscalculic tendency. That is if the mathematical activities are persistently difficult and frustrating for the person. Such symptoms may manifest as: mathematics anxiety and dyscalculia. The observations and research have shown that dyscalculic individuals are troubled by even the simplest numerical tasks such as selecting the larger of two numbers or estimating the number of objects in a display, without counting.

Dyscalculia is a lessor-known of learning disabilities that affects learners. Dyscalculia is a specific learning difficulty in mathematics. Dyscalculia is the name given to the condition that affects our ability to acquire arithmetical/ numeracy skills.  People who are anxious and afraid about all things mathematical and have difficulty learning it have many other symptoms and characteristics. Most people take these coexisting conditions as “the dyscalculia syndrome.” 

B.  History

Schools have long experience of supporting children who experience difficulties with mathematics, but dyscalculia has only recently been identified as a distinct condition for children and adults.  It is a fairly new term to many people.  It also means that there are many adults and children who have never had their difficulties with mathematics formally identified.  Furthermore, while there is currently a great deal of interest in dyscalculia in educational circles, yet there is limited body of research in this area.  To date, research on math disability (MD) is far less extensive than research on reading disability (RD). Yet, like RD, MD is a significant obstacle to academic achievement for many children. There is a need to better understand its nature, its causes, and its manifestations. 

While provision is made to accommodate the needs of pupils with learning problems into the school curriculum, assessment is very largely based on reading difficulties and many times the diagnosis of mathematical problems are overlooked. This is perhaps not surprising in view of the relative scarcity of information about mathematical learning problems. Although, it is becoming a focus of education, most neuro-psychological, and neurological research is concerned with understanding the basic mental processes and their role in mathematical cognition. 

Other research examines the impact of traumatic brain injury on adults and children’s loss of mathematical abilities, although. The resultant mathematical difficulties is called acalculia. Many more people have difficulty in learning mathematics than due to mathematics learning problems or any disabilities. Many of their difficulties are not due to the conditions of learning disabilities. But, many of them assume the presence of due to some learning disability. However, they exhibit the same kinds of symptoms as dyscalculia.  We term them as learning mathematics problems due to environmental factors and call them as acquired dyscalculia. Acquired dyscalculia becomes evident when a student, otherwise able and without learning disability, because of environmental factors—poor standards, poor teaching, lack of practice, frequent and excessive absences from school, etc., shows similar symptoms as dyscalculia. 

While these research directions are increasing knowledge of the development of basic arithmetical skills (counting, addition, subtraction, multiplication and division) and of their epistemological relationships, little research has as yet explore the development of effective and efficient strategies for instruction, intervention, and remediation of fundamental skills underlying the difficulty in learning number concept, numbersense, numeracy and the development of more sophisticated domains such as algebra and geometry. The process of understanding mathematical learning problems is still in its infancy. 

C.  Learning Problems and Mathematics Learning difficulties

In general, the factors responsible for mathematics learning problems fall in the following categories: 

  • Cognitive/Neurological
  • Intellectual
  • Perceptual
  • Language related
  • Pre-requisite skills related

Nature of Mathematics Learning Problems 

The learning problems in mathematics can be popularly categorized as: 

  1. Developmental mathematics learning problems
  2. Carryover mathematical learning problems
  3. Math anxiety

~ Specific math anxiety 

~ Global math anxiety

4. Acalculia, dyscalculia, anarithmetia, dysgraphia

5. Dyslexia and other language related mathematics difficulties

6. Acquired Dyscalculia 

Developmental mathematics learning problems are those where the learner’s preparation for mathematics is not adequate for some developmental cognitive factors.  They have difficulty in acquiring the key developmental ‘milestones’ in mathematics learning—number concept, place-value, fractions, integers, idea of variability, and spatial sense. These problems may manifested in cognitive delay, neurological deficits, lack of preparation in pre-requisite skills for mathematics learning, or lack of experience related to number, quantity and space. The locus of developmental learning difficulties is in the learner. It may either be inherited from a parent – genetic, or the result of a combination of both parents’ genes – congenital and poor developmental patterns.

Carryover problems, on the other hand, are those where the person has difficulty in areas other than mathematics but the difficulty may interfere learning and functioning in mathematics.  The difficulties may relate to language (conceptualization word problems, communication, etc.), psychomotoric problems (handwriting, spatial orientation), and emotional problems (anxiety, fear of failure, etc.). 

Environmental learning difficulties in mathematics (acquired dyscalculia) may be the result of unsatisfactory teaching of basic concepts or of negative social influences on a pupil’s learning. 

1.  Definition of Dyscalculia

Dyscalculia is a term used to describe mathematical learning difficulties. As we know more about how children learn mathematics, why learning problems occur, and how to teach them, we know more about the nature of dyscalculia. In general, dyscalculia or acquired dyscalculia means having: intellectual functioning that falls within or above the normal range and a significant discrepancy between his/her age and mathematics skills (usually two years or more). 

To be diagnosed with dyscalculia, it is important to make sure that mathematics deficits are not related to issues like inadequate instruction, cultural differences, mental retardation, physical illness, or problems with vision and hearing. It is not as commonly diagnosed as dyslexia in schools because of the lack of any strict or measurable criteria.  At present, the diagnosis is by neuro-psycholgists, neurologists, or specialists in dyscalculia.

The definition of dyscalculia is, thus, evolving. In 1968, Dr. Ladislav Kosc, a pioneer in the study of mathematical learning difficulties, defined dyscalculia as follows:

Developmental dyscalculia is a structural disorder of mathematical abilities which has its origin in those parts of the brain that are anatomico-physiological substrate responsible for the maturation of mathematical abilities adequate to age without, however, having as a consequence a disorder of general mental functions. The origin may be either genetic or acquired in prenatal development.  (Kosc, 1986, p. 48-49)

While this definition describes the possible causes of developmental dyscalculia, the destructive impact of poor environment (e.g., acquired dyscalculia) should not be overlooked. We need to also focus on the nurturing role played in this by appropriate education—efficient strategies, effective models and timely interventions. Whatever the cause, the effects will fit into a spectrum of problems detrimental to a child’s schooling.

There are rigorous criteria used to determine if a student has a learning disability based on and guided by special education criteria. When a student’s mathematics difficulties are severe enough to meet that criteria, special education services are indicated. However, dyscalculia has no clearly defined criteria and cannot be assessed reliably, at present. By some educational specialists, a student with any degree of mathematics difficulty may be considered to have dyscalculia.  Because of the ambiguity of categorization, being identified as having dyscalculia may or may not indicate whether special education services are warranted. Nevertheless, substantial number students suffer either from dyscalculia or acquired dyscalculia.

The term learning disabilities is often misused and sometime applied, incorrectly, to students who learn mathematics in different ways or have difficulty learning mathematics. Learning disabilities in mathematics is a generic term that refers to a heterogeneous group of disorders manifested by significant difficulties in learning, acquisition and use of mathematics—reading mathematics text, writing mathematics expressions, reasoning about concepts and procedures, mathematical thinking—seeing patterns and relationships between ideas and concepts, and mastering key developmental concepts in mathematics. Mathematics is not a unary concept or skill; it is complex with multiplicity of concepts, procedures, and branches. Therefore, learning difficulties, learning problems, or learning disabilities span a spectrum. 

Many of the disorders related to mathematics may be intrinsic to the individual and presumed to be due to central nervous system dysfunction. A mathematics learning disability may occur concomitantly with other handicapping conditions such as sensory impairment, mental retardation, social and emotional disturbance. It may occur along with socio-environmental influences such as cultural differences, insufficient or inappropriate instruction, or psychogenic factors, or with attention deficit disorder, all of which may cause learning problems, but a learning disability is not the direct result of those conditions or influences. Dyscalculia, as a mathematics disability, may result from neurological dysfunction and can be as complex and damaging as a reading disability, which tends to be more routinely diagnosed.        

Adults with dyscalculia experience various debilitating problems in handling daily quantitative functions. The difficulty is manifested in conceptual understanding, counting sequences (skip counting forward and backward by 1, 2, 5, 10s), written number symbol systems, the language of math, basic number facts, procedural steps of computation, application of arithmetic skills, and problem solving. Mathematics learning disabilities, because of the complexity and diversity of concepts and procedures, do not often occur with clarity and simplicity. Rather they can be combinations of difficulties, which may include language processing problems, visual spatial confusion, memory and sequence difficulties, and or unusually high anxiety.

Dyscalculia is an individual’s difficulty in conceptualizing number concept, number relationships, numbersense (intuitive grasp of numbers) and outcomes of numerical operations. In this sense, dyscalculia only refers to issues with learning numeracy skills. Dyscalculic children may have difficulty in mastering arithmetic facts, concepts, and procedures (addition, subtraction, multiplication, and division) by the usual methods of teaching arithmetic, particularly, those that are based on counting strategies.  

Dyscalculia affects an individual’s ability to estimate – what to expect as an outcome of a numerical operation and the range of answers. This difficulty manifests in a person having difficulty with estimating time, distance, and money transactions—balancing a checkbook, making change, and tipping. In other words, wherever quantity—number and calculations are involved in daily-to-day living. 

Although it may be co-morbid with other difficulties, dyscalculia relates specifically to problems of mathematical language, concepts and procedure. Pupils are assessed as dyscalculic if their mathematical ability is significantly below their overall cognitive profile as determined by tests such as, the Wechsler Intelligence Scales or similar other cognitive assessments. If cognitive abilities in general are significantly below average the child is likely to be considered as having multiple, rather than specific, learning difficulties and is not dyscalculic. 

2.  Types of Dyscalculia

Dyscalculia can be broken down into four sub-types:

  • Quantitative dyscalculia, a deficit in the skills related to numeracy (e.g., computational skills—counting and calculating). 
  • Qualitative dyscalculia, a result of difficulties in comprehension of instructions or the failure to master the skills, symbols, and concepts (e.g., which is a difficulty in the conceptualizing of math processes) required for an operation. When a child has not mastered the number facts, he cannot benefit from this stored “verbalizable information about numbers” that is used with prior associations to solve problems involving addition, subtraction, multiplication, division, and square roots. 
  • Mixed dyscalculia involves the inability to operate with symbols, shapes, and numbers.
  • Acalculia is difficulty in learning mathematics after an insult or injury to the brain.  The person had intact mathematics skills, but has lost some or many of them after the injury.  The specific nature of difficulty depends on the focal area and the extent of the injury. Acalculics show the similar symptoms as the dyscalculics.

Mathematical calculations are a complex system of skills, concepts, and processes. The understanding, acquisition, and competence depends on the interaction of many abilities and cognitive mechanisms. Dyscalculic pupils are unable to use these (integrating basic mathematics skills and cognitive skills) efficiently and effectively to arrive at the solution of a problem. Many mechanisms – such as those for sequencing and organizing information – are also shared with other non-mathematical processes; consequently developmental dyscalculia frequently accompanies other learning difficulties arising from poor executive functions

  • Memory problems (e.g., short-term–reception, working–manipulation, and long-term memory—retention); 
  • Inhibition control (difficulty maintaining concentration, and focusing on the appropriate concept, procedure, or skill), 
  • Organization (visual-spatial confusion, lack of organization in physical space, working equipment, or ideas, skills, and working scripts, etc.),
  • Flexibility of thought (rigidity in using only a limited strategy—for example, using counting as the only means to derive facts) 
  • Information processing difficulties (e.g., the cumulative nature of mathematics calls for heavy demands on processing information), and,
  • Motor disabilities–dysgraphia (graphomotoric and pscho-motoric issues—poor drawings, writings, lack of clarity in executing procedures).

The natural anxiety of a person may also affect one’s attitude about mathematics and the resultant math anxiety from mathematics failures, in turn, affects further learning in mathematics and further complicates the picture.           

Many people might relate mathematics mostly to numeracy and arithmetic; which is just a small part of a range of widely different concepts constituting mathematical knowledge. There is little in common between rote-learnt multiplication tables, the perspective geometry, coordinate geometry, probability, statistics, or calculus, for example, but they are all aspects of mathematics. Learning difficulties in these areas might be expected to manifest themselves in very different ways. Dyscalculia only affects numeracy skills and the ability to apply numerical competence in other areas of mathematics and other scientific disciplines. 

Dyscalculia may affect just a few skills supporting one, or several, of these branches. For example, successful arithmetic competence requires a sound conceptual grasp of the concept and properties of numbers, their relationships underpinning computations, the decimal system, and arithmetical procedures. These are concepts that are taught to the child. They themselves are built on to other concepts that the child has been taught. Many of the cognitive processes involved in mathematical thought may also serve other non-mathematical purposes. The strengths and weaknesses of ability in each of these areas reflect our developmental history; no one will have the same combination of abilities as anybody else. Mathematics learning is woven into the fabric of the individual and then as related to the complex of mathematics system—language, concepts, procedures, and skills, in different aspects of mathematics. Consequently, mathematics difficulty can be defined with precision. Dyscalculia cannot be defined as a specific difficulty with a clearly identifiable cause or effect, if we do not limit it to the disorder of number concept, numbersense, and numeracy skills.

There are many challenges facing students with mathematical learning difficulties. They may arise from many causes, take many forms and be accompanied by other difficulties which also require intervention and remediation. For example reading difficulties may mask or accentuate accompanying mathematical difficulties; consequently intervention must address both areas to be most effective.

D.  Underlying Causes of Dyscalculia

In our technological society mathematical ability is a valuable asset. From many perspectives, numerical skills are considered to be more important than reading abilities as a factor determining employability and wage levels and possible professional fields.  The interrelationships between mathematical and other learning difficulties lead some authorities to wonder whether dyscalculia is not in fact a symptom of other difficulties, such as dyslexia and dyspraxia. This could have implications for assessment and intervention.

To understand dyscalculia better we need to look at a possible model of the mental mechanisms and processes involved in early mathematical learning, particularly, the number concept, numbersense, and numeracy as the competence in these areas is the basis of dyscalculia. 

Dyscalculia has several underlying causes. One of the most prominent is a weakness in visual processing and visualization. To be successful in mathematics, one needs to be able to visualize numbers and mathematics models and situations. Students with dyscalculia have a difficult time visualizing numberness and estimation and often mentally mix up the numbers, resulting in errors and misconceptions. 

Another problem is with sequencing. Students who have difficulty with sequencing or in organizing detailed information often have difficulty remembering specific facts and formulas for completing their mathematical calculations, particularly, procedural calculations. 

Like dyslexia, dyscalculia can be caused by a visual perceptual deficit. Along with dyslexia, the extent to which one can be affected varies uniquely with the individual.  Like dyslexia there is no single set of signs that characterize all dyslexics, there is no one cause of dyscalculia.  However, dyscalculia refers specifically to the inability to perform operations in mathematics or arithmetic.

1.  What is the incidence of dyscalculia?

There may be more students with and without learning disabilities in any mathematics class, who have problems or difficulty in learning mathematics than we realize. If a class has learners who read numbers backwards, have trouble telling time, confuse part–whole relationships, have difficulty keeping score in a game, and have difficulty remembering arithmetic facts, ideas behind key concepts, strategies/rules in basic operations and formulas, and sequence of steps in key arithmetic procedures, they may be learning disabled. 

Everyone forgets occasionally, but when learning every concept is difficult and the student consistently forgets it, it is a symptom of disability and calls for intervention. According to the National Adult Literacy and Learning Disabilities Center, “it is estimated that 50 percent to 80 percent of students in Adult Basic Education and literacy programs are affected by these learning disabilities,” (1995, p. 1). Some of these have dyscalculia. However, many of them have acquired dyscalculia. The implications of such a staggering statistic for the adult basic education (ABE) teacher are worth further investigation. However, those with specific learning difficulty, dyscalculia, even in this group are much smaller. In school age children it is much smaller. Chinn and Ashcroft (1997) report that from a sample of 1200 children only 18(1.5%) had purely mathematics specific learning difficulties. Many of them, however, will develop acquired dyscalculia, if effective, efficient strategies are not taught and emphasized in regular mathematics instruction.

Since people are just becoming aware of this condition, it is hard to quantify exactly how many people have dyscalculia. Although many people experience difficulty or disability in mathematics, some of the recent studies show that dyscalculia –difficulty with numbers and number operations—afflicts between 5% and 6% of the population, based on the proportion of children who have special difficulty with mathematics despite good performance in other subjects. After a long period of growing awareness it is now widely accepted that dyslexia affects a significant proportion of the population and provisions are made to facilitate their situation as far as possible. Awareness of dyscalculia, however, is far lower although it may be at least as common and as far reaching as dyslexia, longitudinal studies in Europe, Israel and the USA suggest the same—that is about 5-6% of the population are affected by some degree of dyscalculia. However, the proportion of the population with purely mathematical difficulties may be far lower. 

In very simple terms, analogous to dyslexia—where the dysfunction manifests in difficulties in reception, comprehension, or production of linguistic information, dyscalculia can be defined as the dysfunction in the reception, comprehension, or production of quantitative and spatial information.  However, dyslexia may also affect learning mathematics.  Dyslexics frequently have difficulties with certain areas of mathematics; according to the British Dyslexia Association (1982) approximately 40 to 60% of dyslexics experience some mathematical difficulty.

2.  How does dyscalculia develop?

Schools have supported children who experience difficulties with mathematics, but dyscalculia has only recently been identified as a distinct condition for children and adults.  It means that there are many adults and children who have never had their difficulties with mathematics formally identified. 

Our work with children and adults with learning problems in mathematics suggests that there seem to be several factors that may be implicated as the causes of mathematics learning problems: 

  • Cognitive factors,
  • Inadequate and poor teaching—mismatch between mathematics learning personality of a student and teaching style,
  • Lack of pre-requisite skills for mathematics learning,
  • Delay in the development of mathematics language—vocabulary, syntax, and translation ability from mathematics to English and English to mathematics,
  • Inadequate mastery at levels of knowing: movement from intuitive to concrete, concrete to representational, representational to abstract, abstract to applications, and from applications to communication. 

In most cases of dyscalculics, it seems, the pre-requisite skills for mathematics learning are affected.  These prerequisite skills include: following sequential directions, spatial orientation/space organization, pattern recognition, visualization, estimation, inductive and deductive thinking. These prerequisite skills act as “anchors” for mathematics ideas. The degree to which these prerequisite skills are not developed or affected varies from learner to learner. 

3.  Is a dyslexic individual likely to be dyscalculic?

There is some correlational evidence between the co-incidence of dyslexia and dyscalculia.  But a clear link between dyslexia and dyscalculia hasn’t been proved.  The International Dyslexia Association has suggested that 60% of dyslexics have some difficulty with numbers or number relationships.  Of the 40% of dyslexics who don’t have mathematics difficulties, about 11% excelled in mathematics.  The remaining 29% have the same mathematical abilities as those who don’t have learning difficulties. Many dyslexia specialists believe that for many dyslexic people the difficulties, which affect their reading, and spelling also, cause problems with mathematics. 

Since some of the same pre-requisite skills are involved in both language acquisition and mathematics – at least in the early learning concepts and grade levels – the coincidence of dyslexia and dyscalculia is not uncommon. Our observations show that about 40% of dyslexics also exhibit some symptoms of dyscalculia.  However, the group of dyscalculic children/adults, like the group of dyslexics, is not a homogeneous one. Most people with dyscalculia don’t necessarily suffer from any other learning difficulty.  Indeed, they may well excel in non-mathematical areas.

4.  Is dyscalculia widely understood?

All mathematics teachers have encountered children with mathematics learning difficulties and varying degrees of mathematics anxiety.  Most of these teachers have some awareness of the nature of learning disabilities/problems in mathematics. However, few teachers are aware of the causes of these problems—learning disabilities, mathematics anxiety, and dyscalculia. In fact, very few of them are able to recognize and deal with the problems of dyscalculics. 

American Academies of Neurology and Pediatrics have identified dyscalculia as one of the neurological conditions with a cluster of syndromes associated with it.  Similarly, in 2001, as part of the national Numeracy Strategy in the UK, the government published guidance for teachers to provide classroom help to support dyscalculic pupils.  Dyscalculia is likely to be a more familiar condition to people who specialize in learning difficulties such as special needs coordinators and educational psychologists.  In the U.S., many school psychologists, neurologists and neuro-psychologists have begun to diagnose this as a condition. In spite of this, the general public and teachers have limited understanding of the condition of dyscalculia. Early diagnosis of the problems, effective planning of intervention, and effective and efficient remediation support can reduce the number of struggling students in mathematics when this information becomes available to more mathematics and classroom teachers.  

Many students with disabilities have histories of academic failure that contribute to the development of learned helplessness in mathematics. It is important that mathematics instructors recognize the symptoms of dyscalculia and take the necessary measures to help students that are affected. 

E.  Mathematics Symptoms of Dyscalculia

Symptoms of dyscalculia and other mathematics difficulties are manifested in several ways: 

  1. Linguistic
  2. Cognitive/content/conceptual
  3. Procedural
  4. Behavioral (Math Anxiety—global and specific)

Many of dyscalculics students, even when they can produce a correct answer or use a correct method, they may do so mechanically, without conceptual understanding and confidence and using inefficient methods and strategies. Some of the manifested symptoms of dyscalculia are: 

Dyscalculia is a collection of symptoms of learning disability involving the most basic aspect of arithmetical (quantitative and spatial) skills.  On the surface, these relate to basic concepts such as: number concept number facts, estimation, telling time, calculating prices and handling change, and measuring things such as temperature and speed.

Dyscalculia is an individual’s difficulty in conceptualizing numbers, number relationships, outcomes of numerical operations and estimation – what to expect as an outcome of an operation. Math problems begin from number concept.  Math disabilities, therefore, can arise at nearly any stage of a child’s scholastic development. While very little is known about the neurobiological or environmental causes of these problems, many experts attribute them to deficits in one or more of five different skill types. These deficits can exist independently of one another or can occur in combination. All can impact a child’s ability to progress in mathematics. 

(a)  Incomplete Mastery of Numberness and Number Facts

When a student above the age of 8 has to count the dots on a domino or a playing card, this shows that the student has not conceptualized number and may be a prime candidate for the identification of dyscalculia or acquired dyscalculia.  A surprising number of people resort to counting to work out the simplest of quantitative tasks.  

Children who have not mastered numberness are at risk for learning number facts. Numberness is the integration of one-to-one correspondence, visual clustering, sequencing, and decomposition/recomposition of quantity. Number facts are the basic computations (9 + 3 = 12 or 2 × 4 = 8) students are required to master (as defined earlier) in the earliest grades of elementary school. Recalling these facts efficiently is critical because it allows a student to approach more advanced mathematical thinking without being bogged down by simple calculations. 

(b)  Computational Weakness

Many students, despite a good understanding of mathematical concepts, are inconsistent at computing. They make errors because they misread signs or carry numbers incorrectly, or may not write numerals clearly enough or in the correct column. These students often struggle, especially in primary school, where basic computation and “right answers” are stressed. Often they end up in remedial classes, even though they might have a high level of potential for higher-level mathematical thinking. In general, they show

  • Poor mental math computation ability, often fear of and difficulty in common usage of number such as in money transactions—balancing a checkbook, making change, and tipping. 
  • Difficulty with math processes (e.g., addition, subtraction, multiplication) and concepts (e.g., sequencing of numbers). 
  • Difficulty with estimation with and without calculations. 
  • Difficulty with rapid processing of math facts. 
  • Have difficulty using a calculator properly because of absence of concepts and estimation.
  • Difficulty keeping score during games, or difficulty remembering how to keep score in games, like bowling, etc. 

(c)  Difficulty Transferring Knowledge

One fairly common difficulty experienced by people with math problems is the inability to easily connect the abstract or conceptual aspects of math with reality. Understanding what symbols represent in the physical world is important to how well and how easily a child will remember a concept. Holding and inspecting an equilateral triangle, for example, will be much more meaningful to a child than simply being told that the triangle is equilateral because it has three sides with equal length. And yet children with this problem find connections such as these painstaking at best. 

(d)  Recognizing Patterns and Making Connections

Some students have difficulty seeing and extending patterns, making meaningful connections within and across mathematical experiences. For instance, a student may not readily comprehend the relation between numbers and the quantities they represent. For example, realizing that adding two same numbers is same as knowing table of 2. Knowing commutative property of addition/multiplication means, you have to memorize only half as many facts.  If this kind of connection is not made, math skills may be not anchored in any meaningful or relevant manner. This makes them harder to recall and apply in new situations. 

(e)  Incomplete Understanding of the Language of Math

Mathematics is a second language for most children.  It has its own vocabulary, syntax and rules of translation. The vocabulary, syntax and translation from English to math and math to English may impact mathematics learning. 

For someone who has trouble distinguishing letters, a + sign might be confused with ×, for example.  The language of mathematics can also be a problem, they latch onto the first meaning they know for words, for example, distributive property means, “distributing.” 

Therefore, for some students, their math disability is driven by problems with language. These students may also experience difficulty with reading, writing, and speaking. In math, however, their language problem is confounded by the inherently difficult terminology, some of which they hear nowhere outside of the math classroom. These students have difficulty understanding written or verbal directions or explanations, and find word problems especially difficult to translate. 

A student with language problems in math may: 

  • Have difficulty with the vocabulary of math,
  • Be confused by language in word problems and applications,
  • Not know when irrelevant information is included or when information is given out of sequence,
  • Have trouble learning or recalling abstract terms,
  • Have difficulty understanding directions,
  • Have difficulty explaining and communicating about math, including asking and answering questions,
  • Have difficulty reading texts to direct their own learning,
  • Have difficulty remembering assigned values or definitions in specific problems, 
  • It is common for students with dyscalculia to have normal or accelerated language acquisition: verbal, reading, writing, and good visual memory for the printed word, however, some have difficulty with math language, and
  • Mistaken recollection of conceptual names, terms, definitions, and expressions. 

(f)  Perceptual Difficulties 

It deals with difficulty comprehending the visual and spatial aspects of mathematics.  A far less common problem—and probably the most severe—is the inability to effectively visualize math concepts. Students who have this problem may be unable to judge the relative sizes among three dissimilar objects. This disorder has obvious disadvantages, as it requires that a student rely almost entirely on rote memorization of verbal or written descriptions of math concepts that most people take for granted. Some mathematical problems also require students to combine higher-order cognition with perceptual skills, for instance, to determine what shape will result when a complex 3-D figure is rotated. 

(g)  Memory Difficulties

The memory shortcomings associated with dyslexia obviously causes problems with mental arithmetic. Even if children learn their times tables they cannot sequence backwards and forwards – so if you ask them what six times four is, they will have to start again counting through. That does not provide fluency.

Because of the mathematics curriculum and programs and their pace: If a child slips behind it became more and more difficult to catch up. Efficient strategies can make it happen, but for learning strategies, to some extent, depends on memory. The size of working memory and its effective use id critical for mathematic learning. 

(h) Difficulties due to Spatial Orientation/Space Organization

Spatial orientation/space organization is highly correlated with mathematics achievement.  When students have poor spatial skills, they have difficulty in multiple areas of mathematics. For example, 

  • Poor sense of direction (therefore, problems with place-value, aligning numbers, reversals, confusing different forms of numbers because of their spatial locations, etc.), easily disoriented, as well as trouble reading maps, telling time, and grappling with mechanical processes. 
  • Have trouble with sequence, including left/right orientation. They will read numbers out of sequence and sometimes do operations backwards. They also become confused on the sequences of past or future events.  It greatly affects learning and mastering order of operations (GEMDAS—grouping, exponents, etc.) 
  • Poor memory for the “layout” and structure of problems and organization of their work, and  geometrical designs and figures.  Gets lost or disoriented easily. 
  • May have poor eye-hand coordination, difficulty in writing mathematics expressions and problems. 

 (i)  Executive Function related:

  • There is sometimes poor retention and retrieval of concepts, or an inability to maintain a consistency in grasping mathematics computation rules. 
  • Difficulty with abstract concepts of time and direction, schedules, keeping track of time, and the sequence of past and future events.
  • Inability to grasp and remember mathematics concepts, rules formulas, sequence (order of operations), and basic addition, subtraction, multiplication and division facts. Poor long-term memory (retention and retrieval) of concept mastery. Students understand material as they are being shown it, but when they must retrieve the information, they become confused and are unable to do so. They may be able to perform mathematics operations one day, but draw a blank the next. May be able to do classroom and homework and book work but can fail all tests and quizzes.

F. Dealing with dyscalculia: What forms of instruction are most effective? 

Dyscalculia is a special need, and requires diagnosis, support and special methods of teaching.  The support should give the learners an understanding of their condition, and equip them with coping and efficient learning strategies that they can use in the classroom and in their day-to-day encounters with quantity and space. Since this is a heterogeneous group no general or single intervention can be recommended. 

Dyscalculic learners lack an intuitive grasp of numbers, recognizing umber relationships, and have problems learning number facts and procedures by the usual methods of teaching.  Most of their arithmetic facts are derived by counting and procedurally.  Even when these learners produce a correct answer or use a correct procedural method, they may do so mechanically and without confidence; they are anxious about it. 

Therefore, first objective of remedial instruction and intervention is helping them acquire efficient strategies for developing number concept. That success will improve learners’ self-esteem. The same process should be repeated for all of the developmental milestones.  The developmental milestone concepts for learning mathematics are: understanding number concept, number relationships (arithmetic facts)place value (of large and small numbers)fractions (including fractions, percents, ratio and proportions)integersspatial sense and the concept of variability.  Because a person’s mathematics difficulties generally originate from some dysfunction in one of these milestone concepts, intervention should begin with effective and systematic instruction in these areas. For example, a student in third grade with gaps should get extra intervention in the development and mastery of number concept and numbersense. 

Individuals with dyscalculia need help in organizing and processing information related to quantity and space. Since mathematics is a form of language, one should spent time on its vocabulary and syntax and translation from mathematics to English and from English to mathematics. 

These individuals can benefit from tutoring that can accomplish three objectives. 

First, to help them make-up the missing arithmetic concepts—number concept, numbersense, and numerical operations.  However, this should not be done in isolation.  Whatever facts are learned (mastered) should be applied to another mathematics concept or a problem. 

Second, to help them connect these concepts to their current mathematical needs. 

And, third, is to help them develop the pre-requisite skills for future mathematics learning. 

1. The CPVA Model of instruction

Effective teaching combines direct instruction (teacher-directed tasks, discussion, and concrete models) with helping students construct.  The second component is very important.  It means, helping them learn ways to derive and learn efficient strategies that help them in deriving facts. It also means helping them become better learners.  This means learning and memorization techniques for arithmetic facts, making connections, seeing patterns, acquiring study skills and metacognition—learners identify strategies that help them to learn. These are  important principles in instruction for all children, but particularly for students with learning difficulties in mathematics. It is adherence to the six levels of learning mathematics:  

Intuitive  >  Concrete  > Pictorial  >  Abstract  > Applications  > Communication

However, the most important steps in this developmental sequence of a concept or procedure are: Concrete modeling (that are efficient and effective) to Pictorial representation (efficient and effective and congruent to concrete models) to Visualization (picturing the model and rehearsing the script in the mind’s eye) to Abstract (recording it in abstract form—symbols, formula, procedure, equations, etc.) (CPVA). The sequence (CPVA):

Concrete   >   Pictorial   >   Visualization   >   Abstract

is the key to reaching all children including dyscalculic and dyslexic in a mathematics class or individual setting. Example of CPVA: 

Cuisenaire rods or Montessori Colored rods (Concrete)   >            

Visual Cluster Cards or Empty Number Line (Pictorial) >

 Imagining Visual Cluster Cards patterns and constructing Empty Number Line in the mind’s eye (Visualization)   >                           

Recording on paper using mathematical symbols (Abstract).

The concrete model helps develop the concept and language.  Transfer from Concrete to pictorial helps to do the task analysis and helps create the script for implementing and understanding the task. Visualization helps reinforce the understanding and the transfer from concrete/pictorial to abstract and solidifies the script. To be effective the representations (concrete, pictorial, and abstract) should be congruent and the script from one representation to the other should be consistent. Effective questioning from the teacher and proper language helps develop the script efficiently. Such teaching include: 

  • Sequencing and task-analysis (breaking down the task into parts and then synthesizing the parts into a whole, providing step-by-step prompts),
  • Repetition and practice (automatizing arithmetic facts, daily testing, sequenced review)
  • Socratic questioning and responses (structured questioning where teacher asks process or content questions to scaffold learning and develops scripts for tasks and steps under guidance)
  • Control of task difficulty (the teacher provides necessary assistance or tasks sequenced from easy to difficult)
  • Proper use of technology (after mastering facts, concepts, and estimation)
  • Teacher-modeled problem solving before collaboration activities and individual practice
  • Strategy cues (reminders and scripts to be used in strategies)
  • Making connections (each concept and procedure after it has been learnt should be connected to other related concepts, procedures, and skills), and 
  • Frequent practice and assessment of tasks to improve performance and self-esteem. 

Every remedial/instructional intervention session should have the following components:

  • Developing the prerequisite and support skills          
  • Learning arithmetic facts orally
  • Visualizing problems and information 
  • Verbalizing and recording procedures and estimation
  • Helping the child form problems relating to the given concept
  • Counting forward beginning with a given number e.g. begin with 53 and count forward by two’s, by three’s, etc.
  • Counting backwards beginning with a given number, e.g., begin with 97 and count backward, by 1’s, 2’s, 5’s, and 10’s.
  • Showing patterns of number facts, e.g., 4 + 4 = 8, then 4 + 5 = ___; 8 +10 =18, then 18 + 10 = ___, 78 + 10 = ____, etc.

The three components of a mathematical idea: linguisticconceptual and procedural should be carefully included in all mathematical instruction. Before the student performs any arithmetic operation he should be asked about the language, procedure, and the conceptual model of the problem first. When he has completed the problem, he should be asked whether he knows any other problems that are similar to the one he just completed. 

2. Vertical Acceleration

Based on our experiences with many dyscalculics, we find that with the help of a competent tutor, with effort, discipline and structure, and appropriate concrete materials, dyscalculics can make progress in mathematics learning and realize their potential. 

         In the case of students who are several years behind their classmates, we can develop the concepts sequentially—we cannot keep on working on additive reasoning till everything in it is mastered, we need to connect whatever is learned in additive reasoning should be connected  and transferred to multiplicative, proportional, and algebraic reason. 

It should be remembered that problems of dyscalculics are related to numeracy; therefore, the development of number concept, numbersense, and numeracy is the most important part.  However, one should not do it exclusively.  Once one arithmetic fact is mastered, for example, the table of 4, then the student, particularly older student, should practice multi-digit number multiplied by 4 (e.g., 45678 × 4 =?), multi-digit number divided by 4 (44678 ÷ 4), fractions with numerator and denominator that are multiples of 4 (e.g., simplify the fractions 4/20, 20/32, 2x/8x, 240/800, etc. to the lowest term: , etc.), solve the equation: (4x=28, 4y=56, etc.). The choice of problems and topics is dependent on the age/grade level of the student. 

This approach is called Vertical Acceleration. That means start with a lower concept (start with wherever the student is) that means however so the low the student is, start there, and then move vertically. Begin with a concept, fact, procedure that the student does not know, and take the concept to higher levels. For example, multiplication procedure of whole numbers is extended to multiplication of fractions, to decimals, to integers, to variables. This is only possible, if the teacher uses efficient and effective models and materials (e.g., for multiplication, one uses area model), appropriate language, and common script and procedures. In designing instruction, it is important to keep in mind the following aspects of learner characteristics, instructional components, and the nature of the learning problem. 

3. Learner Characteristics and Learner Differences

•  Cognitive preparation 
•  Mathematics learning personality 
•  Prerequisite skills

•  Nature of mathematical problems

The role of a teacher, particularly, the interventionist is not just to help students to acquire the content, but also to help them improve their learning skills. Today, neuroscience research has found that the plasticity of the brain holds possibilities of improving learning capabilities of students in learning material that otherwise was not in the realm of possibilities.  To optimize mathematics learning and for students to acquire competence, teachers should use strategies from the following areas of learning theory: 

  • Cognitive Strategies — Improving children’s cognitive strategies prepares them to learn mathematics more effectively. These strategies may include the use of concrete materials and other models, inquiry techniques, metacognition, and others.
  • Different Ways of Learning — Understanding that individuals may learn differently can help teachers develop appropriate instructional strategies.
  • Prerequisite Skills for Mathematics Learning — Developing appropriate prerequisite skills will help to anchor students’ mathematics learning.
  • Learning Difficulties — Developing an awareness of students’ mathematics learning problems–developmental mathematics learning disabilities, language acquisition, carryover problems, dyscalculia, and mathematics anxiety–will help teachers to mitigate their effects in the classroom.
  • Instructional Design: The focus of the instructional activities should be developing strategies in the areas of: cognition–perceptions, executive functions, memory systems, higher order thinking; mathematical–fundamental thinking skills (decomposition/recomposition, additive and multiplicative reasoning, and mathematical way of thinking. The mathematical models and strategies should be exact, efficient, elegant (can be generalized, extrapolated, and abstracted. 

(a)  Cognitive Preparation and Mathematics Conceptualization

Methods for improving cognitive preparation are:

  • Using appropriate, universal concrete models to introduce mathematics content,
  • Asking many hypothetical questions when students are engaged in learning,
  • Developing metacognition by connecting success to the factors that made the success possible and helping students reflect on their strategies and actions. 

Appropriate universal concrete models help children develop cognitive strategies easily. During this concrete manipulative work, the teacher must pay attention to the development, articulation and application of these strategies. This means:

  • It is not enough to look at a child’s answer, although answer is important. 
  • Examine the strategies by which s/he answers the problem
  • Children always use some kind of strategy 
  • Sophistication of the strategy varies with the child, the concept, and familiarity with the problem.  
  • The more advanced the strategy, the more advanced the thinking. 
(b)  Questioning Process

Teacher’s questions are the mediating link between instructional models and the content that students acquire.  The quality and the proper sequencing of these questions determine the quality and depth of learning.  A question sets off a sequence of cognitive functions, In fact: 

  • Questions instigate language;
  • Language instigates models;
  • Models instigate thinking;
  • Thinking produces understanding;
  • Understanding results in competent performance;
  • Competent performance produces long lasting self-esteem; and
  • Self-esteem is the basis of meaningful learning.

Nature (quantity and quality) of teachers’ questions determines the nature and quality of children’s learning and achievement. Convergent questions produce very little language; divergent questions produce more language and learning. 

Choice of instruction models, type and frequency of questions asked, the level of language used and expected, sequence of tasks selected and designed, and the form, variety, and frequency of assessment determine the effectiveness of the teacher.

  • Mathematics Learning Personality: Ways of Making Sense of Mathematics Information

Each one of us makes sense of mathematics information uniquely using strategies and approaches indicating preference for language, concepts, and procedures.  It is, therefore, important to pay attention students’ mathematics learning personalities and ways of making sense of mathematics information.

Matching teaching approach, conceptual models, and nature of language usage with a student’s mathematics personality results in his/her learning that is easier, deeper, and more productive. 

4.   Prerequisite Skills for Mathematics Learning

  1. Following Sequential Directions
  2. Spatial orientation/space organization
  3. Pattern recognition and its extension
  4. Visualization
  5. Estimation
  6. Deductive reasoning
  7. Inductive reasoning
Teacher Characteristics and Teaching Methods
1.  Mastery of Mathematics Content (n ± 3 grades)

2. Teaching Style and Roles: Methods of Communicating the Content and delivering instruction

a) Socratic Questioning
b) Didactic
c) Coaching

3. Empathy with the Learner and the Content 

4. Teaching models

~ Appropriate to the learner and the content 
~ Universal across concepts 
     • Exact 
     • Efficient 
     • Elegant

PART II

Numbersense: A Window to Understanding Dyscalculia

Acquiring the number concept or numberness—understanding number, its representation, and its applications, is a fundamental skill. It is like acquiring the alphabet of the mathematics language with arithmetic facts as its words.  

Much of the research (Geary, 1993; Robinson et al., 2002) has focused on developing a theoretical understanding of mathematics learning difficulties. This article looks at the role of number concept and numbersense in mathematics learning difficulties and implications for instruction and interventions. Children’s understanding and level of mastery of number concept and numbersense provides a window into their arithmetic difficulties, particularly dyscalculia Dehaene et al., 1998; 1999; Gersten & Chard, 1999). 

Numbersense deals with number concept, number combinations—arithmetic fact, computing and place value. Numbersense is a cluster of integrative skills: number concept, making meaning and ways of representing and establishing relationships among numbers, visualizing the relative magnitude of collections, estimating numerical outcomes, and mastering arithmetic facts and proficiency in their usage (Dehaene et al., 1999; Fleischner et al., 1982). Numbersense is the flexible use of number relationships and making sense of numerical information in various contexts. Students with numbersense can represent and use a number in multiple ways depending on the context and purpose. In computations and operations, they can decompose and recompose numbers with ease and fluency. This proficiency and fluency in numbersense helps children acquire numeracy.  

Numeracy is the ability to execute standard whole number operations/ algorithms correctly, consistently, and fluently with understanding and estimate, calculate accurately and efficiently, both mentally and on paper using a range of calculation strategies and means. Numeracy is the gateway to higher mathematics beginning with the study of algebra and geometry.  

Many individuals encounter difficulties in mastering numeracy. Some because of (a) environmental factors—lack of appropriate number experiences, ineffective instruction and a fragmented curriculum, inefficient conceptual models and strategies, lack of appropriate skill development, and low expectations, and (b) individual capacities and learning disabilities. For example, teaching arithmetic facts by sequential counting (“counting up” for addition, “counting down” for subtraction, “skip counting” on number line for multiplication and division), as advocated by many researchers and educators, is not an efficient strategy for many children including dyscalculics (Gelman & Gallistel, 1978; Gelman & Meek, 1983; Gelman et al., 1986). 

Among those who exhibit learning problems in mathematics, some experience difficulty in specific aspects of mathematics—difficulty only in procedures, in  conceptual processes, or in both. Some have difficulty in arithmetic, algebra or geometry. Some may have general learning disabilities in mathematics while others display symptoms only of dyscalculia.  

Learning disability may manifest as deficits in the development of prerequisite skills: following sequential directions, spatial orientation/space organization, pattern recognition and extension, visualization and visual perception, and deductive and inductive thinking. These deficits may affect learning ability in different aspects of mathematics, for example, a few isolated skills in one concept/procedure or several areas of arithmetic/mathematics. Some learning problems fall in the intersection of quantity, language, and spatial thinking. 

Because of the range of mathematics disabilities, we cannot clearly identify a cause or effect; no one explanation adequately addresses the nature of learning problems in mathematics. Most mathematics problems and difficulties such as carryover problems, dyscalculia, or mathematics anxiety are manifested as lack of quantitative thinking. In this chapter, we are interested in one area of mathematics disabilities, the problems related to numeracy due to dyscalculia or acquired dyscalculia. 

A. Nature of Number Related Learning Problems: Dyscalculia

Difficulties associated with numberness, numbersense, and numeracy are known as dyscalculia. Dyscalculia has the same prevalence as dyslexia (about 6-8% of children) although it is far less widely recognized by parents and educators (Ardilla & Roselli, 2002).

Dyscalculia is manifested as poor number concept, difficulty in estimating the size and magnitude of numbers, lack of understanding and fluency in number relationships, and inefficiency of numerical operations. Dyscalculics depend on immature and inefficient strategies such as sequential counting to solve problems that most children know by heart. At the same time, they find it hard to learn and remember arithmetic facts by sequential methods. Like dyslexics, they need special academic support. When taught with appropriate methods and efficient models, children respond favorably (Cohen, 1968; Dunlap & Brennen, 1982; Shalev et al., 2005).  

A characteristic many dyslexics share with dyscalculics (Light & DeFries, 1995) is limited lexical entries for number and number relationships thus facing problems with automatic labeling the outcome of number relationships—instant recall of arithmetic facts (e.g., multiplication tables). They do not have “sight facts” in their minds for numbers. 

Sight facts are like sight vocabulary, for example, knowing that 7 is 6 and 1; 5 and 2; and 4 and 3. Sight facts are instrumental in achieving automatization, the fluency to produce, for example, the fact 8 + 7 = 15 in 2 seconds or less orally and 3 seconds in writing and understanding (using a non-counting strategy, e.g., 8 + 7 is one more than 7 + 7, therefore, 8 + 7 = 15, or 8 + 2 is 10 and then 5 more is 15). This lack of automatization, in most cases, is an artifact of poor instruction rather than real difficulty or disability.  

Problems that most dyscalculics face in arithmetic are due to poor number conceptualization and numbersense (Dehaene, 1997). Without exposure to efficient and effective methods of learning, children do not acquire proper number concepts, arithmetic facts, and standard procedures and risk not gaining proficiency in mathematics by the end of first grade. Lack of success in the development of number becomes the main reason for a child’s difficulty in learning mathematics and dyscalculia (Jordan, Hanich et al. as cited in Gersten et al., 2005).  

Just as it is possible to build lexical entries for words, letters, and word-parts, it is also possible to acquire strategies to develop lexical entries for numbers, numbers facts, symbols, formulas, and even equations. Although mathematical symbols themselves are not phonetic, each symbol represents a lexical entry whose meaning and interpretation can be understood (Ball & Blanchman, 1991).   

B. Literacy and Numeracy

There are many parallels in the development of literacy and numeracy, which we need to explore. Young children develop literacy through literacy practices (e.g., being read to at bedtime). Similarly, early exposure to the language and symbols of quantity and space creates lexical entries for quantity (number words) and the role of number (size/quantity)— what and how to quantify, what and how to measure, and how to represent and use quantities (Adams, 1990).

The complex process of mastering reading involves a variety of brain components and systems—both localized and global—that perform and integrate tasks such as recognizing and organizing symbols—visual and aural, discerning and analyzing sound patterns, perceiving spatial arrangements—source of speech or location of the symbol, and verbal and non-verbal clues. Some of the same mechanisms are called upon in acquiring numeracy and are related to language, visuo-spatial, sequencing, and working memory. A breakdown and deficits in any of these areas may affect learning letters and numbers alike. Many dyslexics, therefore, show symptoms of dyscalculia (Light & DeFries, 1995). 

While there are important similarities in learning to read and conceptualizing number, there are also important differences. Some unique abilities and systems are needed to learn number and its applications. For this reason, there are people who can read and have poor numeracy skills, but there are very few numerates who cannot learn to read. Keeping in mind these unique differences, we need to design activities for making numeracy accessible to all children. 

1. Phonemic Awareness, Numberness, and Numbersense

Fluent reading and fluency in numberness are analogous. Research (Williams, 1995) in reading shows that phonemic awareness—the insight that words are composed of sounds and the ability to connect fluently grapheme to phoneme and phonological sensitivity—the ability to break words into meaningful “chunks” and then “blend” them fluently—are predictors of early reading performance (better than IQ tests, readiness scores, or socioeconomic level) and essential for reading acquisition. Processes of numberness—one-to-one correspondence, sequencing, visual clustering, and decomposition/re-composition, representation of number orally and graphically— are similar. The ability to associate a number to a cluster is like phonemic awareness and the ability to instantly recognize that a number is made of smaller numbers (decomposition/ recomposition) is equivalent to chunking and blending. Numberness is a predictor for future proficiency and fluency in arithmetic.

Understanding of phonemic awareness has revolutionized the teaching of beginning reading. Numberness and numbersense carry similar implications for instruction for children with or without learning difficulties. The proper definition, and development of numberness and numbersense is the key to planning remediation for dyscalculics and preventing acquired dyscalculia. However, educators and psychologists have taken a narrow view of number concept—e.g., ability to count forward and backward (Gelman & Gallistel, 1978; Klein, Starkey, & Ramirez, 2002; McCloskey & Mancuso, 1995; Moomaw & Hieronymous, 1995). 

In reading, one needs to focus on the phonemes in a word; in math, one needs to see clusters of objects in the mind’s eye. Most children have difficulty forming visual clusters in their minds and sight facts by one-to-one counting. Decoding letters in a word does not lead to reading; similarly, counting individual objects/numbers (concretely or sub-vocally) does not lead to numberness. In fact, one-to-one counting turns most children into counters – that’s all. To conceptualize number, one needs to see clusters (decomposition) in a collection and integrate smaller clusters into larger clusters (recomposition). Associating a number name to the collection and relating this number to smaller clusters (numbers) is forming sight facts. Recognizing clusters (sight facts) is like recognizing phonemes and sight words. With the help of sight facts, children can move beyond counting and learn arithmetic facts at an automatized level. Many LD children have difficulty forming visual clusters in their minds and sight facts by one-to-one counting (Schaeffer et al., 1974).  

The mastery of numberness and proficiency in arithmetic and phonemic awareness and the ability to read with understanding are parallel activities, nevertheless, it is also important to recognize the differences between the two processes. Phonemic awareness involves focus on auditory processes and phonological decoding associates grapheme and phoneme, whereas visual perceptual integration—recognizing clusters, estimating by observation, and decomposition/recomposition of clusters—is fundamental to the development of numberness.    

C. Parallels: Letter Recognition and Number Concept—Numberness A child knows the alphabet when he can

  • Identify the letter (shown M, he recognizes it instantly),
  • Recognize the letter in its variant forms (e.g., M, M, MMM, M, M, M, m. etc.),
  • Recognize letters among other symbols (e.g., M in CALM, MILK, WARMER, $M$569A, etc.),
  • Write the letter and describe the various strokes in the proper order, and
  • Associate a sound to the letter (e.g., M as in monkey).

This should be true for all letters of the alphabet. Mastery of number is similar and is more than just reciting and writing the numbers. A child has number concept when he  

  • Possesses lexical entries for number (knows number names and the difference between number words and non-number words), (Fuson, 1980; Fuson et al., 1982)
  • Can meaningfully count (one-to-one correspondence + sequencing), (Fuson et al., 1982; Piaget, 1968; Pufall et al., 1973; Saxe, 1979)
  • Can recognize and assign a number to a collection/cluster (organized in a pattern up to ten objects) without counting (Resnick, 1993)
  • Can represent a collection—a visual cluster of seven objects  à graphical representation, e.g., 7,
  • Can write the number when heard (hears s-e-v-e-n and writes 7), and
  • Can decompose and recompose a cluster into two sub-clusters (i.e., a number, up to 10, as sum of two numbers and vice-versa).

Images of visual clusters in the mind’s eye provide a child a base of “sight facts.” For example, when one sees the visual cluster of 7 objects, one recognizes the sight facts: 7 = 1 + 6 = 2 + 5 = 3 + 4 without counting. These sight facts, with strategies of addition and subtraction based on decomposition/recomposition, provide a strong base for arithmetic facts mastery beyond 10.

Numberness, thus, is the integration of: 

Mastery of number concept/numberness, arithmetic facts (arrived at by using decomposition/recomposition) and place value is called numbersense.  Lack of proper instruction in numberness and numbersense poses conditions of failure in early mathematics. Instruction in strategies for deriving arithmetic facts and procedures are much more productive when number concept is intact (Geary, 1994; Gelman, 1977).  

D. Language and Number

Early number conceptualization begins with concrete experiences—counting objects in context.

Many children become and remain counters because of this early emphasis on counting. Appropriate concrete experiences accompanied with rich language, on the other hand, help abstract the experience into concepts with labels. Neither concrete experiences alone nor purely language-based teaching develop the concept of number for all children. For abstraction of concrete experiences into numberness, language is essential. Children must transcend the concrete models in order to learn to solve problems and communicate through mathematical symbols. This concurrent thinking of numbers as concrete and abstract is at the core of true number conceptualization and is a real challenge for many children (Baroody, 1992; Brainerd, 1992; Copeland, 1974).

1. Mathematics Language and Native Language

In the child’s native language, numbers function as predicators and qualifiers: five dishes, many books, fewer children, etc. They function like adjectives in a sentence. Most children are quite fluent in this before they enter Kindergarten (Carruthers & Wortington, 2005).

Later numbers function not only as predicators but also as real, concrete objects: six hundred is a big number, an even number, or much smaller than the number six hundred thousand. Thus, in the language of mathematics, numbers are qualifiers as well as ‘real’ abstract objects. Mathematical operations can be performed on numbers when we treat them as real, concrete entities (Williams, 1977). 

Conceptualizing number requires a child to perform two simultaneous abstractions: to translate sensory, concrete representations of quantity into symbolic entities (5 represents any collection of five objectsand to transform a number as a predicate in the native language to its conception as objects in the language of mathematics (Wynn, 1996; 1998). For some children, particularly LD children, these transitions are not easy and need to be facilitated carefully using appropriate language, an enabling questioning process, and efficient instructional models.  

E. Mastering the Concept of Number

1. Lexical Entries and Egocentric Counting to the Cardinality of the Set

Children develop lexical entries for number by hearing others count and copying this process.  Number words are essential but not sufficient for fluent number conceptualization and usage.

Consider the number work of a five-year-old.

Teacher: How many cubes there are? (Points to the collection.)

❒ ❒ ❒ ❒ ❒ ❒ ❒

Child: (Counts by touching each cube) Seven. 

Teacher: What number came just before seven? 

Child: One? Three? Five? I don’t know. 

Teacher: Can you give me six cubes? 

Child: Do I have enough? Maybe I do. (She counts six cubes and gives them to the teacher.) Teacher: That is right.  

Teacher rearranges them. 

Teacher: How many cubes are there now?   

Child: (She counts them) Seven. 

Teacher: Yes! 

Most children can count objects in a rote manner. For many of them, even at age 6, the cardinality of the set is the outcome of their counting process, not a property of the collection. Number is the product of egocentric counting (“These are six blocks. I just counted them.”) rather than the property of the collection (“These are six cubes.”). This is a key step in number conceptualization. Consider number work with another Kindergartener.  Teacher: How many cubes are there? (Points to the collection.) 

❒ ❒ ❒ ❒ ❒ ❒ ❒

Child: (Counts them) Seven. 

Teacher: Yes! You counted them from left to right (points to the direction). Do you think you will have the same number if you counted them from right to left? (Points to the direction.) Child: I do not know. Let me try.   

Child: (Counts them) Seven.   

Teacher: That is right.  

Child: (Counts them again) Seven. It is always seven.  

The child associates a number to the collection as the property of the collection. Scaffolding questions resulted in converting a child’s concrete experiences and egocentric counting into the cardinal number.  

2. Development of Visual Clusters

For number conceptualization, child must transcend counting (Turner, 2003; Sophian & Kailihiwa, 1998). Young children spontaneously use the ability to recognize and discriminate small numbers of objects. This is called subitizing (Klein & Starkey, 1988, Clements, 1999). Subitizing is instantly seeing how many in a small collection of objects. But some young children cannot immediately name the number of objects in a collection. It is important for number conceptualizing. Work with dominos, dice, and playing cards helps in the process (Clements & Callahan, 1986). However, for efficient number concept, subitizing must be extended to numbers up to ten. We term that process as forming visual clusters in the mind. Visual Cluster CardsTM (VCC), with modified arrangements of clusters, are especially effective for developing visual clustering and then number concept.  

A VCC deck (60 cards) consists of 4 cards with 1 to 10 pips in 4 suites (heart, diamond, club, and spade); two cards with no pips represent zero; and 2 jokers (can be assigned any value). Numbers 3, 8, 9, and 10 have two representations. For example, number 3 is represented as: 

The pips on higher number cards are organized so that the sub-clusters of smaller numbers can be instantaneously recognized. For example, on the 7-card, one can see clusters of 4 and 3; 5 and 2; and 6 and 1. No number names are displayed on the cards. 

Creating images of visual clusters and developing the decomposition/recomposition process of numbers are at the heart of number conceptualization and arithmetic facts. Many children may achieve the decomposition/recomposition skill through counting; however, many children, particularly those with special needs, have difficulty achieving this with counting. Robinson et al. (2002) proposed that interventions for students with poor mastery of arithmetic combinations should include two aspects: (a) interventions to help build more rapid retrieval of information, and (b) concerted instruction in any areas of numbersense that are underdeveloped in a child.  VC cards help achieve both. 

The teacher introduces the VC cards for 1, 2 and 3. She identifies the cards by counting the pips. The card with two pips is identified as 2; 1 and 1; or two ones. Children learn that each visual cluster card is made of sub-clusters. For example, the teacher displays the card with three pips.  

Teacher: Look at the card. How many diamonds are on the card? 

Child One: Three. 

Teacher: What three numbers make 3? 

Child Two: 1 + 1 + 1. 

Teacher: Look at the card with a circle around the diamonds. How many diamonds are circled? 

Teacher: How many are not circled? 

Children: One. 

Teacher: What two numbers make 3? 

Child One: 2 and 1.  

Child Two: 1 and 2. 

Teacher: Right! 2 + 1 makes 3 (traces the two circled diamonds and the one diamond); 1 + 2 also makes 3 (traces the one individual circled diamond and then the two diamonds); 1 + 1 + 1 also makes 3 (traces the three individual circled diamonds).  

Once children have created the image of number 3 in the standard form, they do the same with the three objects organized in another form. For example: 

The same process is used for developing the cluster images for higher numbers. For example, the teacher introduces the card representing number 5.  

Teacher: How many diamonds are there?  

Children count the diamonds on the card and say: Five.  

Teacher: We will call this the 5-card. How many diamonds are there in the first column of the card? (She traces the first column.) 

Children: 2. 

Teacher: Yes. It represents the number 2. 

Teacher: Now, how many diamonds are there in the last column? (She traces the last column.) 

Children: 2.  

Teacher: Good! It also represents number 2.   

Teacher: Look at the middle column. How many diamonds are in the middle column? 

Children: One! 

Teacher: What if the middle diamond was not there, what number will the card represent? 

Children: 4. 

Teacher: Very good! (If a child is unable to answer, the teacher displays the card by covering the middle pip or show the 4-card again.) What if the first column was not there, what number will the rest of the card represent?  

Children: 3. 

Teacher: Very good! What if the last column was not there, what number will the rest of the card represent? 

Children: 3.  

The teacher continues till children have created the image of the cluster of number 5 in their minds. Every child should be able to identify the card in less than two seconds (without counting). They also know that the cluster of 5 has component sub-clusters of 2 and 3; 4 and 1; of 2, 2, and 1.  

Teacher: Remember the card we have been looking at? I am going to show the card, but a portion of the card will be hidden. You need to tell me the missing number.  (She hides the first column.) How many are visible? 

Children: 3. 

Teacher: How many are hidden? 

Children: 2. 

Teacher: What two numbers make 5? 

Children: 2 and 3. 

Teacher: Great!  (She uncovers the hidden part of the card and shows the 5-card). Yes, 2 and 3 make 5.    

Finally, each child has formed images of the number 5 as a visual cluster and its relationship with other numbers (decomposition/recomposition) as 4 + 1; 3 + 2; 2 + 2 + 1; and 1 + 1 + 1 + 1 + 1. Then she asks them to write these relationships.

Thus, knowing a number means an ability to write the number, use it as a count, recognize the visual cluster, and that it is made up of smaller numbers. This is true for all ten numbers:  

  • 2 = 1 + 1
  • 3 = 2 + 1 = 1 + 2  
  • 4. = 3 + 1 = 1 + 3 = 2 + 2  
  • 5 = 4 + 1 = 3 + 2 = 2 + 3 = 1 + 4 
  • 6 = 5 + 1 = 4 + 2 = 3 + 3 = 2 + 4 = 1 + 5 
  • 7 = 6 + 1 = 5 + 2 = 4 + 3 = 3 + 4 = 2 + 5 = 1 + 6 
  • 8 = 7 + 1 = 6 + 2 = 5 + 3 = 4 + 4 = 3 + 5 = 2 + 6 = 1 + 7 
  • 9 = 8 + 1 = 7 + 2 = 6 + 3 = 5 + 4 = 4 + 5 = 3 + 6 = 2 + 7 = 1 + 8
  • 10 = 9 + 1 = 1 + 9 = 8 + 2 = 2 + 8 = 3 + 7 = 7 + 3 = 6 + 4 = 4 + 6 = 5 + 5  

As the table above show, there are a total of 45 sight facts. Without the idealized image of these numbers, dight facts, and the decomposition/ recomposition process, children have difficulty in developing fluency in number relationships. Sight facts and decomposition/recomposition play the role in numberless and arithmetic as sight words and phonemic awareness plays in acquiring reading skills. Most dyscalculics and many underachievers in mathematics have not learned number concept properly.  

Cuisenaire rods are another efficient tool for developing and extending the decomposition/ recomposition of numbers achieved through visual cluster cards. For example, the number 10 can be shown as the combination of two numbers as follows (the same process is used for other numbers):  

Both Visual Cluster Cards and Cuisenaire rods help children to create and learn these decompositions.  

F. Concept of Addition

Early mathematics interventions should focus on building fluency and proficiency with basic arithmetic facts as well as more accurate and efficient use of addition strategies (Gersten et al., 2005; Siegler, 1991; 1988). When children achieve fluency and efficiency in arithmetic combinations, teachers can assume that children are able to follow explanations of concepts or procedures.

Once children conceptualize idealized images of the ten numbers in the ‘mind’s eye’, they form sight facts and then easily learn addition facts. For example:

7 is made up of 4 and 3; 5 and 2; and 6 and 1.

Addition of numbers is facilitated through strategies of decomposition and recomposition of numbers. For example, to add 8 and 6 it is much easier to take two from 6 and give it to 8 so that the number combination can easily be seen as: 8 + 6 = 8 + (2 + 4) = (8 + 2) + 4 = 10 + 4 equals 14. Decomposition (the breaking of 6 as 2 + 4 and thinking of 10 as 8 + 2) and recomposition (thinking of 10 + 4 as 14) are key strategies for learning addition and subtraction facts.  

When children do not automatize facts, they are unable to apply their knowledge to newer situations. To find the answer to a number problem, they digress from the main problem to generate the facts needed for solving problems. Because of the use of inefficient strategies, such as counting, their working memory space is filled in the process of constructing these facts, then it is not available to pay attention to instruction, observe patterns, or focus on concepts, nuances, relationships, and subtleties involved the concepts.   

A child’s struggles with arithmetic facts in Kindergarten and first grade reflect a difficulty in transitioning from concrete to abstract number relationships and should trigger an intense intervention program in numberness and numbersense that focuses on visual clustering and decomposition/recomposition skills (Van Engen & Steffe, 1970). The use of tools such as Visual Cluster Cards and Cuisenaire rods can achieve that goal, prevent the development of acquired dyscalculia, and mitigate the effects of dyscalculia. 

References: 

Adams, M. J. (1990). Beginning to read: Thinking and learning about print. Cambridge, MA: MIT Press. 

Ardilla, A., & Rosselli, M. (2002). Acalculia and dyscalculia. Neuropsychology Review12(4), 179-231.

Ball, E. W., & Blanchman, B. A. (1991). Does phoneme awareness training in kindergarten make a difference in early word recognition and developmental spelling? Reading Research Quarterly26, 49-99. 

Baroody, A. J. (1992). Remedying common counting difficulties. In J. Bideaud, J. P. Fischer, C. Greenbaum, & C. Meljac (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 307-324). Hillsdale, NJ: Erlbaum. 

Brainerd, C. J. (1979). The origins of the number concept. New York, NY: Praeger. 

Carruthers, E., & Worthington, M. (2005). Children’s mathematics: Making marks, making meaning. London, UK: Sage. 

Clements, D.H. (1999). Subitizing: What Is It? Why Teach It? Teaching Children Mathematics (December), Reston, VA: NCTM  

Clements, D. H., & Callahan, L. G. (1986). “Cards: A Good Deal to Offer.” Arithmetic Teacher 34(1986): 14-17. Cohn, R. (1968). Developmental dyscalculia. Pediatric Clinics of North America75(3), 651-668.

Copeland, R. (1974). How children learn mathematics: Teaching implications of Piaget’s research. New York, NY: Macmillan. 

Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York, NY: Oxford University Press. 

Dehaene, S., Dehaene-Lambertz, G., & Cohen, L. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neuroscience21, 355-361. 

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science284(5416), 970-973. 

DfEE (1999). The National Numeracy Strategy. London, UK: Department of Education and Employment. 

Dunlap, W., & Brennen, A. (1982). Blueprint for the diagnosis of difficulties with cardinality.  Journal of Learning Disabilities14(1), 12-14.

Dunlap, W., & Hynde, J. (1981). The effects of grouping patterns on the perception of number. Focus on Learning Problems in Mathematics3(4), 13-18. 

Fleischner, F., Garnett, K., & Shepard, M. (1982). Proficiency in arithmetic basic fact computation by learning disabled and nondisabled children. Focus on Learning Problems in Mathematics, 4(2), 47-55.  

Fuson, K. (1980). The counting word sequence as a representation tool. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education. Berkeley, CA: University of California. 

Fuson, K., Richards, J., & Briars, D. (1982). The acquisition and elaboration of the number word sequence. In C. Brainerd (Ed.), Children’s logical and mathematical cognition. New York, NY: Springer Verlag. 

Geary. D. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin114, 345-362. 

Geary. D. (1994). Children’s mathematical development. Washington, DC: American Psychological Association. 

Gelman, R. (1977). How young children reason about small numbers. In N. J. Castellan, D. B. 

Pisoni, & G.R. Potts (Eds.) Cognitive theory, 2. Hillsdale, NJ: Erlbaum. 

Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. 

Gelman, R., & Meek, E. (1983). Preschooler’s counting: Principles before skills. Cognition13, 343-360.

Gelman, R., Meek, E., & Merkin, S. (1986). Young children’s numerical competence. Cognitive Development, 1, 1-30. 

Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education33(1), 18-28.  

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal Of Learning Disabilities38(4), 293-304. 

Hughes, M. (1986). Children and number: Difficulties in learning mathematics. Oxford, UK: Blackwell. 

Klein, A., Starkey, P., & Ramirez, A. B. (2002).  Pre-K mathematics curriculum. Boston, MA: Scott-Foresman. 

Light, J., & DeFries, J. (1995). Comorbidity of reading and mathematics disabilities: Genetic and environmental etiologies. Journal of Learning Disabilities28(2), 96-106.  

McCloskey, M., & Mancuso, P. (1990). Representing and using numerical information. American Psychologist50, 351-363.  

Moomaw, S., & Hieronymus, B. (1995).  More than counting: Whole mathematics for preschool and kindergarten. St. Paul, MN: Redleaf Press. 

Nesher, P., & Katriel, T. (1986). Learning numbers: A linguistic perspective. Journal for Research in Mathematics Education, 17(2), 110-111. 

Piaget, J. (1965). The child’s conception of number. New York, NY: Norton. 

Piaget, J. (1968). Quantification, conservation, and nativism. Science162, 976-979. 

Pufall, P., Shaw, R., & Syrdal-Lasky, A. (1973). Development of number conservation: An examination of some predictions from Piaget’s stage analysis and equilibration model. Child Development, 44(1), 21-27. 

Resnick, L. (1993). A developmental theory of number understanding. In H. Ginsberg (Ed.), The development of mathematical thinking (pp. 109-151). New York, NY: Academic Press. 

Robinson, C., Menchetti, B., & Torgesen, J. (2002). Toward a two-factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice17, 81–89.  

Saxe, G. (1979). A developmental analysis of notational counting. Child Development48, 15121520. 

Schaeffer, B., Eggleston, V., & Scott, J. (1974). Number development in young children. Cognitive Psychology6, 357-379. 

Shalev, R.S., Manor, O., & Gross-Tsur, V. (2005). Developmental dyscalculia: A prospective six-year follow-up, Developmental Medicine and Child Neurology, 47(2), 121-125.  

Siegler, R. (1991). In young children’s counting, procedures precede principles. Educational Psychology Review, 3, 127-135.  

Siegler, R. (1988). Individual differences in strategy choices: Good students, not-so good students, and perfectionists. Child Development59, 833–851.  

 Sophian, C., & Kailihiwa, C. (1998). Units of counting: Developmental changes.  Cognitive Development13, 561-585. 

Turner, M. (2003). Tally marks: How to visualize and develop first skills in mental maths (report). Essex, UK: SEN and Psychological Services. 

Van Engen, H., & Steffe, L. (1970). First-grade children’s concept of addition of natural numbers. In R. Ashlock & W. L. Herman (Eds.), Current research in elementary school mathematics. New York. NY: Macmillan. 

Williams, J. (1995). Phonemic awareness. In T. Harris & R. Hodges (Eds.), The literacy dictionary (pp. 185-186), Newark, DE: International Reading Association. 

Williams, R. (1977). Ordination and cardination in counting and Piaget’s number concept task. Perceptual and Motor Skills45, 386. 

Wynn, K. (1996). Origins of numerical knowledge. In Butterworth B. (Ed.), Mathematical cognition (vol. 1) (pp. 35 – 60). London, UK: Erlbaum, Taylor & Francis Ltd. 

Wynn, K. (1998). Numerical competence in infants. In C. Donlan (Ed.), The development of mathematical skills (pp. 2 – 25). Hove, UK: Psychology Press. 

Suggested Reading: 

Chomsky, N. (2006). Language and mind. Cambridge, UK: Cambridge University Press.  

Sharma, M. (2008). How to master arithmetic facts easily and effectively. Framingham, MA: CT/LM. 

Sharma, M. (2008). Mathematics number games. Framingham, MA: CT/LM. 

Sharma, M. (1981). Prerequisite and support skills for mathematics learning. The Math Notebook, 2(1).  

Sharma, M. (1981). Visual clustering and number conceptualization. The Math Notebook2(10). 

Sharma, M., & Loveless, E. (Eds.) (1986). Developmental dyscalculia. Focus on Learning Problems in Mathematics8(3 & 4). Framingham, MA: CT/LM 

Shipley, E., & Shepperson, B. (1990). Countable entities:  Developmental changes. Cognition, 34, 109-136. 

Skemp, R. (1971). Psychology of learning mathematics. Harmondsworth, UK: Pelican. 

Vygostky, L. (1962). Thought and language. Cambridge, MA: MIT Press. 

CENTER FOR TEACHING/LEARNING OF MATHEMATICS

LIST OF PUBLICATIONS 2019

Mathematics For All 

Programs and ServicesCT/LM has developed programs and materials to assist teachers, parents, therapists, and diagnosticians to help children and adults with their learning difficulties in mathematics. We conduct regular workshops, seminars, and lectures on topics such as: How children learn mathematics, why learning problems occur, diagnosis, and remediation of learning problems in mathematics.       How does one learn mathematics? This workshop focuses on psychology and processes of learning mathematics—concepts, skills, and procedures. Participants study the role of factors such as:  Cognitive development, language, mathematics learning personality, prerequisite skills, and conceptual models of learning mathematics.  They learn to understand how key mathematics milestones such as number conceptualization, place value, fractions, integers, algebraic thinking, and spatial sense are achieved.  They learn strategies to teach their students more effectively. 2. What are the nature and causes o problems in mathematics? This workshop focuses on understanding the nature and causes of learning problems in mathematics.  We examine existing research on diagnosis, remedial and instructional techniques dealing with these problems. Participants become familiar with diagnostic and assessment instruments for learning problems in mathematics. They learn strategies for working more effectively with children and adults with learning problems in mathematics. 3. Content workshops.  These workshops are for teachers and parents on teaching mathematics milestone concepts and procedures. For example, they address questions such as:  How to teach arithmetic facts easily?  How to teach fractions to students more effectively?  How to develop the concepts of algebra easily?  In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a simple concept from arithmetic and take it to the algebraic level. We offer individual diagnosis and tutoring services for children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia, in particular. We provide: Consultation with and training for parents and teachers to help their children cope withand overcome their anxieties and difficulties in learning mathematics. Consultation services to schools and individual classroom teachers to help themevaluate their mathematics programs and help design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics. Assistance for the adult student who is returning to college and has anxiety about his/her mathematics. Assistance in test preparation (SSAT, SAT, GRE, MCAS, etc.)Extensive array of mathematics publications to help teachers and parents tounderstand how children learn mathematics, why learning problems occur and how to help them learn mathematics better. www.mathematicsforall.org
Current Publications
Dyslexia and Mathematics Language Difficulties                             $15.00 Dyscalculia                                                                                                    $15.00 Guide for an Effective Mathematics Lesson                                        $15.00 Games and Their Uses: The Number War Game                                $15.00 How to Teach Arithmetic Facts Easily and Effectively                    $15.00 How to Teach Fractions Effectively                                                       $15.00 How to Teach Number to Young Children                                           $15.00 How To Teach Subtraction Effectively                                                 $12.00 Literacy & Numeracy: Comprehension and Understanding          $12.00 Math Education at Its Best: Potsdam Model                                        $15.00 The Questioning Process: A Basis for Effective Teaching               $12.00  Visual Cluster Cards (Playing Cards without Numbers)                 $15.00     

eBooks:
The Questioning Process: A Basis for Effective Teaching. $10.00
How to Teach Number Effectively $15.00

DVDS         
How Children Learn: Numeracy                                                        $30.00 (One-hour long video interviewing Professor Mahesh Sharma on his ideas about how children learn mathematics)        
Teaching Place Value Effectively                                                       $30.00  

Numeracy DVDs 
(Complete set of six for $150.00 and individual for $30.00)
(Teaching arithmetic facts,Teaching place value,Teaching multiplication,Teaching fractions,Teaching decimals and percents, andProfessional development: teachers’ questions)
Most children have difficulty in mathematics when they have not mastered the key mathematics milestones in mathematics. The key milestones for elementary grades are: Number conceptualization and arithmetic facts (addition and multiplication), place value, fractions and its correlates—decimal, percent, ratio and proportion. These videos and DVDs present strategies for teaching these key mathematics milestone concepts. They apply Prof. Sharma’s approach to teaching numeracy. These were videotaped in actual classrooms in the UK.  
Please add 20% of the total for postage and handling with your order: CENTER FOR TEACHING/LEARNING OF MATHEMATICS 
754 Old Connecticut Path,
Framingham, MA 01701 
508 877 4089 (T), 508 788 3600 (F) mahesh@mathematicsforall.orgwww.mathematicsforall.org 

The Math Notebook (TMN) 

Articles in TMN address issues related to mathematics learning problems, diagnosis, remediation, and techniques for improving mathematics instruction.  They translate research into practical and workable strategies geared towards the classroom teacher, parents and special needs teachers/tutors. Topics covered range from K through College mathematics instruction. Selected Back Issues of The Math Notebook: 

  • Children’s Understanding of the Concept of Proportion – Part 1 and 2 (double)
  • A Topical Disease in Mathematics: Mathophobia  (single)
  • Pattern Recognition and Its Application to Math  (double)
  • Mathematics Problems of the Junior and Senior High School Students  (double)
  • Mathematically Gifted and Talented Students  (double)
  • Types of Math Anxiety  (double)
  • Memory and Mathematics Learning  (double)
  • Problems in Algebra – Part 1 and Part 2 (special)
  • Reversal Problems in Mathematics and Their Remediation  (double)
  • How to Take a Child From Concrete to Abstract  (double)
  • Levels of Knowing Mathematics  (double)
  • Division:  How to Teach It  (double)
  • Soroban: Instruction Through Concrete Learning  (double)
  • Mathematics Culture  (double)
  • Mathematics Learning Personality  (double)
  • Common Causes of Math Anxiety and Some Instructional Strategies  (double)
  • On Training Teachers and Teaching Math  (double)
  • Will the Newest “New Math” Get Johnny’s Scores Up?  (double)
  • Dyslexia, Dyscalculia and Some Remedial Perspectives For Mathematics Learning Problems (special)
  • Place Value Concept:  How Children Learn It and How To Teach It  (special)
  • Cuisenaire Rods and Mathematics Teaching  (special)
  • Authentic Assessment in Mathematics  (special)

FOCUS on Learning Problems in Mathematics

FOCUS has been an interdisciplinary journal. For the last thirty years, the objective of FOCUS was to make available the current research, methods of identification, diagnosis and remediation of learning problems in mathematics.  It published original articles from fields of education, psychology, mathematics, and medicine having the potential for impact on classroom or clinical practice.  Specifically, topics include reports of research on processes, techniques, tools and procedures useful for addressing problems in mathematics teaching and learning:  descriptions of methodologies for conducting, and reporting and interpreting the results of various types of research, research-based discussions of promising techniques or novel programs; and scholarly works such as literature-reviews, philosophical statement or critiques.  The publications in Focus have real contribution in the field of mathematics education, learning problems in mathematics and how to help children and adults in dealing with their mathematics difficulties. 

  Back issues are available from 1979 to 2009 on request. 

About the Author
Professor Mahesh Sharma is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts and Berkshire Mathematics in England. Berkshire Mathematics facilitates his work in the UK and Europe. He is the former President and Professor of Mathematics Education at Cambridge College where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. He is internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics.   He is an author, teacher and teacher-trainer, researcher, consultant to public and private schools, as well as a public lecturer. He was the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children.  He provides direct services of evaluation and tutoring for children as well as adults who have learning disabilities such as dyscalculia or face difficulties in learning mathematics. Professor Sharma works with teachers and school administrators to design strategies to improve mathematics curriculum and instruction for all. Contact Information: Mahesh C. Sharma  mahesh@mathematicsforall.org508 494 4608 (C) 508 788 3600 (F) 
Blog:  www.mathlanguage.wordpress.com
Center for Teaching/Learning of Mathematics 
754 Old Connecticut Path Framingham, MA 01701 mahesh@mathematicsforall.org
http://www.mathematicsforall.org
Mathematics Education Professional Development 
Workshop Series
Framingham State University
with
Professor Mahesh Sharma
Academic Year 2019-2020  
Several national professional groups, the National Mathematics Advisory Panel and the Institute for Educational Sciences in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings–number conceptualization, arithmetic facts, place value, fractions, and integers–may be overwhelming to many students and teachers.
Being proficient at arithmetic is certainly a great asset when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for mastery, Common Core State Standards in Mathematics (CCSSM), set for arithmetic for early elementary grades are specific: students should have (a) understanding (efficient and effective strategies), (b) fluency, and (c) applicability and will ensure that students form strong, secure, and developmentally appropriate foundations for the algebra that students learn later. The development of those foundations is assured if we implement the Standards of Mathematics Practices (SMP) along with the CCSSM content standards.
In these workshops, we provide strategies; understanding and pedagogy that can help teachers achieve these goals.  
All workshops are held on the Framingham State University campus from 8:30am to 3:00pm
Cost is $49.00 per workshop and includes breakfast, lunch, and materials. 
PDP’s are available through the Massachusetts Department of Elementary and Secondary Education for participants who complete a minimum of two workshops together with a two page reflection paper on cognitive development.    

A. Creating A Dyscalculia Friendly ClassroomLearning Problems in Mathematics (including math anxiety)For special education, regular education teachers, interventionists, and administrators 
October 11, 2019 
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. The major aim is to deliver mathematics instruction that prevents learning problems in mathematics from debilitating a student’s learning processes in mathematics. 

B. Number Concept, Numbersense, and Numeracy SeriesAdditive Reasoning (Part I):
How to Teach Number Concept Effectively
For K through grade second grade teachers, special educators and interventionists
November 1, 2019
Number concept is the foundation of arithmetic. Ninety-percent of students who have difficulty in arithmetic have not conceptualized number concept. In this workshop we help participants learn how to teach number concept effectively. This includes number decomposition/recomposition, visual clustering, and a new innovative concept called “sight facts.” 

Additive Reasoning (Part II):
How to Teach Addition and Subtraction Effectively
For K through grade third grade teachers, special educators and interventionists
November 22, 2019
According to Common Core State Standards in Mathematics (CCSS-M), by the end of second grade, children should master the concept of Additive Reasoning (the language, concepts and procedures of addition and subtraction). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students.  

Multiplicative Reasoning (Part III):
How to Teach Multiplication and Division Effectively
For K through four second grade teachers, special educators and interventionists
December 13, 2019
According to CCSS-M, by the end of fourth grade, children should master the concept of Multiplicative Reasoning (the language, concepts and procedures of multiplication and division). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students. 

C. Proportional Reasoning Series
How to Teach Fractions Effectively (Part I):
Concept and Multiplication and Division
January 24, 2020 
For grade 3 through grade 9 teachers and special educatorsAccording to CCSS-M, by the end of sixth grade, children should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and multiplication and division of fractions and help their students achieve that. 

How to Teach Fractions Effectively (Part II): Concept and Addition and Subtraction
For grade 3 through grade 9 teachers
February 28, 2020
According to CCSS-M, by the end of sixth grade, students should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and operations on fractions-from simple fractions to decimals, rational fractions and help their students achieve that. 

D. Algebra
Arithmetic to Algebra: How to Develop Algebraic Thinking
For grade 4 through grade 9 teachers
March 20, 2020
According to CCSS-M, by the end of eighth-grade, students should acquire algebraic thinking. Algebra is a gateway to higher mathematics and STEM fields. Algebra acts as a glass ceiling for many children. From one perspective, algebra is generalized arithmetic. Participants will learn how to extend arithmetic concepts to algebraic concepts and procedures effectively and efficiently. On the other perspective, algebraic thinking is unique and abstract and to achieve this thinking students need to engage in cognitive skills that are uniquely needed for algebraic thinking. In this workshop we look at algebra from both perspectives: (a) Generalizing arithmetic thinking and (b) developing cognitive and mathematical skills to achieve algebraic thinking. 

E. General Topics

Mathematics as a Second Language: Role of Language in Conceptualization and in Problem Solving
For K through grade 12 teachers
April 3, 2020
Mathematics is a bona-fide second language for most students. For some, it is a third or fourth language. It has its own vocabulary, syntax and rules of translation from native language to math and from math to native language. Some children have difficulty in mathematics because of language difficulties. Most children have difficulty with word problems. In this workshop, the participants will learn how to teach effectively and efficiently this language and help students become proficient in problem solving, particularly, word problems. 

Learning Problems in Mathematics (including dyscalculia)
For special education and regular education teachers 
May 15, 2020
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning.  

Standards of Mathematics Practice:
Implementing Common Core State Standards in Mathematics
For K through grade 11 teachers (regular and special educators)
June 12, 2020
CCSS-M advocates curriculum standards in mathematics from K through Algebra II. However, to achieve these standards, teachers need to change their mind-sets about nature of mathematics content; every mathematics idea has its linguistic, conceptual and procedural components. Most importantly, these standards cannot be achieved without change in pedagogy-language used, questions asked and models used by teachers to understand and teach mathematics ideas. Therefore, framers of CCSS-M have suggested eight Standards of Mathematics Practice (SMP). In this workshop, we take examples from K through high school to demonstrate these instructional standards with specific examples from CCSS-M content standards. For registration, PDPs, Parking, and other information, please  Contact:
Anne Miller:  508 620 1220
Continuing Education Department
Framingham State University
Framingham, MA 01701
Dyscalculia and Other Mathematics Learning Difficulties