Teaching mathematics as second language 


Teaching mathematics as second language 

TEACHING MATHEMATICS AS A SECOND LANGUAGE MATHEMATICS AS A SECOND LANGUAGE A ONE-DAY WORKSHOP  BY MAHESH C. SHARMA  Professor Emeritus and Past-President, Cambridge College President, Center for Teaching/Learning of Mathematics Framingham, MA  April 12, 2019 Organized by Continuing Professional Education  Framingham State University Framingham, MA Mathematics As a Second Language God wrote the universe in the language of mathematics Galileo Mathematics is truly a second language for all children. It is a complex language. It has its own alphabet, symbols, vocabulary, syntax, and grammar. It has its own structure.  Numeric and operational symbols are its alphabet; number and symbol combinations are its words; and equations and mathematical expressions are the sentences of this language. Acquiring proficiency in mathematics and solving mathematics word problems means learning this language well. For all grade levels This workshop will cover:
  • Building a Mathematics Vocabulary
  • Use of Syntax and Voice in Mathematics Language
  • The Role of Language in Developing Conceptual Schemas
  • Developing Translation Ability—Solving Word Problems
  • The Role of Questions in Acquiring Mathematics Language
  • Understanding Instructions to Mathematics Problems
  • Teaching Mathematics Language
In studying the nature of mathematics disabilities, professionals have looked at the problem from several different perspectives. Psychologists and neuro-psychologists have examined the nature of mathematics learning problems from the perspective of the learner characteristics that relate to the underlying processes and mechanisms involved in mathematics skills.  Special educators and teachers have looked at the problem from the perspective of making modifications in teaching based on the nature of the disability. On the other hand, mathematics educators have looked for causes of children’s difficulties by focusing on the nature of the mathematics content itself. To improve mathematics instruction and learning and to address children’s difficulties in learning and achieving in mathematics, we need to consider and integrate what we know from the following:
  • Thenature of mathematics content and learning,
  • The learner characteristics and skill sets,
  • The teaching models those are effective for particular content and skills for particular groups of children.
A considerable amount of research in the area of mathematics learning disability has been conducted by extending the hypotheses and results from the field of reading and reading disability.  The reason for this is found in the key similarities that exist between reading and mathematics learning processes.  In particular, the similarities are in
  1. a) Symbols—in both reading and mathematics, children must recognize and comprehend the message being conveyed by the words or numerical and operation symbols;
  2. b) Vocabulary—for children to understand the symbols in both reading and mathematics, it is necessary to learn the vocabulary associated with each area.  In effect, they must develop the language whether it is mathematical language or native language;
  3. c) Skill hierarchy—both reading and mathematics contain hierarchies of skills.  Children show progress to attain mastery in these skills through several intermediate stages: intuitive, concrete, pictorial, abstract, applications, and communication (Sharma, 1979a);
  4. d) Readiness for learning—both reading and computation skills have prerequisite skills.  Readiness for certain mathematical skills can be viewed from at least four theoretical points of view—cognitive, neurological, instructional, and ontological;
  5. e) Sensory processes—since children must rely on input in both mathematics and reading, it is important that one pays attention to the modalities (visual, auditory, and tactile) of information processing;
  6. f) Decoding—decoding and encoding processes are crucial to a child’s success in mathematics and reading. When children encounter an event involving numerical quantities or spatial information, they must be able to express (decode) these numerical and spatial relationships in symbolic notations; and
  7. g) Comprehension—as in reading, the four factors—word recognition, comprehension, rate and accuracy—are important, similarly, in mathematics, the four factors of basic facts – recognition, comprehension, rate and accuracy – constitute good computational ability.
These and other similarities between reading and mathematics have given rise to comparable postulates in both disciplines. The common elements can help us understand some of the issues in mathematics disabilities. However, to assume that the same processes underlie the acquisition of reading skills and learning mathematics concepts, skills, and processes would be erroneous. There are many substantial differences. The differences lie in the nature of mathematics content, concepts, and in the diversity of skills and procedures involved in learning mathematics. To understand the nature of learning problems in mathematics means understanding the nature of these processes.
  • Linguistic component,
  • Conceptual component, and
  • Procedural component
MATHEMATICS         —->                           ENGLISH (Forming number stories from mathematics expressions.) ENGLISH                     —->                        MATHEMATICS (Translating word problems into mathematical expressions.) In relating any two languages, it is important to understand the interplay of three elements:
  1. Vocabulary{words, terms, and phrases, such as: multiplication, product, sum, quotients, least common denominator, rational number, proportional reasoning, etc.} and Symbols {= -, x, +, %, ( ), etc.}
To develop vocabulary, the teacher should, on one side of the board develop and define, by the help of students, terms when talking about vocabulary: For example:  Product Product:The outcome/result of a multiplication operation. What is the product of 3 and 4? This is a mathematical sentence written in English language.  Now, students translate it into mathematics symbol form: 3×4, 3Ÿ4, 3(4), (3)4, (3)(4). Sum: The outcome/result of an addition operation.  What is the sum of 2 and 4? 2 + 4. Rational number: What is a rational number? A rational numberis a numberthat can be written as a ratio: a/b, where aand bare integers(a whole number, its opposite, and 0), b ≠ 0, and and bare relatively prime(the greatest common factorof aand bis 1). Note 1: The number of terms involved in the definition of rational numbers. The concept of rational numbers will not be understood if the phrases and terms involved in the definition are not understood and mastered. Note 2: The words, terms, and expressions that connote the mathematics ideas behind them do not help children to remember the mathematics idea. The words, terms, expressions, and symbols are language containers for concepts.
  • Syntax: The order of wordsor order of operationsused in mathematical expressions (e.g., difference of a and b (a –b) is not the same as difference of b and a (b – a)).
Note: The syntax in mathematics language is governed by strict rules. For example, How would you read?  27 ÷ 3 The expression is written as “3, procedural division operation sign, and then 27,” but is read as “27 divided by 3,” “3 divides into 27,”  “how many groups of 3 are there in 27?” “How many 3s can fit into 27?” Many students misread and translate division problems incorrectly. can also be written as: 27 ÷ 3 and is read as: “27 divided by 3” and as a quotient, it will be read as: “twenty-seven thirds.” And, as a ratio, it will be read as: “ratio and 27 and 3.”
  • Translation: To conceptualize mathematics ideas and solving word problems, students need competence in “two-way” translation: English to Mathematics as well as Mathematics to English. In learning and teaching mathematics we are constantly moving from one language to the other. For example: If we have:  17 − 9 = ?  How can we write this as an English sentence? Teacher should generate one or two statements and then, ask students to write more statements with their partners.
(a) 17 subtract 9 equals what? Let students work together and then share out. Record them on the board. Discuss them and then reorganize them according to the pattern. For example: (b) 17 minus 9 equals what? (c) 17 take away 9 equals what? (d) 17 decreased by 9 equals what? Here the syntax is direct: number, operation, number. (e)If 9 is taken away from 17, what do you have? (f) 9 less than 17 is what? (g) Subtract 9 from 17 is what? Or, What is when we subtract 9 from 17? Here the syntax is indirect: the order of numbers is changed without changing the meaning.  The placement of the question does not change the syntax of mathematics operation.  (h) What is the difference between 17 and 9? (i)17 is how much more than 9? (j) 9 is how much less than 17? Here the syntax is more complex than the other two patterns. It is the beginning of algebraic reasoning. Similarly, the teacher should spend some time writing statements for addition, multiplication, and division expressions. Now let’s do the reverse translation – let’s go from English language to mathematical expressions. Example: 8 decreased by 5 translates into 8 −5 as an arithmetical expression. Example: 9 increased by 7 translates to 9 + 7. Example: 12 increased by 6 translates to 12 + 6 At this point, the teacher should define terms such as: Arithmetical expression, numerical expression, algebraic expression, equation, inequality,etc.  On one side of the board, with the help of students, develop and define:  Arithmetical expressionis a mathematical expression that involves numbers, arithmetic operations, and arithmetic symbols.  It becomes an algebraic expressionwhen the expression involves numbers, operations, symbols and variables. Have students try the next few with a partner and then share back out to the class.  Collect them and then discuss the relationship between words and mathematics counter parts.  Now consider, Example: 8 less than 15 translates to 15 –8. This one might be hard for some students to see. Devote some time on the order of words.  Relate it back to the list previously made that showed the different orders and wordings.  Also bring attention to which number is being operated on, and which number is being operated at by what operation. In the expression: “8 less than 15” 15 (operand) is being operated on by the number 8 (operator) and the operationis subtraction. So, we begin with the 15 (operand, it will come first) and the 8 is the operator so 8 is subtracted from 15. Example: 2 more than 32 translates into 23 + 2.  Here: 32 (operand), 2 (operator), and addition (operation). Example:15 decreased by xtranslates into 15− x.   This expression introduces a variable and therefore becomes an Algebraic Expression. A variable (we defined already) is a symbol that may take different values or roles according to the situation/context. Now the teacher, shuld begin with some concrete examples. For example, a concrete example of a variable that might help students is: Who sits in this seat?  (Point to a student chair.)   Today it is student J’s seat, but next week it might be student M and next month it might be student A.  The person to whom this seat belongs varies on the day or week or month. However, who sits in this seat? Point to the teacher’s chair. Only the teacher sits on it.  No one else is allowed to sit in that chair. Therefore, this chair is not variable but constant. Once the idea of variables and the translation process is understood, one can introduce the concept of equations. Example:  Write, “5 more than 2x is decreased by 3 less than twice a number” into a mathematics expression. This is an example of a compound expression.  Two algebraic expressions are connected by the phrase “decreased by.”  Let us take each expression separately.  The expression “5 more than 2x” is: 2x + 5, and the expression “3 less than twice a number” is: 2y −3, where yis the number. Combining the two, we have: (2x + 5) – (2y – 3).   An equation is also a compound statement.  It joins two algebraic expressions.  A mathematics expression is called an equation when two algebraic expressions are equated. 
  1. Language Difficulties, Dyslexia, and Mathematics Learning
The nature of reading and language disability sheds light on a child’s mathematics learning. Some children are late in learning to read and have persistent difficulty in remembering spellings in contexts without any issues of intelligence or lack of opportunity to practice.  Some of them experience other difficulties also—long struggles over learning to tell time (on non-digital clock), uncertainty over left and right, confusion over times and dates, inability to recite without stumbling, any but the easiest arithmetic tables. These issues spill over into the area of mathematics.
  1. Dyslexia
A child of average ability who is late at learning to read and has special difficulty over spelling, and if, in addition, shows confusion over getting things into spatial and temporal order, whether in language or arithmetic, has a distinctive disability called dyslexia. A person with this disability (or group of disabilities) has difficulty in receiving, comprehending, and/or producing language. Dyslexia, which affects three to four boys to every girl, has been associated with slow speech development, speech and language difficulties, delay in motor development, sequencing problems (usually the inability to remember the days of the week or the months of year is noticeable), impairments in temporal or spatial awareness, and visual perceptual deficits.          Most dyslexics are late in learning to read and have considerable difficulty in learning to spell.  Most dyslexics remain slow readers, and although some speeding up is possible, any task, which calls for the processing of symbolic material or speed, is likely to cause them trouble. Dyslexics have difficulty with phonology—with the remembering and ordering of speech sounds.  Problems over left and right also persist in their work. Learning mathematics for a dyslexic is just like learning Latin as a second language (L2). Since, it is a dead language, it is mostly learned by writing and in written form. It is not orally supported, as it is not spoken outside the classroom.  Research has shown that individuals with dyslexia have difficulty in reading not only in their native language but also in a second language (L2). The considered L2, however, has always been a language acquired through exposure to both written and oral forms. A recent study examined the case of Italian adolescents reading in Latin as an L2, which is the special case of a dead language with very limited use of orality. As the learning of Latin is mainly based on the acquisition of grammar, this study also examined the relationship between grammatical proficiency and reading ability in Latin. Results suggested that, compared with control peers, students with dyslexia had difficulty in reading words and non-words in Latin. Interestingly, in spite of Latin being learnt mainly through written language, the extent of their difficulty was no larger than they encountered when reading in their native language. Also, despite the fact that students with dyslexia showed relatively less severe difficulties with Latin grammar (as compared to reading), this did not support them when reading Latin words, unlike typical readers. The theoretical and educational implications of these results of this study are profound when it comes to learning mathematics as a second language. Mathematics calls for many different and some similar kinds of abilities. In general, dyslexics tend to be slow at certain basic aspects of mathematics—learning and recalling arithmetic facts, particularly multiplication tables, adding up columns of figures, etc. –but once they have understood the symbols, they may be quite creative. At the same time, based on their profile, some dyslexics can be quite strong in certain aspects of mathematics.  It is important to distinguish dyslexic children form slow learning children. For individuals with dyslexia, learning mathematical concepts and vocabulary and the ability to process and use mathematical symbols can be impeded by problems similar to those that interfered with their acquisition of the written language. Too frequently and too readily, individuals with dyslexia who have difficulty with mathematics are misdiagnosed as having dyscalculia – literally trouble with numberness, knowing number relationships, calculating, and a neurologically based disability. Around 40 percent of dyslexics have difficulty with basic mathematics. Some dyslexics are only numerically dyslexic—having difficulty only in the numerical aspects of mathematics, but this can also be most embarrassing. Difficulties with math for dyslexics can be identified by the following symptoms:
  • The dyslexic may have a problem with numbers and calculations involving adding, subtracting, and timetables.
  • He may be confused by similar—looking mathematical signs: + and ×; –, :, ¸and = ; < (less than) and > (greater than).
  • He may not grasp that the words ‘difference’, ‘reduce’ ‘take away’ and ‘minus’ all suggest ‘subtraction’.
  • He may understand the term ‘adding’, yet be confused if asked to ‘find the total or sum’.
  • The dyslexic may reverse numbers, and read or write 17 for 71, or 2/3 as 3/2.
  • He may transpose numbers i.e., 752 to 572 or some other arrangement of digits.
  • He may have a difficulty with mental arithmetic.
  • He may have a problem with telling the time.
Individuals with dyslexia may have problems with the language of mathematics and the concepts associated with it. These include spatial and quantitative references such as before, after, between, one more than, and one less than. Mathematical terms such as numerator and denominator, prime numbers and prime factors, and carrying and borrowing may also be challenging.          In many ways, some of the symptoms of dyscalculia closely parallel the behaviors exhibited by students with language dysfunctions; therefore, many neurologists believe that dyscalculia does not exist as a separate dysfunction but is a manifestation of a brain lesion which is causing language and mathematics dysfunctions simultaneously. This view has some relevance, as mathematics is also a language—the language of quantity and space. Some mathematics problems are extensions of language difficulties such as alexia—an inability to read, and/or agraphia—an inability to write.  For example, half of the students with difficulties in mathematics also have difficulties in spelling. The terms dyscalculia and acalculia have been used in the literature interchangeably.  Dyscalculia is the lack of or delay in the development of numberness, number relationships, calculations and other related mathematics difficulties, whereas acalculia is the loss of these abilities because of insult or injury to some specific part or regions of the brain. We will use the term dyscalculia referring to both, except when the problem relates to pure dyscalculia—difficulty with numberness, number relationships and outcome of numerical operations. In contrast to what neurologists think, several recent researchers have argued that dyscalculia should be considered as separate from language related problems and different from the more general category of learning disabilities and even from learning disabilities in mathematics. In other words, dyscalculia or acalculia can exist independently of all other learning problems in mathematics and language.  That is, a number of individuals may manifest dyscalculia and no other learning disabilities.  Some others may exhibit difficulty in mathematics and even dyscalculia or acalculia in the presence of dyslexia. Not all dyslexics have problems with mathematics, and not all dyscalculics have difficulty with reading and other language skills. Of course, there are many differences based on strengths and weaknesses of dyslexics and dyscalculics. Pure dyscalculia (the difficulty in conceptualizing number, number relationships, and outcome of numerical operations) and dyslexia are independent. Dyscalculia and dyslexia mutually influence each other when we consider dyscalculia as poor performance in mathematics including the failure of the number and calculation mechanisms. For many dyslexics, the difficulties that affect their reading and spelling also cause problems with mathematics. There is some correlational evidence emerging between the coincidence of dyslexia and dyscalculia. The International Dyslexia Association has suggested that more than 40% of dyslexics have some difficulty with numbers or number relationships. In several studies, they found that almost 51% of dyscalculics also show signs of dyslexia. Of those whom do not have mathematics difficulties, about 11% excelled in mathematics.  The remaining have the same mathematical abilities as those who do not have learning difficulties. Arithmetic and mathematics learning deficits can be caused by a variety of factors, sometimes because of reading difficulties and at other times with no connection to reading difficulties. Children with specific arithmetic difficulties and children with combined arithmetic-and-reading difficulties represent two different underachievement subtypes whose problems may be underpinned by qualitatively different cognitive and neuropsychological deficits. This is consistent with evidence from clinical and experimental studies suggesting that children of normal intelligence who experience difficulties with arithmetic can be divided into specific and combined subtypes. The existence of a group of arithmetic-and-reading difficulty individuals shows that some arithmetic difficulties result from difficulties with reading. On the other hand, a number of children with normal reading scores obtain low scores on the arithmetic test, which shows that not all arithmetic difficulties can be attributed to a general deficit in language-related processing.  In addition, there exists a substantial group of poor readers of normal intelligence whose performance on the arithmetic test is higher. Therefore, mathematics learning problems in general and even the types of problems exhibited by dyscalculics, acalculics, and or dyslexics are not homogeneous in nature. Because of the heterogeneous nature of mathematics learning problems, language related problems in mathematics fall into three major domains:
  1. Mathematics problems related to and originating from language processing difficulties,
  2. Mathematics problems that have the same basis as the reading problems because of the underlying learning mechanism, such as sequencing, visual-perceptual integration, working memory, organization, spatial orientation, etc.
  3. Mathematics problems that originate from the combination of language and reasoning deficits.
III. Mathematics Problems Related to Language  Since formal and informal language plays varying roles in learning concepts and applying mathematics, some of the difficulties emanate from the interaction of the systems responsible for number, calculations, procedures, and language. In the case of language, it may be vocabulary, syntax, the ability to translate from mathematics to language and from language to mathematics, and reading. Many children with language related problems do not have problems in mathematics. However, since some of the same prerequisite skills are involved in both language acquisition and mathematics learning–at least in the early years, the coincidence of dyslexia and dyscalculia is not uncommon.  Many dyslexics can solve computational and spatial problems easily. They have difficulty with only language related problems in mathematics as they do not have the facility to receive, comprehend, and produce the quantitative and spatial language (words, symbols and expressions) properly. They are not able to solve problems that have heavy language involvement. Research studies have shown that many elementary-age students who perform poorly in mathematics also have basic language deficits. Dyslexics’ language related problems in mathematics are of two kinds: primary and secondary.  Primary problems are directly contributed to mathematics by language difficulties in reading, spelling, etc.  Dyslexic children sometimes also manifest problems in arithmetic that are of a secondary nature.  That is, since the dyslexic child is provided extra instructional experiences in reading and language areas, his experiences in and exposure to arithmetic may be limited. As a result, he may begin to do poorly in arithmetic. Consequently, in many cases, once the child’s primary problem in reading and other language areas is remedied, he may begin to do well in arithmetic unless the failure in arithmetic has by then affected the child’s self-esteem.  Then the problem translates into mathematics anxiety. Mathematics anxiety is a person’s negative emotional response to consistent failure in mathematics.  The unsuccessful experiences in mathematics create negative feelings for anything mathematical—from mild distaste and aversion to strong hatred for mathematics.  The symptoms of this anxiety are the same as symptoms for general anxiety—fidgetiness, dilation of pupil, restlessness, sweating of palms, etc. Mathematics anxiety, up to about age ten, is only a symptoms of mathematics difficulties or disabilities. Up to this point, it may not have been internalized, and as a result, it may not have yet affected the child’s self-esteem.  Once anxiety has been internalized, it begins to be a causative factor.   Because of mathematics anxiety, the person avoids mathematics and does not do well. Mathematics anxiety is a problem of secondary type. Both dyscalculics and dyslexics can suffer from it.  Sometimes even a student without any disability may do poorly in arithmetic and mathematics and may develop anxiety. The nature of math anxiety—global or specific, depends on whether it has an emotional or cognitive base. If the basis of mathematics anxiety is emotional, we will have to significantly improve that child’s self-esteem by providing successful experiences so that he may begin to participate in quantitative experiences and do better in arithmetic. For some students, their mathematics disability or difficulty is driven by problems with language.  The very same difficulties that the child experiences in reading and other language concepts interfere in learning mathematical concepts. In mathematics, however, the language problems are confounded by the inherently difficult terminology, some of which children hear nowhere outside of the mathematics classroom, and the language based logic of mathematics. The way that numbers are represented linguistically is significant. For example, twelve hundred and one thousand two hundred appear to be handled differently in the brain. Twelve hundred is understood as the product of twelve and a hundred (whether consciously or not), one thousand two hundred as the sum of one thousand and two hundred. Asking for the solution to fifteen hundred plus one hundred most frequently brings the response sixteen hundred, but one thousand five hundred plus one hundred is one thousand six hundred. Although the underlying numerical values are the same, the brain appears to process the numbers differently according to their linguistic representation. Unlike language, where one engages only with words and their combinations, in the case of mathematics one uses symbols, words, and their combinations. In mathematics, words appear as operators and as operand. This differentiation has a major impact on a child’s ability to learn mathematics.          Operation (mathematical symbols +, —, =, ´, ÷, ( ),  <, £, , etc.): Each symbol is packed with concept and meaning. In mathematics, both symbols and words act as operators (mathematical operations) that are to be performed on quantity, words, and space. Examples: Solve for x: 3x + 7 = 11. Find the product of .34 and 67. Divide the circle into halves. The area of the circle is given by: A = , where r is the radius of the circle.          Operand:Number (quantity) and shapes are the arguments and objects of the operators. Example: “Find the square root of 144.”  “Find the product of 23 and 15.”  “Circle the figure that represents half of the circle.”  “Circumscribe the square in the diagram.”  “Integrate the function sin (3x + 5).” To complicate matters, sometimes there is a combination of the two.  For example, find the square root of the sum of 19 and 6. Operator: Mathematical symbols andnumbers, both can act as operators.  Fro example, 3 increased by 7, here number ‘3’ is the operand, ‘+ or increased by,’ is an operation, and 10 is 1o an operator. Difficulty in understanding the operatoroperation, and operandcan be a problem in learning mathematics that is different from the simple but related problems of the dyslexic. For many students, it is difficult to separate the operator, operation, and the operand.  For example, when it comes to integer problems such as: -3 +7, they cannot determine the roles of signs  ‘-’ and ‘+’ before 7 as operator, operation, or operand. Students face the same issues when several operations are involved in an expression or an equation.  For example, in simplifying the expression: 4y(-2y+3y), using the distributive property of multiplication over addition, many students have difficulty deciding which signs are being used as operands, operations, or operators. Examples of this type abound in mathematics, more once operations have been introduced.
  1. Lexical Entries (Naming, Labeling and Language Containers)
One of the major characteristics of dyslexics that they share with dyscalculics is the inability to recall arithmetic facts at an automatized level because of a smaller number of lexical entries for numbers.  The reading skill is acquired by creating images of letters, words, combination of words, objects, and ideas in our minds.  They are like names and labels or language containers for words, numbers, combination of words and numbers, combination of numbers (language containers), and information and concepts.  This naming process facilitates the recall of immediate knowledge. Just as it is possible to build lexical entries for words and word-parts as well as for single letters, so, too, it seems reasonable to suppose that there are lexical entries, which represent numbers, combinations of numbers (facts), and symbols. It is becoming evident that for number conceptualization of numbers up to 9 or 10, we form visual clusters (lexical entries for numbers—numberness) in our brain. It is reasonable to assume that there are lexical entries for arithmetic facts also. When the number of lexical entries is small, dyslexics face a major problem of labeling (tasks of visual and auditory discrimination). In the absence of lexical entries, the student constructs the facts each time he encounters them. Most dyslexics are less proficient than mental age matched controls on tests of object naming and slower on rapid automatized naming of visual and verbal stimuli. The reading disabled children are less accurate in labeling the objects and have particular difficulty with low frequency and polysyllabic words. For example, young dyslexic children, aged between 7 and 9 years, are no different from the matched controls in tasks that do not require labeling.  They are appreciably weaker than the controls in word analysis tasks and labeling, and in some of these tasks, they are weaker than the poor readers believed not to be dyslexics.  This disability affects mathematics learning. They may understand the logic of arithmetic operations, but they show difficulty and inability to perform simple calculations because they cannot recall the needed facts automatically. This also applies to naming geometric shapes and describing terms in arithmetic and mathematics. It seems dyslexics cannot rapidly access verbal labels and arithmetic facts as they have problems in retrieval from long-term memory or even from working memory. They have difficulty holding intermediate steps in calculations in their minds as they have problems with short-term memory function and with the retrieval of stored memory traces. There seems to exist a strong association between mathematics performance and response time on rapid automatized naming. Because dyslexics have smaller number of lexical entries, they have problems in most common areas of arithmetic: difficulty in memorizing and recalling simple addition and subtraction facts, difficulty in learning multiplication tables by rote, and difficulty arising from uncertainty over sequencing and direction both in space and time. This inability to retain complex information in the memory system over time gets in the way of learning mathematics normally.  Accordingly, one would expect a lesser range of immediate knowledge of facts and information by memory on the part of a dyslexic person. They have fewer number facts available to them than do non-dyslexics: thus, if the question is, “What is 6 ×7?” a non-dyslexic of suitable age and ability can respond “42” in one response whereas many dyslexics can reach the answer only by working it out. Most non-dyslexics are aware intuitively of the difference between situations when they can instantly give the answer (72 – 9 = 63 or 72 ÷ 9 = 8) to a calculation and when they need to work this out (127 × 23 = 2921). The non-dyslexic will learn, for instance, that 8 ×7 = 56 after a relatively small number of exposures to stimuli whereas the dyslexic, because of his slowness at naming and labeling, will be unable to make use of the presented learning opportunities.  Consequently, he needs more exposures to stimuli before these stimuli take on symbolic significance. The dyslexic’s slowness results in longer naming times, which has consequences for the recall of digits, facts, terms, formulas and therefore creates difficulties in learning and using calculation systems. The development of lexical entries is dependent on two factors: the underlying prerequisite skills necessary and the amount, type, and quantity and quality of early training in number conceptualization. This is expected to vary considerably among individuals with dyslexia since the amount and type of compensatory early training and the level of mastery of prerequisite skills are likely to be different. This does not mean dyslexics or dyscalculics cannot learn arithmetic and other mathematics facts and recall them fast; they just need special methods and efficient strategies. Ordinarily, for automaticity and faster recall we focus on rehearsal—increasing the number and frequency of exposures.  The non-dyslexic will learn, for instance, that 6 ×7 = 42 and automatize this fact, whereas the dyslexics, because of their slowness at naming will construct the fact and will have to pay attention to other intermediate information. This construction may take the form of laborious counting or recall of the sequence: 6, 12, 18, 24, 30, 36, and 42. The construction of facts distracts and dilutes the cumulative effect of the exposure to stimuli.  The habits of construction of facts carry into higher mathematics.  When they are older, these children have difficulty memorizing even the simplest of formulas in algebra and geometry. The longer naming time problem, if not treated at the appropriate age, has consequences later on for the recall of digits and facts, a difficulty with calculations, and generating immediate knowledge for use in problem solving. The extent to which number-sense and mathematical conceptualization are impaired and these facts (lexical entries) and appropriate procedures are missing can be expected to vary considerably. Many of the problems that dyslexics face in arithmetic can be put right with proper teaching; and in particular, the use of concrete instructional materials may help to generate the appropriate lexical entries.  Just as a multi-sensory and special approaches, such as the Orton-Gillingham reading program, help in the building of lexical entries for letters, combination of letters, and words, and teach efficient strategies so can the visible and tangible presence of say, seven objects in a visual cluster, the mark on paper,‘7’, the written letters ‘seven’ and the sound ‘seven’ jointly contribute to the formation of the same lexical entry for the number 7. In some cases, better and extra exposures and more meaningful stimuli that match students’ mathematics learning personalities are needed.  For example, many dyslexics may also have sequencing difficulty, finding addition or subtraction facts by sequential ‘counting up’ and ‘counting down’ are difficult for them. Memorizing the tables in the usual sequential order and teaching them in the usual way may not work for them. Simply providing exposure to this counting process will not result in a successful experience. We have found that breaking the 100 (10 x 10) multiplication facts grid into meaningful chunks based on clearly identified patterns helps dyslexics to learn multiplication tables faster and to have better recall (How to Teach Arithmetic Facts Easily,Sharma 2005). For instance, the two hundred facts of addition and multiplication are reduced to almost half once we introduce the idea of commutative property of addition(If I know 6 + 7, then automatically I know 7 + 6.) and multiplication (If I know 6 ×7, then I automatically know 7 ×6.).  Similarly, if we know 10 facts of doubles (1 + 1, 2 + 2,  …, 9 + 9, and 10 + 10), then we know 18 facts of near doubles (1 + 2, 2 + 1, 2 + 3, 3 + 2, …, 8 + 9, 9 + 8, 10 + 9, 9 + 10); if we know the sum of the two numbers that make 10, then we know the sums that are near tens, and so on.  With these patterns and efficient strategies, we have found almost all children are able to memorize arithmetic facts with fluency.
  1. Mathematics Disability Subtypes 
Acquiring the skill of reading, in itself, is a complex matter. The reading in a specialized content area such as mathematics, is even more demanding. Many children have difficulty mastering it. However, extensive research in this area has clearly identified a core set of skills. For example, phonological decoding deficits—processes in which grapheme-to-phoneme conversion rules are applied to ‘sound out’ a word’s spoken representation, have been identified as core symptoms of reading disabilities (RD). Reading aloud of novel words is achieved by phonological decoding. These core deficits are evident across the various subtypes of RD that have been described. Understanding these core deficits leads diagnosticians to identify RD and special educators to provide effective remediation.   Understanding RD helps curriculum planners and classroom teachers to design programs and provide preventive instruction. The reason it is possible to identify a set of core skills and related symptoms, as an explanation of RD is that reading skills are well defined. We know what constitutes fluent reading and when to expect its presence with mastery. Unfortunately, no such core symptoms and skills in the case of mathematics disabilities (MD) have yet been identified, as research on mathematics disability is less well developed than RD research. There are several reasons for the absence of defining core skills in order to diagnose MD.
  • First, we do not have any agreement about what constitutes the core skills in mathematics. There is no one particular skill that is at the core of every mathematical operation. Despite a general agreement on the wider contours of mathematics concepts to be mastered by children in elementary school, there is no agreement on the specific nature and type of skills, the level of mastery and fluency, and the timetable for achieving the skills for mathematics achievement.
  • Second, because of the cumulative nature of mathematics, we are not able to identify the core skills. Unlike the key basic processes that underlie reading achievement, mathematical achievement is cumulative and comprehensive throughout and beyond the elementary school years, with quantitative and qualitative changes occurring within and across grade levels. Almost 30% of curriculum material at each grade is new or expanded substantially.
  • Third, after acquisition of the key skill (number conceptualization), learning and applying mathematics depend on a diverse set of skills.  These skills are spread over several different domains of functions.
The mastery of arithmetic and mathematics skills, concepts, and procedures and poor math achievement, therefore, is linked to several factors and skills: language (native and mathematics), memory, visuospatial skills, affective, prerequisite skills such as: sequencing, pattern recognition, and/or executive skills. For reasons mentioned earlier, mathematics disability (MD) is emerging as a collection of subtypes that cluster around several problem areas: (a) related to number, (b) related to language, (c) conceptual/ procedural, (d) visuo-spatial, (d) related to executive function, and (e) affective/behavioral. Because of the complexity of mathematics skills and their varied nature, it is highly unlikely that MD subtypes will share a unifying core deficit. As mentioned earlier, phonological processing/lexical labeling, have been associated with computational math skills in children, in earlier grades, with poor math achievement. However, additional factors, other than reading, also influence MD outcome. For example, language-specific difficulties in children with MD and RD have been reported relative to children with MD only. Many children with only MD outperform their peers with both MD and RD on exact arithmetic tasks, whereas both groups demonstrate comparable difficulty on estimation tasks. Math performance levels are also linked to executive function skills (working memoryinhibitionorganization, and flexibility of thought).  Different components and aspects of executive functions appear to account for some of the variability in children’s math performance levels, with strong contributions of poor inhibition and poor working memory, particularly as it relates to visualization. Visualization takes place in the working memory. Therefore, working memory plays a critical role in mathematics learning and performance. For example, in working out a problem, a simple calculation, 12 × 8, may be involved as a subsidiary problem that needs to be resolved mentally before we can solve the main problem.  If the student knows the fact, there is no digression. When he does not have the fact automatized, he has to construct it.  If he must construct, he digresses. That construction takes place in the working memory space or on paper.  To do so in the mind, the student has to keep several pieces of information in his mind: 10 × 8 is 80 and 2 × 8 is 16, so 12 × 8 is 80 plus 16, which is equal to 96.  Therefore, 12 × 8 = 96.  The student has to mentally manipulate this information, which requires visualization. Therefore, working memory deficits in children with learning disabilities, including children with reading or math difficulties, also exist. There is consistency across reports that both reading and executive skills are associated with math achievement levels. However, there is not enough information to explain the extent to which these cognitive and neuro-psychological correlates underlie one or more specific MD subtypes.
  1. Mathematics Problems Related to Reading
One of the obvious connections between dyslexia and mathematics difficulties is reading. Yet, many children with reading problems may not have problems in acquiring mathematical concepts. This is particularly so with straight computational and procedural aspects of mathematics and where the instructions are straightforward. However, mathematics problems involving language, particularly reading, pose problems for them, for they may not have the facility to comprehend the words and expressions properly. For this reason, some children with poor reading skills are also below average in arithmetic skills. Below-average performance can exist in different areas, from very simple language-based symbolic conceptualization to complex problem solving such as word problems, making conjectures, writing definitions and proofs in algebra and geometry, as well as communicating mathematics. They frequently perform below average on the arithmetic subtests of Wechsler Intelligence Scale for Children (WISC). When developmental dyscalculics who were good readers and those who were poor readers are compared, the good readers misread signs, align rows and columns inappropriately, and miss entire calculation steps. The poor readers avoid unfamiliar words, word problems, and operations; they have problems with tables and in recalling appropriate calculation procedures. We find specific areas of difficulties in which failure in mathematics is coupled with reading disability. These students have
  • Difficulty with the vocabulary and terminology of mathematics, understanding directions and explanations or translating word problems,
  • Difficulty with irrelevant information included in the word problem or out of sequence information,
  • Trouble learning or recalling concepts, definitions and meanings of abstract terms,
  • Difficulty reading texts to direct their own learning and communicating mathematics, including asking and answering questions,
  • Lack of information concerning mathematical facts due to the failure of the child to make normal school progress (since, the child with reading problem may be taken out of the mainstream class or placed in special classes where the emphasis is on reading progress, the child may not get enough instruction in mathematics and therefore has limited exposure to mathematics), and
  • Emotional blocking due originally to reading disability but eventually extended to mathematics.
The demands on reading in mathematics extend far beyond story problems. They include reading equations, mathematics conceptualization, definitions, etc. Research in the area of problem solving demonstrates that there is clear relationship between reading of algebraic symbols, instructions, and concepts and performance accuracy.  In our remedial work in mathematics (Sharma, 1980, 1988 & 2004) with children and adults, we have found that many students do not read and comprehend the vocabulary of algebra, nor do they read and comprehend an expression such as 3x + 7 > 2(x + 5) fluently and accurately. Although the symbols themselves are not phonetic, each symbol does represent a vocabulary word whose meaning must be understood (Lerner, 1993).
  1. Role of Instructions in Mathematics Learning and Problem Solving
Mathematics concepts and problem solving subtests (or even computational subtests) on standardized national, state, local, and classroom tests and examinations contain sentences which must be read, comprehended, and followed by the test taker in answering the problems.  Understanding these instructions is the key to student success on any assessment.  From the outset, one can say that mathematics instructions are easier to understand when one knows the mathematics content, properly.  However, the processes of reading, comprehending, understanding, and executing instructions are the key to answering questions and problem solving. Each problem or a set of problems in a mathematics textbook has a set of instructions written in English but uses special terms and symbols. Many of our students do not read or understand instructions in these computational and/or word problems. They decide what to do from the context or information gleaned about the problem by superficial reading of the problem. Sometimes, it happens because they donot understand the instructions as the words and phrases are not familiar to them. Other times, they can read them, but they do not have the conceptual schemas invoked by these words or phrases. Some times, they do not have the ability to execute them, as they have not mastered the procedure invovled in the problem. And, other times, they do not comprehend them, as the instructions are not clear. Writing effcient and effective instructions and explaining instructions and their role are the mark of a good teacher and a textbook. This does not mean all instructions should be non-technical or overtly simple. It is that classroom instructional strategies should have definite emphasis on developing, explaining, and processes of understaning and executing instructions involved in problem solving. Reading, comprehending, and understanding instructions, and then executing these insructions properly requires that, in their lessons, teachers emphasize and help students master the three major components of a mathematics idea. Students need to have:
  • The mastery of mathematics language in order to be able to read, comprehend and conceptualize the problem (The use of language is to create ideas, receive and communicate ideas);
  • The presence of and facility in recognizing, and relating the language to appropriate conceptual schemas(arithmetical, algebraical, geometrical—definitions, diagrams, formulas, and relationship); and,
  • The ability and facility in executing appropriate procedures(involved explicitly and implicitly in the language of the instructions and in the problem) in an effective and efficient manner.
A mathematical idea is received or constructed by a student through language, explorations using concrete and visual (pictorial—iconic and diagrams) models, or through discussions and problem solving.Solving problems and discussions solidify new and old learning and help integrate them. Teacher’s instructional approach, langauge usage, and setting of insructional activities facilitate and accelerate this learning process.  The quality of language and questions used by the teacher are the most important factors in learning mathematics and problem solving. Language creates language containersin the student’s mind for mathematics ideas. These, in turn, help create and hold conceptual schemasfor these mathematics ideas. Conceptual schemas and language help us derive, construct, develop procedures, and the ability to apply and execute these procedures and algorithms. Instuctions are the conencting links/bridges between these different components of mathematics ideas. They play an important role in learning mathematics and solving problems. Understanding and mastery in execution of instruction is dependent, first, on the emphasis and distribution of these components in the lesson and then on the clarity of the given instructions in the problem. In the absence of clear instructions or lack of understanding of the instructions, students often ask teachers and tutors: What am I supposed to do here?”  How do I solve this problem?”  What formula or operation should I use?”  Can you tell me whether it is ‘division’ or ‘multiplicaiotn’?” Etc. Others ask:  “Why do we have these instructions in a mathematics problem? Just tell us straight what to do?” “Why do they hide the instruction in the problem, I can’t even find them?” “Why can’t they make them clear?” “Why can’t you just tell us what to do?”   Many teachers (regular and sepecial educators) and tutors out of exasperation or exisgency provide the formula, operation, or other related information without explaining. These questions sound simple, but many of our students’ mathematics difficulties can be traced to misunderstanding or lack of understanding of these instructions. The casue of their difficulty goes directly to the heart of mathematics teaching. Before students can solve problems, they need to understand the instruction(s); they need to connect the language with the problem.
  1. What is a Mathematical Instruction?
Each mathematics problem is a mathematical expression or a collection of mathematics expressions. A mathematical expressionis a combination ofnumbersand/orvariablesterms(words, phrases) andsymbols(knowns and unknowns, simple and complex) in the form ofexpressions,equationsinequalitiessystemsorformulas.For example, anequationis the outcome of equating two mathematical expressions. Each word problem is interplay between native, academic, and mathematics languages.  In word problems, we have words, phrases, and expressions that provide information and instructions to translate words into mathematics expression(s), construct a mathematical problem, and to solve that problem(s). The words and their combination in mathematics, and in instructions, from the persepctive of their functions, fall in several categories: some words in the instructions are used as:
  • Identifiers (i.e., The shape in the diagram is called…; the digit in the hundered’s place in the number 45,678.12 is ___. );
  • Verbs (i.e., multiply the fractions: ⅘ and½; differentiate the function …;reciprocate the fraction …;find the square root of ….; locate the point √2 on number line; reduce the fraction …. to the lowest term; it implies that 2 is an even number; deduce that every square is a rectangle;  prove that 8 is not a prime number; show that 7 is a prime number; determine the nature and number of factors of square numbers; etc.);
  • Concepts/nouns (i.e., place-value, arithmetic sequnece, multiplication, ratio and proportion, exponential function, addition, etc.);
  • Qualifiers/adjectives(i.e., 123 is a 3-digit number; 24, and 2n are even numbers, where nis an integer; √(n)is an irrational number, for any non-square, positive, whole number number; the equation: y = mx + bis called the slope-intercept form of a linear equation; the least common multiple of 8 and 12 is 24; the greatest common factor of 8 and 12 is 4; y = x2is a continuous function for all x; etc.);
  • Objects/noun (i.e., triangle, quadratic formula, parabola, focus of a conic section, square-root symbol, etc.);
  • Outcome of operations (i.e., sum, difference, product, quotient, ratio, differential coefficient, square root, etc.);
  • Cognitive and mathematics thinking functions (i.e., compare, analyze, relate, recognize the pattern, extend the pattern, make a conjecture, conclude, arrange, organize, focus, visualize, manipulate the information in the mind’s eye, spatial orientation/space organization, logical connectives: all integers…, every squareis…; if and then, if and only if, etc.).
Theverbsand thinking functionsin the problems make demands or give commands to do something. These commands, or requests, are called the “instructions.” Before a student could answer the question(s) in the problem, the student must understand the role, meaning and purpose of these instructions. When a student encounters mathematics problems, including word problems. Thefirst stepis to read the words, terms, and phrases. That is purely a reading skill combined with recognition of mathematics symbols. Success in this step is dependent on reading skills and knowledge of academic language. Some knowledge on the part of mathematics teachers about the key elements of the reading process and manifest difficulties is important, so that a teacher can decide whether she can help the student or the support of reading/special/support teacher is warranted. However, not giving no word problems to solve to students with reading difficutlies is not the answer. Language and reading gets better with support and practice. The second step is to know the meanging and role of all the words in the problem, including logical connectives invovled. This cannot happen without comprehending the conceptual meanings behind them and their interrelationships. This step requires the mastery of academic langauge and mathematics language—particularly the knowledge of related conceptual schemas. For example, multiplication does not just getting the product of two numbers by counting or memorizing.  It means that it is: “repeated addition,” “groups of,” “an array,” or “the area of a rectangle.” After successfully reading, comprehending, and understanding the text of the problem, the third stepis to identify the unknowns and knowns in the problem. What is being asked in the question? This gives rise to the identificaiton or defining of the variable(s). At the elemetntary level, it may involve only the identification of the operation or relationship asked for. The fourth stepis to identify the type and nature of the problem.  For example, some part of the instruction may ask for constructing or articualting a relationship between knowns and unknowns resulting in a formula, equation, inequality, table, pattern, or graph. This may also result a main problem and subsidiary problems. The solution of the subsidiary problems may answer the main question asked in the problem. Other times, the instruction may call for executing a procedure or a formula or forming and solving an equation. The words and symbols or collection of words and symbols in the instruction or the problem invite the student to translate them to develop and form mathematical expressions (e.g., geomerical, algebriac and numerical) and, then to perform operation(s) or action(s) on them—from simple to complex, from single step to multi-steps to answer the question(s) posed in the problem. Many students and teachers may focus only on this last step, by disregardig the earlier steps.  That robs students of forming new concepts and relating new concepts with old concepts.
  1. Why are instructions such a challenge for many students? 
Understanding instructions is learning and mastering the language of mathematics.  Therefore, in any approach to helping students to understand instructions, the emphasis should be on understanding the role of language in mathematics problem solving. In this context, we as mathematics educators have to ask: Could the instructions in the problem be given differently? Better, easier language, or more succinctly?  As in desgining test items, we ask for validity and reliabiity of the content, we need to use the same criteria in the case of writing instructions for problems. A teacher/tutor must always ask: What do the instructions mean for the student in this problem?   Have we achieved that goal? We need to constantly ask our students what do they do and understand when they read an instruction to a problem. And the student has to ask: What are they asking me to do in this problem?  Do I understand the meaning of the words and expressions in the problem? What do I know here?   What do I not know in this problem?   Can I find an entry point to the problem?  But, just focusing on the instructions is not the answer.  We need to go to the root—the development of the language of mathematics during our teaching. Many children do not read the instructions, as they do not understand them. One of the reasons is the language and unfamiliar phrases used in the instruction.  Most students determine such a task based on the context. They think: We have been solving problems like this in this chapter.  This problem must use the same method.”  The simple solution to these kinds of problems by changing the difficult terms with simpler ones has limited implications. This decision solves the problem of instructions to some extent. For example, many questions in mathematics are preceded by the term “evaluate.”  Most people never use this term in their day-to-day conversations, or even by mathematics teachers during their instruction.  Rather than using “evaluate,” in a problem, we can use “find the value of.” This expression is easier, but the larger problem of not understanding instruction due to lack of mastery of content remains. Every discipline and field of study has technical and specific terms, words, and expressions to describe ideas, concepts, and procedures. To be competent in the field, it is important to know them and use them well. For example, on a test for tenth graders in 2002, as part of the Massachusetts Comprehensive Assessment System (MCAS), the term “represent” appeared 17 times and in one question it appeared five times (released items).  Several places, it was the only term that could do justice to the question. When I asked students to explain or define the term “represent” few students could define it clearly and that too only in a non-mathematical context. It will be better, if instructors/parents helped the children to understand the instructions first, before they help them to attack problems. Once, while I was tutoring a high school student in algebra, I asked her to read the following problem: 4p + q(q-3)          Evaluate:  where p = 3and q = 7 (This was one of the problems on the exercise set given to the students on a test.) Question:  What does the word “evaluate” mean here in the problem? Answer:    I do not know. Question:  Have you ever seen this word before? Answer:    Maybe? Question:  Can you guess the meaning of this word? Answer:  I think it means, “subtract”? No, no!  It means “addition” as there is an addition sign. I think it means, “solve” as this is an equation.  Am I right? But, wait. I do not really understand what are they asking – the question has already given me the values of  “p” and “q”.  It is alreeady solved. I am really confused.” These types of answers are repeated quite frequently in every mathematics class and in every tutoring session.  How can the student be expected to solve these types of problems if the instructions are not clear to him/her?  It is very important to have the instructions made clear in order for the student to work through the problems effectively.  This means that one should emphasize the language (vocabularysyntax, and translationof mathematical terms into mathematical symbols), and understanding the meaning of words, both linguistically and conceptually—mathematically. Some times, the difficulty arises as questions are embedded in the problem. Such problems elicit different levels of complexity of thinking according to the words (particularly the verbs) used. A particular verb elicits a specific level of thinking. Some only ask for recognition of information. Some prompt students to do analysis. Some want them to develop a conjecture, hypothesis, pattern, a relationship, or a thesis. Some call for synthesis of ideas. Each word expects different level of engagement and a type and level of action from the learner. For example, the question, ‘Is this a polygon?’ requires from a student a yes or no answer. The question, “Which one of these figures is a polygon?” This question requires the student to analyze the shapes, to separate them into two types of figures (polygon or non-polygon) and then to compare the catgoriezed figures (what is common to the identified group). Whereas, the question, “Describe, why is this figure called a polygon?’ elicits a different level of language production. The question: “Determine which one of these statements is true?” (a) “Every rectangle is a square.” (b) “Every square is a rectangle.” (c) “Both statements are true.” “Justify your answer.” The instructions of this type prompt students to talk further, elaborate, make connections, and even elicit questions from each other. The answer calls for the integration of academic language, mathematics langauge, and deductiv reasoning. It is asking for a lot. That is what creates the difficulty.
  1. Types of Instructions:  Explicit and Implicit
Instructions in mathematics problems are of two types: explicitor implicit.In explicit instructions, the student is clearly instructed on what to do. Action or operation is already determined by the problem. In the case of mathematics problems where the instructions are explicit, most students find it easier to determine what to do. The demands/commands in explicit instructions are quite clear(e.g., find theaverage of the data; multiply the numbers; write the equation for a line in slope-intercept form; etc.). In implicitinstructions, the demands are indirect and sometimes hidden in the problem (e.g., What is the value of the ☐in the equation: 9 + 4 = ☐+ 5; find the maximum area under the curve with the conditions given; find the dimensions of a rectangle, if the length is 3 more than twice the width and the perimeter is 96 cm; Given two angles of the traingle, find the third angle; etc.). When these demands/commands are explicit, a student knows what is being asked in the problem, but when they are not explicit, many studentsare at a loss. They give up easily. Many teachers construct problems with explicit instructions only, as a result students think, if the answer is not easily forthcoming, it must be an impossible problem.  It is important that chidlren experience a range and different types of problems, with explicit to implicit instruction from the very beginning of their schooling. When the instructions are explicit, probability of a student solving the problem is increased. When they know the content and when the instructions are explicit,students know what to do and can solve the problem. For example,
  • Multiply the numbers 1.2 and 1.3,compared with the instruction: find the product of 1.2 and 1.3.
  • Multiply the binomials (2a + 3)and (3a + 4) using ‘FOIL’ or distributive property of multiplication, compared with the instruction: find the quadratic expression with binomials (2a + 3) and (3a + 4) as its facotrs.
  • Find the value of the function f(x) = (2x + 3)(7x + 5)for x = −3, compared with the instruction: find f(a), if f(x) = (2x + 3)(7x + 5).
  • Differentiate the function: f(x) = (2x + 3)(7x + 5) at x = 2, compared to the instruction find f(a) for the function: f(x) = (2x + 3)(7x + 5).
  • Differentiate the function f(x) = 2x3sin(5x) by parts, compared to the instruction find df/dx for the function: f(x) = 2x3sin(5x).
  • Subtract 7 from 10, compared with find the difference of 7 and 10. 
The instruction, in the first part, in each of these problems, is familiar and direct.  It is simple. It is almost an order to execute the operation. It is in common language. If the student knows the procedure associated with the term, she can execute it. In the second part, the instruction is clear, but the language, in each problem is not common. It is technical.  It has a specific meaning in the context of mathematics. For example, in the first problem, ‘find the product’ may not be present or recent in the student’s mind. It is specific to the operation of multiplication.  And, since it may not be familiar to some children, it becomes difficult. This siuation happens when mathematics being taught in the classroom is taught just procedurally. In (b) the instruction in the second part is related with several concepts, although the procedure for answering the problem is the same.  In (c) the instruction, on the surface, is clear if the student is familiar with these kinds of problems, otherwise it is unclear.  In (e) the instruction is clear if the student knows the process of differentiating by parts. Similarly, in (f) the first part is straight forward, whereas, the second part, many adults write it incorrectly. If the instructions in the problem are explicit, and if students know the meaning of terms in the question—product, factors, quadratic expression, function, differentiation, or differentiation by parts, they can provide the answer to these specific problems. In the case of problems with explicit instructions, the success is dependent on knowing the content.  The hindrance is not in the instructions. If the teacher has focussed on teaching the curriculum only from the persepctive of procedures, the children in that classroom will have more problems with these kinds of problems. Many students have difficulty understanding instructions because they do not know the content of the problem—the vocabulary, the concept, and executing procedures. It is not just one or the other. Knowing the meaning of words in the problem is not enough.  For example,
  • using the long division procedure, find the quotient of 7.25 divided by .025,
  • Find the product of fractions: 3½ and 2¾, or 
  • find the greatest common factor of numbers 6, 48, and 54. 
In (a) the students may know what the words division and long division, however, they may not be able to find the quotient, if they do not know the process of long division, particularly when decimal numbers are invovled in the dividend and divisor, In (b), they may know the meaning of the product, but may not know how to multiply two mixed fractions.  In (c), they may know the meaning of the term the greatest common factor of a set of numbers, but may not have the procedure for finding the greatest common denominator of three numbers. Here, the vocabularyis known, but the problem may be with not knowing the appropriate concept and/orprocedurein the context of the problem. A similar situation may exist when instructions are implicit.  In many implicit instructions, the information is assumed; the information is indirectly given or embedded in the problem. Some problems may be made complicated by items, which ask students to supply information, which are not stated in the problem but are nonetheless necessary to solve the problem.
  • Fred is 63 inches tall.  What else must you know to find out how much he   has grown in the past year?
    •   How much did he weigh a year ago?
    •   How tall will he be next year?
    •   How old is he this year?
    •   How tall was he last year?
(b) What is the measurement of the smallest angle of a triangle if the two angles of the triangle are 70°and 80°? Many word problems do not ask for the operation, algorithm, or procedure directly, but they are embedded in the problem. For example, (c) Simplify the expression: -4y{xy3-2xy3+23(y3x +2)} +7xy3; (d) John practices the piano 1.5 hours each day. His coach said: he needs to practice at least 30 hours before the next concert.  At least, how many days of practice does he need to be prepared for the next concert? In these examples, the instructions are indicated either by mathematical symbols or by words. The knowledge of the symbols, their role in the context, and the conceptual schema embedded in the words and phrases are the key for understanding and responding to the instructions correctly and efficently.  In other words, instructions are implicit and knowledge of the content is needed. Students who have had practice with these types of problems are more likely to answer correctly than students who have not practiced a particular type of problem.  Here the vocabularyis simple, but the conceptis hidden and, therefore, important to know to resolve the problem. Just as we need to understand how to read a map before we can use it, before we can solve a problem, we need to understand the instructions. Students cannot solve problems if the instructions are not clear.  Understanding instructions is dependent on students’ mastery of the language of mathematics(vocabulary, language containers, syntax, and translation from English to mathematics and from mathematic to English), the content of mathematics(conceptual and procedural aspects), the mathematical way of thinking(organizing, classifying, seeing patterns, reasoning, critical thinking and communication skills).   Sometimes, even problems with explicit instructions are difficult for many children. For example, even though the reading level of the instructions and the concept in the problem may be at a lower level, still many students particularly those with reading problems, with limited academic langauge, and/or without an appropriate mathematics vocabulary, find themselves at a disadvantage in word problems.  Similarly, some students may infer the procedure involved in the problem and could solve the problem if they knew how to execute or apply that procedure in the context of that problem.  Others, for example, even if they were able to read the instructions, they may have difficulty in understanding the instructions that lead to the particular concept and/or procedure, as they may not have the conceptual schema behind the words, therefore, may not be able to solve the problem. Ability to read the problem is a necessary conditon for solving the problem, but it is not a sufficient condition. A student may be able to read the problem without any difficulty, but may not be able to translate the technical words into mathematics concepts and procedures because of poor mathematics vocabulary and lack of conceptual schemas. Therefore, may not be able to arrive at the procedure or strategy to be applied. Only the proper mathematics language and its understanding will lead to the construction of cocneptual schemas and efficent concpetual schemas lead to procedures.
  1. Role of Questioning in Understanding Mathematics Instructions
Following instructions, first, is a task in reading: vocabulary, comprehending, and understanding. The mathematics language plays a big role in it and learning how to read instructions and following them is an important part of mathematics learning. Once the reading task is performed, then, cognitively, it is connecting the vocabulary with the concepts, procedures, and mathematical way of thinking. Vocabulary for mathematics ideas, concepts, and procedures should emerge through discussion and experimentation and then formalized, connected to what is already known rather than transpalnted by giving vocabulary lists to memorize. Because words should result from a need to describe our world—this is where they gain their power.Therefore, the type and the number of questions we ask in the mathematics classroom determine how the students are going to do on mathematics tasks. Questions elicit different levels of complexity of thinking according to the words used (especially verbs). Particular verbs elicit a specific level of thinking to prompt: analysis of data and ideas; developing conjectures, hypotheses, and then a thesis about the problem; synthesizing different types of problems, strategies, and procedures; type of thinking—recognition, comparing, contrasting, and constructing; or, calling for levels of engagements from the learner. For example: The question “Is this a polygon?” requires a student to say either yes or no. In contrast, the request “Describe why this shape is called a polygon” elicits a different level of language production. Instructions of this type prompt students to elaborate, make connections, and even form questions. Thus, questions and the art of questioning are critical to learning. Questions commence a cascade of actions in the brain:
  • Questions instigate language;
  • Language instigates models;
  • Models instigate thinking;
  • Thinking instigates understanding;
  • Understanding produces competent performance;
  • Competent performance is the basis of long lasting high self-esteem; and
  • High self-esteem contributes to motivation for learning and engagment. 
The type and number of questions we ask in the mathematics classroom determine how successful students will be in mathematics. Mathematics language plays a critical role, and learning how to read instructions and following them is an important part of mathematics learning. Students should also know the type of questions that will appear on a test, and teachers should feel comfortable giving them such information since it will focus students’ study efforts. For example, in the case of multiple-choice questions, they will be required to identify rather than generate information.  Although identification formats are generally less difficult than producing information, multiple choice tests often focus on the identification of a large amount of less important information.  These formats have different implications for language requirements in test taking. Standardized tests have become a permanent feature of education.  A student should know there are great varieties of test item formats on standardized achievement tests.  There are also varieties of difficulty levels in different types of questions. To understand the instructions, a sufficient level of reading comprehension on the part of students is required.  Formats used for mathematics tests are usually relatively straightforward, but they could also vary.  To succeed on tests, it is important to know these formats; the knowledge of mathematics content alone is not enough.  Because teacher made tests may not follow the same format and structure, the teacher should check the manuals of the standardized tests to determine whether she can help her students on the reading part or the structure of the test items.  She should also know how content questions are formulated on these tests. Similarly, some of the test items may require the use of charts and graphs.  It is important that students have experience with these before taking the test.
  1. Strategies for Improving the Understanding of Instructions 
Good teaching in mathematics, at the elementary level, requires that students be taught key number concepts associated with computation. They should have:
  • practiced arithmetic facts to the automatization level with efficient and effective strategies (arithmetic facts are best derived using decomposition/recompsotion strategies);
  • know a concept in its different models(e.g., multiplication as: repeated addition, groups of, an array, and area of a rectangle);
  • and have applied computational procedures in a variety of different formats(e.g., division as partial quotient, long division, and short division).  
Students with learning problems in mathematics and in special education settings are presented with a restricted range of mathematics with the mistaken belief that they experience less complex material to minimize confusion and frustration.  That kind of strategy works initially but in the long term it is detrimental to their progress and their development as learners.  It does not provide exposure to meaningful mathematics and limits their cognitive development. The goal of special education support for children should be two-fold:  to help them improve cognition and to expose them to meaningful content, in meanginful ways. For that purpose, we can begin with simpler language, simple instructional models and with a narrow range of the content and setting, but then we need to increase the complexity of language, range of material and models, and content.  All students benefit from learning the range of problem formats and terminology and sufficient exposure to the rich vocabulary of mathematics. Again, effective teaching strategies, which employ a large vocabulary and a variety of formats, are the most helpful practices to acquire flexibility of thought. In addition, knowledge of key vocabulary terms and use of multiple visual representations of mathematical information are important for a conceptual understanding of mathematical ideas. In light of this, when students are confronted with an item in an unfamiliar format or context, teachers should encourage them to use scripts such as: “What is the key information here?” “Do I know a related word or expression?” “How else could one ask this question?”  “Can I think of another example for this?”  “Do I remember another problem like this?”  “Will I able be to solve this problem if I substitute simple numbers?” Students should practice answering questions by replacing vocabulary they use less commonly and rewriting them in formats that are more familiar.  Once stated in more familiar terms, the student is more likely to answer the question correctly.  For example, consider the following:          Which set has both odd and even numbers that are not square numbers as its members?
  1. {9, 11, 15, 3, 5}
  2. {6, 10, 4, 2, 8}
  3. {6, 10, 7, 5, 8}
  4. {25, 49, 225, 144, 9, 400}
This problem could potentially confuse students who are aware of odd and even numbers but uncertain of the meaning of the word set or square numbers. Or, they may have difficulty in understnading the phrase: both odd and even that are not square numbers.”  If a teacher shows students how to actively reason through each item and to temporarily set aside unfamiliar terms, they should be able to see that one of the four answer choices has both odd and even numbers and therefore “c” is the best answer. Once, we have used proper reasoning to find the correct answer, we could introduce the term setand square numbers. We could also expand the idea by giving a few more examples of this type.  Using this problem, we can extend the discussion by asking: “What kind of numbers are in the set described in the option ‘d’?” and, then extending it to further discussion: “Can a square number have digits 2, 3, 7, and 8 in the one’s place?” Using the area definition of multiplication and identifying the product as the area of the rectangle and the sides as the factors, one can conncet the nature of numbers—even, odd, square, prime, and composite and properties of operations—zero property, comutative, associateive and distributive. When students do not read or understand instructions for a problem, they cannot show what they may know.  Over the past decade, teaching approaches have changed in the mathematics classroom. Some have resulted in improved learning and some have contributed to the detriment of learning.  One approach that has recently been neglected is time on task. The more time spent on being directly engaged in learning mathematics language, the better students will understand the instructions in problems involving mathematics language. It is important to increase the amount of actual time on task for students to work on mathematics language, in context of problem solving.  Time on task is the amount of real time spent on teaching and learning mathematics language and the related instructions in the context of word problems. Another critical variable is the amount of contentmastered raher than just covered.  If the content is not covered, students will not have the opportunity to learn enough information, but if it is not mastered, they will not be able to apply it.  In such a situation, working on comprehension of instructions is ineffective.  Further, if the content is covered too rapidly, students may not have the opportunity to master the information sufficiently. Teachers need to develop a clear scope and sequence.  This should be planned at the beginning of the year, not during the year.  During the year, one can make adjustments to the content.  In countries where students achieve higher levels in mathematics have an unwritten pedagogy—they plan the scope and seuqence for a time that is at least one to month less than the school year.  The last two months are devoted to review, reinforcement, practice, and integration of content. Time to teach skills to understand instructions should be included in each lesson, and this should begin early in year.  Including test-taking strategy instruction with examples from standardized tests enables students to practice and apply them throughout the year. Since standardized tests are typically administered in the spring, training for these tests should take place several times a year. Intensive practice should take place prior to the administration of tests. Throughout the school year, students should be taught to extract meaning from a variety of graphic displays and tables in mathematics contents (e.g., relief maps, topographic maps, weather maps, maps of ocean currents, timeline of historic events, scientific tables and charts, population charts and maps, and other graphic displays) and answer questions based on these displays in a multiple choice format.  Newspapers and the Internet are a good source of this information. Additionally, when they read passages describing mathematics content, students should use the same strategies found in reading comprehension tests.  These are valuable exercises because students can come to understand that the strategies they use to answer reading tests can be used for readings wherever they occur. All of these strategies relating to mastering mathematics content and solving problems depend on children practicing the three components of mathematics—linguistic, conceptual and procedural.  Instructionas are only label on the package involving mathematics contet.  If students do not know the content, instructions cannot be blamed for lower acheivment.
  1. Examples of Instruction to Mathematics Problems
The following is an attempt to identify key vocabulary words, expressions, and symbols used as part of the instructions, generally used in mathematics texts, tests and examoination (in most cases few examples are used):
  • About/Approximately/Rounding: (a) About how many miles is 66.5 million feet? (b) The value of the number √(145) is close to what integer in value? (c) Nate says:“The value of the fraction ⅛ is about.13 when approximated to the hundredth place.”  Is he right? Did he round to the hundredth’s place correctly? (d) What will be the value of ⅛, if rounded to the tenth’s place? (e) Is rounding is same as approximation?
  • Add/subtract/multiply/divide:What is the value of , (a) if we add the other numbers of the set: {, 1, 8, 5 and 34}. (b) if we multiply other members of the set?  (c) What is the smallest quotient, if we divide any two members of the set {1, 6, 5, and 30}
  • Apply:(a) Apply the graphing method for solving the set of equations: 3x + 4y = 12and 4x + y = 29. What does mean to solve this system of equations? (b) Which of the following shows an application of the distributive/ associative/commutative property? (c) Apply any of the Prime Factoriazation methods to find the Greatest Common Factor (Least Common Multiple) of 24 and 40.                                        
  • Assume:(a) Assume that this triangle is equilateral. (b) Assume that the numbers mn in the fraction m/n are prime. Is the fraction, expressed in the lowest term? (c) What is the value of n, if we assume that the line passing through the points P(n, 5) and Q(2,7) is horizontal? 
  • Compare:Compare the following numbers:  and .24. Write a number relationship between these two numbers.
  • Compute/Calculate/Perform the operation: (a) Compute 35.2 ÷.574. (b) Which number in the box makes the number sentence (15 – 3) × (2 +3) = ÿ× 5 true? (Choices: 5, 15, 12, 30). (c) Perform the indicated operation in the following calculation(s) ….
  • Conclude:What pattern do you see in the data? What do you conclude from the result you derived from the data?  Write your pattern as a relationship between the two variables? When you compare your pattern relationship, with this equation: y = mx + b? What do you conclude by the slope in your equation? 
  • Consider: Consider that this pentagon is a regular figure, what does the term ‘regular’ indicate here? What is a regular triangle called?
  • Compare and contrast:  Compare and contrast the members of the set by their properties: {2, .2, 2%, , ½, (.2)−2, and 22}. 
  • Decide: (a) Decide which is the largest number in the set: {2, , .2, 2%, , ½, (.2)−2, and 22}.  (b) Decide which is the smalleest number in the set: {2, .2, 2%, , ½, (.2)−2, and 22}.
  • Describe:Describe the pattern that can be used to predict the height of the bounces of a ball that bounces back half as much as the previous bounce.
  • Determine: (a) Determine the relationship among the values of the coins from the following clues ….. (b) How can you determine if a rectangular array can be built for an expression …..? 
  • Distinguishbetween: (a) An even numberand an odd number; (b) a prime number and a non-prime number; (c) a polygonand a non-polygon, (d) an integerand a rational number; (e) a continuous functionand a non-continuous function;  …
  • Envision/visualize/picture/think:(a)Envisionyou rotated the diagram (rectangle, a square, an equilateral triangle, and a regular hexagon) by 90°clockwise. What will the figure look like after the rotation?; (b) What amount of rotation (and about what axis of rotation, or what point) will tranform the first diagram to the second diagram?  
  • Estimate:(a) What is the best estimate of how many more times Cathy jumped than Wilson? (b) Which arrow on the radio dial below is closest to 96.3?  (c)What is the closest degree measure of he angle formed between the hour and the minute hands of a clock at 3:40 PM?  (d) Which graph below most likely shows the outcome? (e) Three friends plan to equally share the cost of a video game that costs $38.89 including tax.  Which is the best estimateof the amount each will have to pay? (e) Using estimation, decide which sticker below has the greatestperimeter. (f) The value of  is closest to … (g) 2Ö5 is between what whole numbers. 
  • Evaluateeach expression: (a) 3xy2+ 5x2y -4x2y2, where x = −2and y =−.5.
  • Explain/express:Explain your reasoning in your words why a prime number has odd number of factors.
  • Extrapolate: (a) From the data given extrapolate the nature of the graph. (b) Assuming that her income and expenses continue to grow at approximately the same rate, estimate her income and expenses for the month of may.  Explain or show how you found your estimates. 
  • Find the value of: (a) 3xy2+ 5x2y -4x2y2, where x = ]−2and y =−.5.
  • Generalize: (a) Use the sequence of numbers 1, 3, 7, 15, 31, 63, …to find the general pattern/formula/expression.  
  • Graphing: (a) Graph/plot on a number line/coordinate plane. (b) Construct two line graphs using the given data. (c) Draw a circle with radius 5 and center (3, 4).
  • How many/long/much/much more/much less: (a) How many millimeters of iodine are in 1,000 ml of solution? (b) How many times greater is the surface area of the cube with side 2 inches and the cube with side one inch? (c) How long will each column of names be? (d) How long will take him to travel this distance? (d) How do a and b compare? 
  • Identify:  (a) Identify the reciprocal of .25. (b) Identify the inverse of the function f(x) = 3x + 4. (c) Identify the property of the equality used in the equation: 3(x + y) = 3x + 3y. Identify the shape that is: 
  • Interpret the graph: A graph is given.
  • Interpret the definition:  In what ways the definition of a prime number: “A whole number is called prime, if its only factors are 1 and itself”  differs from “A whole number is called prime, if it has exactly two factors, namely I and itself.” Which definition is accurate? Why 1 is not a prime number?
  • Model the information:  Show the distributive property of multiplication over addition and subtraction using the area of a rectangle definition of multiplication. 
  • Name: (a) Name one of the shapes you chose.  Make a list of four different things that describe this shape.  (b) Name another one of the shapes you chose. Make a list of four different things that describe this shape. (c) Name the last shape you chose.  Make a list of four different things that describe this shape.  
  • Notice the list of numbers/formula/diagram/data
  • Observe the following information and:
  • Pattern: (a) When we multiply 37 by multiples of 3, we see a pattern. 37 × 3 = 111; 37 × 6 = 222; 37 × 9 = 333; 37 × 12 = 444; … If the pattern continues this way, then 37 × 21 =  ? (b) What is the next number in the pattern below?
  • Predict: (a) Predict the chances of getting a red balls out of the container that contains 3 red balls and 7 balls of different colors.  (b) Predict the height of the fifth bounce. (c) is it more likely that …. (c) Which is the BEST way for Bridgett to show this information? 
  • Prove: Which of the following statements gives enough additional information about the figure above to prove that DABC is similar to DDEC.  
  • Rewrite each expression in a simpler form:  (a) 48/128 (b)  (x-2)/(x-2)(x-3).
  • Remember: (a) Remember a polygon has more than three sides. Define a quadrilateral as a polygon. (b) Using estimation, decide which sticker below has the greatest perimeter. (Remember: Perimeter is the distance around a figure.) 
  • Represent:  (a) Represent this point on the coordinate graph. Do these points represent a circle? (b) Which graph below most likely represents Ms. Hall’s class on Tuesday? (c) Which point represents the intersection between the lines: 3x + 4 y = 7and 4x + 3x = 7
  • Show/describe:Show or describe how you found your answer.  (a) Use pictures, numbers, or words to show or explain how you found your answer.  (b) Use pictures, numbers, or words to show or explain how you know. (c) Which shows a slide of Y. (d) Show how to build rectangular arrays, if possible, for each of the following expressions using the math tiles.
  • Simplify: (a) Simplify the expression  … (b) Simplify the numerical expression …. (c) Use the expression 2x – 3(5x –8)to answer the question: What could be the first step in simplifying the given expression? 
  • Solve:(a) (Direct instruction) Solve the following equation for x ..(b)  (Indirect instruction) If 4 + 2 (3x – 4) = 8, then 3x – 4 equals…. (c) 24   ×3is the same as …. (d) Find allthe values of xthat satisfy the following equation. (e) The expression 4 x2+ 2x – 6 – x(3 – x)is equivalent to ….
  • Summarize:  Write a proof of the Pythagoras theorem for a right triangle.
  • Suppose that (see assume that): 
  • Tell:  (a) Tell whether each statement is true or false. (b) Which number sentence tells how much milk is in all the glasses? 
  • True:  (a) Which of the following statements is always/sometimes/never true?
  • Use: (a) Use the information in the scatter plot/graphic method/equation/process/table/chart to answer the question. (b) Use the balance scale to answer the following question. (c) Use t−2, 4). (b) Write a rule for the table shown below. (c) Write four different number sentences that follow these rules. Each number sentence must show a different way of getting the number 42.  Each number sentence must contain at least two different operations. Use each of the four operations at least once. An example is shown below.  You may not use this example as one of your four number sentences.  Example: (8 ¸4) + 44 − 4 = 42. (d) Write a number sentence to show how much money Ralph spent for stamps.  Be sure to include the answer in your number sentence.  
  Numeracy & Literacy Analogies The fields of mathematics and reading both have basic skills.  In both subjects, all students must master all of the basic skills to work productively.  But, while the skills necessary to read well are well known since the publishing of the Report of the National Reading Panel[1]in the year 2000, the equivalent skills to learn mathematics are less well understood.  The table below puts the two side by side to help teachers see the parallels.
Mathematics Literacy
Number Concept – an understanding of the concept of number in language, in orthographic symbols, and in visual clusters (create 3 part Venn diagram) 1.     Phonemic Awareness – an understanding of the sounds in their language and how they form words
Decomposition/Recomposition– the ability to manipulate numbers to see number relationships and fluently solve unfamiliar problems using numbers 2.     Decoding – the ability to figure out unfamiliar words, and to learn to read them fluently
Language of Numbers – mastery of the words and phrases used to describe numerical operations 3.     Vocabulary – mastery of an adequate number of words to understand text passages
Fluency – Automatic knowledge of basic arithmetic facts without counting using: a) sight facts, and b) strategies 4.     Fluency – transforming vocabulary into sight vocabulary through practice to automaticity
Understanding – the ability to understand the questions in a problem, apply appropriate facts and strategies to solve them, and explain the solution to others 5.    Comprehension– the ability to understand the direct meaning of text, and also its implications and intention, and finally the ability to perform analysis on text
Communication – the ability to explain to others: a) the choice of numerical processes, concepts and procedures in solving problems; and b) explaining the nature of the solution.  C) These concepts and procedures may be expressed concretely, orally, pictorially or symbolically. 6.     Writing – the ability to explain ideas to others so that they understand: a) the meaning of the text; and b) the implication of the text. C) This ability may be expressed in outline, expository, story-telling or other forms.
  Center for Teaching/Learning of Mathematics  CT/LM has 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 lectureson topics such as:
  1. How does one learn mathematics?This workshop focuses on psychology and processes of learning mathematics—concepts, skills, and procedures. The role of factors such as: Cognitive development, language, mathematics learning personality, pre-requisite skills, conceptual models, and key developmental milestones (number concept, place value, fractions, integers, algebraic thinking, and spatial sense) in mathematics learning. Participants learn strategies to teach their students more effectively.
  2. What are the nature and causes of learning 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 in 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, such as: dyscalculia and math anxiety.
  3. Content workshops.  These workshops are focused on teaching key mathematics milestone concepts and procedures. For example, How to teach arithmetic facts easily and effectively?  How to teach fractions more effectively?  How to develop the concepts of algebra easily? Mathematics As a Second Language.In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a very simple concept from arithmetic and take it to the algebraic level.
  4. What to look for in a results-oriented mathematics classroom: This is a workshop for administrators and teachers to understand the key elements necessary for an effective mathematics classroom.
We offer individual diagnosisand tutoring servicesfor children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia. We provide:
  1. Consultation with and training for parents and teachers to help their children cope with and overcome their anxieties and difficulties in learning mathematics, including dyscalculia.
  2. Consultation services to schools and individual classroom teachers to help them evaluate their mathematics programs and teaching and help design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics.
  3. Assistance for the adult studentwho is returning to college and has anxiety about his/her mathematics.
  4. Assistance in test preparation (SSAT, SAT, GRE, GMAT, MCAS, etc.)
  5. Extensive array of mathematics publications to help teachers and parents to understand how children learn mathematics, why learning problems occur and how to help them learn mathematics better. 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. Selected back issues ofFocus: Volume 3, Numbers 2 & 3: Educational Psychology and Mathematical Knowledge Volume 4, Numbers 3 & 4: Fingermath: Pedagogical Implications for Classroom Use Volume 5, Number 2: Remedial and Instructional Prescriptions for the Learning Disabled Student in Mathematics Volume 5, Numbers 3 & 4: Mathematics Learning Problems and Difficulties of the Post Secondary Students Volume 6, Number 3: Education of Mathematically Gifted and Talented Children Volume 6, Number 4: Brain, Mathematics and Learning Disability Volume 7, Number 1: Learning Achievement:  Implications for Mathematics and Learning Disability Volume 7, Numbers 3 & 4: Using Errors as Springboards for the Learning of Mathematics Volume 8, Numbers 3 & 4: Dyscalculia Volume 9, Numbers 1 & 2: Computers, Diagnosis and Teaching (Part One and Two) Volume 11, Numbers 1 & 2: Visualization and Mathematics Education Volume 11, 3 (1989): Research on Children’s Conceptions of Fractions Volume 12, Numbers 3 & 4: What Can Mathematics Educators Learn from Second Language Instruction? Volume 13, Number 1: Students’ Understanding of the Relationship between Fractions and Decimals Volume 14, Number 1: The Psychological Analysis of Multiple Procedures Volume 15, Numbers 2 & 3: Vygotskian Psychology and Mathematics Education Volume 17, Number 2: Perspective on Mathematics for Students with Disabilities Volume 18, Numbers 1-3: Gender and Mathematics:  Multiple Voices Volume 18, Number 4: The Challenge of Russian Mathematics Education: Does It Still Exist? Volume 19, Number 1: Components of Imagery and Mathematical Understanding Volume 19, Number 2: Problem-Solution Relationship Instruction: A Method for Enhancing Students’ Comprehension of Word Problems Volume 19, Number 3: Clinical Assessment in Mathematics: Learning the Craft Volume 20, Numbers 2 & 3: Elements of Geometry in the Learning of Mathematics Volume 22, Numbers 3 & 4: Using Technology for the Teaching and Learning of Mathematics Volume 23, Numbers 2 & 3: Language Issues in the Learning of Mathematics Volume 28, Number 3 & 4: Concept Mapping in Mathematics Index of articles in Focus on Learning Problems in Mathematicsfrom Volume 1 to 30 available on request.  Individual issue                                                                                      $ 15.00 Double issue                                                                                             $ 20.00 Each Volume (four issues)                                                                     $ 30.00   Whole set of 30 volumes                                                                        $400.00 Math Notebook: Single issue ($3.00); Double issue ($6.00); Special issue ($8.00) Other Publications    Dyslexia and Mathematics Language Difficulties             $15.00    How to Master Arithmetic Facts Easily and Effectively    $15.00    Guide for an Effective Mathematics Lesson                      $15.00    How to Teach Fractions Effectively                                   $15.00    Math Education at Its Best: Potsdam Model                      $15.00    How to Teach Number to Young Children                         $15.00    Dyscalculia                                                                       $15.00    How to Teach Subtraction Effectively and Easily               $12.00   Literacy&Numeracy:Comprehension and Understanding $12.00    The Questioning Process: A Basis for an Effective Lesson   $12.00    The Games and Their Uses in Mathematics Learning      $15.00            Visual Cluster cards without numbers                              $12.00         DVDs     How Children Learn: Numeracy                                              $30.00 (An interview with Professor Sharma on his ideas about how children learn mathematics) How To Teach Place Value                                               $30.00 (Strategies for teaching place value effectively) Numeracy DVDs (Complete set of six for $150.00 and individual for $30.00)
  1. Teaching arithmetic facts,
  2. Teaching place value, 
  3. Teaching multiplication, 
  4. Teaching fractions,
  5. Teaching decimals and percents, and
  6. Professional 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, regular classrooms in the UK.    Please mail or fax order to (add 20% extra for postage and handling): CENTER FOR TEACHING/LEARNING OF MATHEMATICS 754 Old Connecticut Path, Framingham, MA 01701 508 877 4089 (T) 508 788 3600 (F) Mahesh Sharma Professor Mahesh Sharma is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts, USA and Berkshire Mathematics in Reading, England. Berkshire Mathematicsfacilitates 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. 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 has been 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 students (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 Center for Teaching/Learning of Mathematics 754 Old Connecticut Path Framingham, MA 01701 Mathematics Blog [  

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