CHAPTER 11

MATHEMATICS

1. INTRODUCTION
1. There is a close link between developmental variation and acquisition of math skills and daily performance.
2. Success in math demands
1. Precision
2. Consistency
3. Attention to detail
4. Conceptual agility
5. Problem-solving flexibility
6. Speed of processing and recall
7. Cumulative learning
3. Major requisites for accomplishment in math include:
1. Number concepts
2. Basic operations
3. Graphomotor implementation
4. Transfer of knowledge
5. Mathematical linguistics
6. Visualization
7. Problem solving
8. Active working memory and mental arithmetic
9. Higher order cognition and proportion
10. Higher order cognition and abstraction
11. Higher order cognition and proof
12. Higher order cognition and equation
13. Depth of knowledge and access to knowledge
14. Attention to detail and self monitoring



2. REQUISITES AND DEVELOPMENTAL FUNCTIONS
1. Number Concepts
1. The representation of quantity by numbers is a significant conceptual acquisition.
2. Through early experience children begin to understand that quantitatively similar objects can differ in size or dimension.
3. Piaget describes four underlying concepts which must be present in order to develop an understanding of number concepts. These include:
1. Classification (develops between 5 and 7) - is the capacity to perceive categories and their relationships. It involves the ability to group objects into categories. Some children with higher order difficulties may have trouble with classification, therefore delaying their acquisition of math skills.
2. Ordering - is the capacity to organize materials in a logical sequence (i.e. placing things in alphabetic order). Children with sequential disorganization may have difficulty with this concept which will impact their ability to acquire math skills.
3. One-to-one correspondence - a particular object has a fixed value despite their characteristics (i.e. a pile of five automobile tires and a stack of five pieces of paper contain the same number of objects).
4. Conservation - this is closely linked to one-to-one correspondence. It is the concept that volume or quantity remain the same regardless of its spatial arrangement.
4. It is unusual for children to show significant delays in the mastery of these concepts. Even those children who have some difficulty with these concepts usually have mastered them by the age of nine. Because the opportunities for acquiring these skills are multisensory even children with processing difficulties (such as poor visual processing or sequencing weaknesses) will still be able to grasp the concepts even if only on a superficial level
2. Basic Operations
1. In order to master these operations, children need to arrive at four distinct levels of awareness. These include:
1. Acquiring a basic understanding of each process (i.e understanding that adding to a quantity makes it greater).
2. Understanding when a particular process is called for (i.e. plus sign means to add).
3. Learning and recalling basic number facts.
4. Learning the series of steps required to solve a particular type of problem. This involves the understanding of place values.
2. A wide variety of developmental dsyfunctions can interfere with a child's ability to understand and apply basic math operations.
1. Children with attention deficits may be unable to attend to the detail required to successfully complete math operations
2. Children with spatial or sequencing processing difficulties may have trouble reading length numbers or arranging numbers in columns
3. Children with language difficulties may have trouble understanding verbal explanations of math operations.
4. Children with graphomotor dysfunction may have trouble arranging numbers on a page, as well as copying legible number symbols.
5. Children with spatial visualization difficulties may have trouble computing problems when numbers are arranged in irregular columns.
6. Higher order cognition difficulties may limit a child's ability to fully understand math concepts
7. Children with dysfunctions of memory may have difficulty mastering facts (i.e. multiplication tables, etc).
3. Graphomotor Implementation
1. This involves the transcription of numbers on paper.
2. Children who have difficulty with motor memory may have trouble writing legible numbers. This phenomenon is described as strephosymbolia.
3. Impulsivity, poor tempo control, and poor attention to detail may also lead to illegible number formation.
4. Students with motor planning problems have difficulty lining numbers up in columns. These students often benefit from worksheets with examples of properly aligned numbers.
5. Some students are so preoccupied with legibly writing and lining up numbers that they forget the steps necessary to solve the problem. This "eclipse phenomenon" is common in the early elementary grades, By late elementary school most students have automatic number formation, but their delay in gaining this automatization may have caused them to fall behind in the acquisition of skills.
4. Transfer of Knowledge (Equilibration)
1. This is the ability to transfer knowledge from one context to another. It allows children to easily move from math computations to informal everyday applications.
2. Cognitive weakness, memory deficits, and cultural factors are all likely to affect a child's ability to transfer knowledge.
5. Mathematical Linguistics
1. Communications in math may cause a young child a lot of difficulty. This is because:
1. There is an inconsistency between verbal labels and number representation (i.e. sixteen and 16).
2. In reading and explanations words are constantly combined with abstract symbols.
3. Math has a very specialized vocabulary which must be learned.
2. In math the vocabulary and semantics depends almost totally on the prior entries into memory. Thus students who struggle with semantics in other settings may also have difficulty with it in math. For example a child who has difficulty with semantics may become confused when a teacher interchangeably uses the words minus and subtract.
3. Several differences between mathematical English and nonmathematical English have been described. These are:
1. Words that are common in math are very uncommon in nonmath English.
2. Words that in non-math English may have several meanings (i.e. point, and set) only have one meaning in math English.
3. Because there a few verbal context clues to help students, they must learn the precise meaning of words.
4. Math reading is full of abstract graphic symbols that students must memorize.
5. Many math expressions are hypothetical references. Because students find hypothetical reasoning difficult before the age of 14 this is problematic.
6. Many expressions in math refer to concepts which are new, thus requiring students to learn new words before they can work through the concepts.
4. The language of word problems often cause children particular difficulty because they are more compact and "conceptually dense" than ordinary prose. Specific aspects of word problems which cause difficulty are:
1. The presence of extraneous information
2. Presentation of numbers in the problem in an order different than what is required the solve the problem.
5. Students with language disabilities may have extreme difficulty solving word problems, because they have difficulty processing essential information.
6. In addition the multi-step explanations present in some math problems may cause particular problems for children with language disabilities and dysfunctions of sequential processing and memory.
6. Mental Imagery and Spatial Appreciation
1. Some students bypass the linguistic difficulties of math by forming mental images of concepts.
2. The capacity to visualize math concepts is an asset, thus children whose nonverbal cognition and visualization are poor are at a significant disadvantage in math. They are likely to experience extreme difficulty with geometric concepts
7. Problem Solving
1. Almost all levels of math require problem solving
2. In math students must be as concerned with using the correct method to solve a problem as they are with arriving at the correct answer.
3. To be a good problem solver requires a number of capacities that are susceptible to disruption in children with developmental dsyfunction.
4. Bloom and Broder highlighted the following differences between good and poor problem solvers. These include:
1. Poor problem solvers have trouble getting started. Unlike good problem solvers they have trouble pulling out the key words or ideas that suggest the best process to use.
2. Poor problem solvers tend to offer correct solutions to a problem, but it is not the problem they were asked to solve.
3. Good problem solvers are able to remember and apply relevant knowledge in solving a problem.
4. Good problem solvers are very systematic and organized. They often reorganize the data provided in the problem to make it easier to solve.
5. Good problem solvers are able to follow extended chains of logical reasoning, whereas poor problem solvers tend to give up.
6. Poor problem solvers are less confident about their problem solving abilities, and easily become discouraged.
5. Children who lack cognitive flexibility, who fail to recognize recurrent themes, and who are impulsive tend to have difficulty with problem solving.
8. Estimation
1. Competent math students are able to estimate the answer to a problem
2. The ability to estimate requires flexibility, a strong sense of place value, good spatial visualization abilities, and an ability to translate numbers into a more usable form.
9. Memory, Computation, and Problem Solving - Some students who have difficulty in math also have problems with memory
1. Students have problems holding the elements of a problem together while they are solving it and forget vital components
2. Word problems involve active working memory and an extended process of reasoning that requires holding on to several important elements in order to screen out irrelevant data, then to select the necessary operations and carry them out
3. Some children with attention deficits also have difficulty retaining the components of a problems while solving it. Their distractibility and inattention to detail could impede active working memory
4. Rapid retrieval memory is critical for the efficient recall of math facts and procedures
10. Reasoning Processes: Abstraction, Proportion, Equation, and Proof
1. Abstraction - Even early stages of math development require students to operate on an abstract symbolic level and manipulate numbers. Middle stages require abstract manipulation of symbols to represent processes (+, -, x, /). Later stages require manipulating numbers and symbols in algebra
2. Proportion - In the early stages students are taught to recognize simple proportions (pies, ratios, fractions). Later children must compare ratios
3. Equation - the concept of equation means to have two equal quantities, one on each side of the =. This is essential for algebra and other forms of math reasoning.
4. Proofs - these require students to consider several different solutions and reason from premises to conclusion
1. requires several levels of thinking
1. recognition (of shapes, vocabulary)
2. analysis (of the properties of the figures)
3. categorization (of shapes)
4. deduction (relationship of postulates and theorems to proof)
5. rigor ( metacognitive awareness of need for precision)
2. problems that may interfere
1. poor conceptual understanding
2. inattention to detail
3. defective sequential organization
4. lack of problem-solving flexibility
5. superficial prior learning
11. Depth of Knowledge and Facility
1. Math is cumulative and there must be a progression of basic skills to advanced skills. Basic skills must be come increasingly automatic.
2. When math knowledge is first acquired there are variant degrees of assimilation
1. Children can have a poor take - they are unable to make use of explanations either because of the complexity of the math language, the nature of the concepts, the burden on memory, or the failure to have mastered earlier basic skills
2. Children can have a tenuous grasp - they understand the new information but not enough. They cannot fully elaborate on the new information and have a tendency to forget what they have learned
3. Some students use rote mimicry - they master and remember the math but they rely too heavily on memory and not on math reasoning. Tendency to simply regurgitate
4. Some math learners engage in true learning - they acquire new math skills that they can store, retrieve and apply. They are prepared to use them for their benefit when learning science
5. The ultimate learning level is elaborate capacity. children master new math concepts and are able to remember and use them spontaneously and to apply them to a variety of situations
12. Attention
1. Attention problems that interfere with math include: poor attention to detail, intentional goal setting, reflecting on alternative strategies, monitoring, and vigilance
2. But some children with attention problems do well in math because they have strong conceptual and reasoning abilities. Also if math is motivating it may override impulsivity



3. ERROR PATTERNS AND CONTRIBUTING DEVELOPMENTAL DYSFUNCTIONS
1. NOTE: This chapter does not have a separate section on developmental dysfunctions. Rather they have been discussed through out the section above.
2. For lists and examples of common error patterns see table 11-1, p. 402-3 and table 11-2, p, 415-6. Both of these tables indicate the possible processing problems that are involved in each error pattern



4. STAGES OF LEARNING
1. Acquisition - students learn to perform a skill (initial) and then through practice develop accuracy and consistency (advanced)
2. Proficiency - students at this stage become fluent and develop automaticity in executing skills while maintaining accuracy. This enables them to progress to more advanced levels and perform more difficulty math tasks better.
3. Maintenance - students continue with regular and periodic practice for continued mastery
4. Generalization - students apply a skill in different situations
5. Adaptation - students apply their understanding of skills and concepts for problems solving, reasoning, and decision making. Estimation skills are important at this stage.



5. ASSESSMENT
1. Preschool and Kindergarten-
1. During the preschool years, a child should
1. Develop a sense of number
1. Have a sense of meaning and constancy of numerals
2. Begin to develop counting skills
3. Count by rote
4. Compute the number of objects in a croup
5. Have some understanding of 1-1 correspondence
2. Develop spatial-mathematical skills
1. Sort objects by size and shape based on nonverbal understanding of similarities and differences
2. Develop mathematical vocabulary: prepositions indicating spatial directions, words for size and quantity (more, less, equal), terms for fractions and part-whole relationships (half)
3. Left-right discrimination (not necessarily accurate)
3. Develop sequencing skills
1. Sequence objects by size and quantity, also numbers from 1-10
2. Appreciate time concepts and vocabulary
3. Appreciate simple money concepts and vocabulary
4. Follow 2 and 3 step commands in correct order
4. Develop skills for using number symbols
1. Recognize number symbols and associate them with concrete representations of quantity
2. Copy and later write numbers
5. Understand concepts of addition and subtraction and apply counting skills to perform simple computations
2. An assessment of academic readiness should evaluate these abilities.
1. Preschool readiness or development tests contain these items
2. Informal checklists, skill inventories, and observations can be used
2. Early Elementary School
1. During these years a child should
1. Develop computations skills
1. Simple addition and subtraction
2. Basic facts
3. Operational signs
4. Writing legible numbers and arrange them in columns
2. Develop arithmetical concepts
1. Place value
2. Regrouping
3. Multiplication and times tables
3. Practical applications
1. Telling time
2. Money and making change
3. Measurement
4. Estimation skills
2. Questions to Ask - when assessing error patterns ask the following questions
1. Has the child applied the wrong operation?
2. Was the error due to inattention, lack of understanding the operational sign, incomplete grasp of the procedural steps?
3. Has the child used an incorrect arithmetic fact and was this due to inaccurate fact learning and recall or to inadequate counting strategies?
4. Has the child missed an algorithm?
5. Does the child overindulge in guessing?
6. Are the guesses wild guesses or do they represent reasonable estimates of the answer?
7. Does the child have difficulty understanding the material or do they understand bu have trouble retrieving or applying what they have learned
3. Developmental problems that appear at this stage
1. Conceptual understanding
2. Memory and/or retrieval of concepts, steps in algorithms, facts
3. Spatial/ nonverbal problems with estimation, setting up computations, geometry
4. Language problems -difficulty selecting the important data and appropriate operations in word problems
5. Attention problems with inconsistent errors, arranging columns, math detail, attention to signs, wild guesses, "careless errors," monitoring work.
3. Late Elementary and Middle School
1. During these years a child should
1. Develop more advanced computational skills
1. automatize math facts
2. mastery of multi-step problems
3. competence with fractions, decimals, percent and the relationships between them
2. Develop facility with word problems
3. Refine estimation skills
4. Refinement of geometric concepts
5. Automatize algorithms
6. Acquire basic computer and calculator skills
2. Questions to ask when assessing word problems
1. Is the child able to distinguish relevant from irrelevant information in word problems
2. Can the child discern relationships between content of word problem and appropriate algorithm
3. Can the child identify the order in which the information is needed in the algorithm
4. Can the child generalize from one word problem to another
3. Questions to ask when assessing computation
1. Does child plan his work
2. Does child establish an appropriate work tempo
3. Does child proofread or self-monitor
4. Does child have automatized facts
5. Does child work neatly
6. Can child engage in mental math
7. Can they maintain steps in memory
4. Problems that interfere
1. Language problems interfere with teasing out relevant information from math problems
2. Higher cognition problems interfere with word problems
3. Memory and sequencing can interfere with computation
4. Language problems interfere with learning specialized vocabulary
5. Higher cognition interferes with more abstract and symbolic areas such as algebra
4. Secondary School
1. At this level a child should
1. Develop formal operations to meet the demands of algebra, geometry, trigonometry, and calculus
2. Remember specialized vocabulary, formulae, notation signs, steps in proofs
3. Translate between different forms and symbols for the same concept (percent, fraction, decimal)
4. Develop ability to graph and plot equations
5. Develop ability to solve practical problems (tax, interest, rates,)
6. Use calculators and computers
7. Understand statistics, data collection and analysis
2. Questions to ask
1. Does child plan and self-monitor
2. Does child use appropriate work tempo
3. Dopes child have automatized facts, algorithms
4. Does child have solid conceptual vase
5. Does child understand proportion and ratio
6. Can child illustrate and explain concepts
5. Parameters of Math Assessment
1. Skills
1. Number knowledge
2. Math facts
3. Math notation
4. Written computation
5. Math vocabulary and verbal concepts
6. Advanced abstract concepts
7. Problem solving
8. Mental calculation and estimation
9. Applications (money, time, measurement)
10. Calculator use
2. For a breakdown of component skills in these areas see table 11-3, p, 423
6. Assessment methods
1. Types of assessment
1. Standardized tests
1. Are useful to measure skill acquisition
2. Measure accuracy only (not process) and so may penalize children with attention problems
3. May be out of synch with a school's curriculum
2. Reports and observations - direct conversation with math teacher, or questionnaire
3. Interview with student
2. Assessing attitude
1. Anxiety, math phobia
2. Resilience
3. relationship with math teachers



6. MANAGEMENT
1. General Recommendations - for a list see Levine p. 424-426
2. Management for Specific Developmental Dysfunctions
1. Attention Deficits - p. 427
2. Spatial Ordering Deficits - p. 427-8
3. Sequential Ordering Deficits - p. 428
4. Memory Dysfunctions - p. 428-9
5. Language Dysfunctions - p. 429
6. Higher Order Cognition - p. 430
7. Graphomotor Dysfunctions - p. 430
3. Math Survival
1. Use writing, journals, essays to teach and assess math
2. Focus on practical life applications