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Good Morning! Christopher Kaufman, Ph.D. (207) 878-1777 e-mail: [email protected] web: kaufmanpsychological.org 1 Mind Over Math The Neuropsychology of Mathematics and Practical Applications for Instruction I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally. ~ Calvin Trillin Agenda Morning Afternoon 8:30 - Neuroanatomy 101 (A Quick User’s Guide to the Brain) 12:30 Practical/Implications Strategies for Classroom and Remedial Instruction 9:00 - The Brain on Math (AKA: The Neuropsychology of Mathematics) 2:00 Mini-Break 2:15 More Strategies 2:45 Q&A 3:00 Adjourn 10:30 Break 10:45 When Brains and Math Collide! The Neuropsychology of Math Disorders (With a Side Trip into Math Anxiety) 11:30 Lunch Math refusal from an FBA perspective . . . The student who hides his head under his hood or exclaims, “This is BORING!” is usually saying, “I hate this repeated feeling of not being successful, and I don’t ever want to have to feel it again.” David Berg, Educational Therapist Author of, Making Math Real Your Turn . . 1. Choose a kid from your caseload who struggles significantly with math. 2. Take a few moments to complete the first part of the Personal Case Study Form Neuroanatomy 101: A Quick User’s Guide to the Brain DA’ BRAIN: Its two hemispheres and four lobes The Hemispheres Fancifully Illustrated . . . Sequential, Factual Processing Integrative, ‘Big Picture’ Processing Left Hemisphere • Where spoken and written language are primarily processed (greater hemispheric specialization in boys) • Where language originates (language-based thoughts develop in the left hemisphere) • Where phonemes, graphemes, grammar, punctuation, syntax, and math facts are processed • Where routine, overlearned information is processed Right Hemisphere • Has greater capacity for handling informational complexity because of it’s interregional connections • Has greater capacity for processing novel information • Tends to be more dominant for processing creative, imaginative, flexible thinking • Tends to be more dominant for emotional aspects of writing • More common source of spatial/visual-motor deficits Your Turn . . . Take a moment to consider . . • Which elements of math functioning would be more likely processed in the left hemisphere? • Which elements of math functioning would be more likely processed in the right hemisphere? Why? The Four Lobes FRONTAL LOBE PARIETAL LOBE OCCIPITAL LOBE TEMPORAL LOBE The Neuropsychology of Math (AKA: The Brain on Math) The Nature of Math • It’s sequential and cumulative (earlier skills continually form the basis for newer skills across the grade span) • It’s conceptual (lots of ideas and themes must be understood and ‘reasoned’) • It’s procedural (lots of rules and algorithms must be mastered to calculate – perform ‘numerical operations’ • It’s highly variable from a skill perspective (math is a many varied thing!) Arithmetic Skill: An Intrinsic Capacity? Research suggests . . • Infants demonstrate number sense early in development (Sousa, 2005) • 8-month olds can reliably distinguish individual objects from collections (Chiang and Wynn, 2000) Has math sense been selected for by evolution? (Sousa, 2004) Our most ancient ancestors were best able to pass on their genes if . . . • They could quickly determine the number of predators in a pack • They could determine how much to plant to feed the clan Math Ability & the Neurodevelopmental Functions (Portions adapted from the work of Mel Levine) Spatial-Motor Visualizing problems/procedures, comprehending angles (and other elements of geometry), creating charts, graphs, etc., and maintaining sufficient grapho-motor accuracy to solve problems correctly on paper Attention Maintaining sufficient cognitive energy and attention on work Executive Functioning Planning, organizing, monitoring the quality of work (also determining what is/is not important for problem solving) MATH Temporal-Sequential Following sequences and multiple steps (Levine) Memory Recalling facts, procedures, and rules, recognizing patterns, and problem solving Language Processing written language and spoken information – in directions, problems – and understanding/recalling technical math vocabulary Attention Controls There is no single math processing center! Working Memory Executive Functioning Neuromotor Functions Spatial Comprehension Language Memory (LTM) Left vs. Right Brain Math Skill In general terms . . • Left Hemisphere: More responsible for processing of arithmetic (tasked to determine exact answers using language processes) • Right Hemisphere: Responsible for estimating approximate magnitude using visual-spatial reasoning skills Verbal Functioning and Math Ability Related to the language centers of the temporal lobe and posterior frontal lobe The ability to store and fluidly retrieval digit names and math facts is mediated by the temporal lobe Frontal and temporal language systems are used for exact computations because we tend to ‘talk our way’ through calculations How much language is required to solve this? 1013 - 879 Side Bar Issue: Vocabulary Deficits and Math • Math is replete with technical terms, phrases, and concepts (i.e., “sum,” “factor,” “hypotenuse,” “perimeter,” “remainder”) • Math also requires the following of often detailed verbal instructions • Students with limited language comprehension skills can struggle greatly with math, even if they have no difficulty recalling math facts and the specific terms related to them! Visual/Verbal Connections Related to Math Functions • Also temporal lobe areas related to language functioning • Occipital-Temporal Convergence links the visual element of digits to their verbal counterparts • This area allows for the attaching of fixed symbols to numerical constructs (Feifer & Defina, 2005) Visual-Spatial Functioning and Mathematics • We’re talking primarily about processing in the parietal lobe (site of spatial processing) and occipital lobe (the site of visual processing) • Left and right hemispheres are involved, with the left being associated with arithmetic/sequential/factual processing and the right related to simultaneous/spatial/holistic processing Left Parietal Lobe: Center of Arithmetic Processing? Area associated with arithmetic processing 15% bigger In Einstein’s Brain! Side Bar Issue: Einstein’s Brain • Actually weighed a bit less than the average for brains of it’s time/age; • But, had greater neuronal density than most brains and was about 15% wider in the parietal lobe region (and had fewer sulci in this area) • Thus, he had somewhat greater brain capacity in the areas associated with arithmetic and spatial reasoning ability More on Right Hemisphere Functioning and Math Skills A (not the) visual-spatial processing center (left parietal also processes visual-spatial information) Approximations of magnitude are largely made in the right parietal lobe Mental rotation and similar spatial reasoning tasks tend to be processed in the right hemisphere Math concepts are ‘reasoned’ in the right hemisphere (the brain’s ‘big picture,’ ‘integration center’) Novel stimuli are processed in the right hemisphere Many aspects of math are visualspatial in nature • Visualization and construction of numbers • Visualizing of the ‘internal number line’ • Visualizing of word problems (easier to determine the needed operations if one can picture the nature of the problem) • Geometry (duh . .) Are boys intrinsically better at math than girls? • NO (pure and simple) • Boys do have better mental rotation skills • This may give them greater confidence in attacking certain kinds of math problems (Feifer & DeFina, 2005) • Overall, though, there is growing consensus in the field that any advantage boys have over girls in math is a product of cultural/societal convention Your Turn . . . Which figures to the right match the ones to the let? A closer look at the frontal lobe Central Sulcus (or ‘Fisure’) Math strategies and problemsolving directed from here! Frontal Lobe Specifics (Adapted from Hale & Fiorello, 2004) Dorsolateral Prefrontal Cortex Planning Strategizing Sustained Attention Flexibility Self-Monitoring ------------------------------- Orbital Prefrontal Impulse Control (behavioral inhibition) Emotional Modulation Motor Cortex Executive Skill and Math Math’s Changing Face (It’s new again) Out with the explicit teaching of facts and standard algorithms . . And in with constructionist math curricula that emphasize discovery learning and the selfconstruction of math know-how Executive dysfunction impacts: • • • • Self-directed learning Discovery-based learning Self-initiated strategy application Collaborative learning This is why so many kids with EFD have struggled with constructionist math curriculums BREAK TIME! Impact of Executive Dysfunction on Math Working memory problems lead to poorly executed word problems Impulse control problems lead to careless errors (e.g., misread signs) =? Attention problems lead to other careless errors (i.e., Forgetting to regroup, etc.) Organizational/planning deficits lead to work poorly organized on the the page (or work not shown) The Three Primary Levels of Memory: • Sensory Memory (STM): The briefest of memories – information is held for a few seconds before being discarded • Working Memory (WM): The ability to ‘hold’ several facts or thoughts in memory temporarily while solving a problem or task – in a sense, it’s STM put to work. • Long-Term Memory (LTM): Information and experiences stored in the brain over longer periods of time (hours to forever) The Brain’s Memory Systems Working Memory: Some kids have got ‘leaky buckets’ • Levine: Some kids are blessed with large, ‘leak proof,’ working memories • Others are born with small WM’s that leak out info before it can be processed Your Turn . . . A Working Memory Brain Teaser! I am a small parasite. Add one letter and I am a thin piece of wood. Change one letter and I am a vertical heap. Change another letter and I am a roughly built hut. Change one final letter and I am a large fish. What was I and what did I become? WM capacity tends to predict students’ ability to direct and monitor cognition. How Large is the Child’s Working Memory Bucket? Case 1: Rachel Recallsitall algorithm Case 2: Nicky Normal Case 3: Frankie Forgetaboutit fact directions fact algorithm facts directions algorithm 42 42 Working memory: A fundamental element of math functioning So much of learning and academic performance requires the manipulation of material held in the mind’s temporary storage faculties • Mental math (classic measure of working memory skill) • Word Problems • Recalling the elements of algorithms and procedures while calculating on paper • Interpreting and constructing charts/graphs The majority of studies on math disabilities suggest that many children with a math disability have memory deficits (Swanson 2006) Memory deficits affect mathematical performance in several ways: •Performance on simple arithmetic depends on speedy and efficient retrieval from long-term memory. •Temporary storage of numbers when attempting to find the answer to a mathematical problem is crucial. If the ability to use working memory resources is compromised, then problem solving is extremely difficult. •Poor recall of facts leads to difficulties executing calculation procedures and immature problem-solving strategies. •Research also shows that math disabilities are frequently co-morbid with reading disabilities (Swanson, 2006). Students with co-occurring math and reading disabilities fall further behind in math achievement than those with only a math disability. However, research shows that the most common deficit among all students with a math disability, with or without a co-occurring reading disability, is their difficulty in performing on working memory tasks. Let’s Look at a Classic Word Problem . . • Sharon has finished an out-of-town business meeting. She is leaving Chicago at 3:00 on a two-hour flight to Boston. Her husband, Tom, lives in Maine, 150 miles from Boston. It’s his job to pick up Sharon at the airport as soon as the flight lands. If Tom’s average speed while driving is 60 miles per hour, at what time (EST) must he leave his house to arrive at the airport on time? Math Anxiety Mathematics is the supreme judge; from its decisions there is no appeal. ~Tobias Dantzig Math Anxiety on a Brain Level (or, ‘When the amygdala comes along for the ride’) Amygdala Temporal lobe Hippocampus Bottom line: It’s crucial to keep kids from getting overly anxious during math instruction (or they may always be anxious during math instruction!) Research (and common sense) clearly indicates . . . As anxiety goes up . . Working memory Capacity goes down! The best math anxiety limerick ever? There was a young man from Trinity, Who solved the square root of infinity. While counting the digits, He was seized by the fidgets, Dropped science, and took up divinity. ~Author Unknown When Brains and Math Collide! Subtypes of Math Disabilities and Their Neuropsychological Bases Can you say, “Dyscalculia?” Sure you can!! Occur as often As RD’s!! Developmental Dyscalculia defined: DD is a structural disorder of mathematical abilities which has its origin in a genetic code or congenital disorder of those parts of the brain that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities adequate to age, without a simultaneous disorder of general mental functions (Kosc, 1974, as cited by Rourke et al., 2005) Huh?! Said more simply! Dyscalculia refers to any brain-based math disability! Epidemiology of Math Disabilities • Occur in about 1 - 6% of the population (Rourke, et al., 1997; DSMIV-TR); Geary (2004) says 5 – 8%. A recent Mayo Clinic study suggested the incidence in the general population could be as high as 14% (depending upon which definition of math LD is used . .) • Like all LD’s, Math LD occurs more often in boys than girls • MD’s definitely run in families (kids with parents/siblings with MD are 10 times more likely to be identified with an MD than kids in the general population) • Important take home point: Math disabilities (‘MD’s’) occur just as often as reading disabilities (‘RD’s’) – this has big implications for the RTI process!! Types of Math Disability (MD) 1. Verbal/Semantic Memory (language based, substantial co-occurrence with reading disabilties) 2. Procedural (AKA: ‘anarithmetria;’ substantial overlap with executive functioning and memory deficits) 3. Visual-Spatial (substantial overlap with NLD) Semantic/Language-Based MD’s • Characterized by poor number-symbol association and slow retrieval of math facts (Hale & Fiorello, 2004) • Commonly co-occur with language and reading disorders (Geary, 2004) • Are thought to relate to deficits in the areas of phonological processing and rapid retrieval/processing of facts from long-term memory • Math reasoning skills (i.e., number sense and ability to detect size/magnitude) are generally preserved (Feifer & DeFina, 2005) Error Patterns Associated with the Verbal/Semantic Subtype • These kids tend to struggle recalling and processing at the ‘what’ (as opposed to the ‘how’) level. • They’ll forget (or will have great trouble learning) the names of numbers, how to make numbers, the names/processes of signs (i.e.,might often confuse ‘X’ with ‘÷’), and multiplication facts • They’ll make counting errors and other errors related to the ‘exact’ nature of math (always have to ‘rediscover’ the answer to problems such as 8 + 4 and 7 X 3). • May arrive at the right answer, but have trouble explaining how they got there. The Procedural Subtype of MD (Feifer & DeFina, 2005; Hale & Fiorello, 2004) • Disrupts the ability to use strategic algorithms when attempting to solve math problems • That is, kids with this subtype of MD tend to struggle with the syntax of arithmetic, and have difficulty recalling the sequence of steps necessary to perform numerical operations (leads to lots of calculation errors!) • Often seen in conjunction with ADHD/EFD subtypes, because the core deficit is thought to relate to a frontal lobe/executive functioning weakness (particularly working memory difficulties and slow processing speed) • These kids tend to rely fairly heavily on immature counting strategies (counting on fingers and through the use of hash marks on paper) Working Memory and the Procedural Type of MD • How much working capacity and sequential processing skill is needed to solve the following? An elementary school has 24 students in each classroom. If there are 504 students in the whole school, how many classrooms are there? I forget how you do . . . Error Patterns Associated With the Procedural Subtype of MD • Like kids with verbal/semantic MD, kids with the procedural subtype make errors related to ‘exactness’ (as opposed to estimating magnitude or comprehending concepts) • Errors are not related to the ‘what,’ but are instead related to the ‘how’ (e.g., How do you subtract 17 from 32? How do you calculate the radius of a circle?) • These kids know their facts (e.g., might easily recall addition & multiplication facts), but struggle greatly with recalling the steps/procedures involved in subtraction with regrouping and multiple digit multiplication. • Often do better on quizzes of isolated basic facts, but struggle with retrieval of the same facts to solve word problems or longer computations The Visual-Spatial Subtype of MD • Heavily researched by Byron Rourke (leading researcher in the field of nonverbal learning disabilities – ‘NLD’) • This subtype relates to deficits in the areas of visualspatial organization, reasoning, and integration • Difficulties with novel problem solving generally compound math reasoning struggles • At a brain level, the deficits are thought to relate to processing deficiencies in the right (and, to some extent, left) parietal lobe (were visual-spatial-holistic processing occurs) Error Patterns Associated with the Visual-Spatial Subtype • Fine-motor problems incorrectly formed/poorly aligned numbers • Strong fact acquisition, but struggles with comprehending concepts • New concepts and procedures are acquired slowly and with struggle (must first understand visual concepts on a very concrete level before they can grasp the abstraction) • May invert numbers, or have difficulty grouping numbers accurately into columns • Tend to have marked difficulties grasping the visual form of mathematical concepts (i.e., may be better able to describe a parallelogram than to draw one) • Often have difficulty seeing/grasping ‘big picture ideas’ (get stuck on details and struggle with ‘seeing the forest for the trees’ Key Facts Related to Math Disabilities Across the Grade Span • The verbal/semantic subtype is usually most obvious in the early primary grades, given the emphasis on math fact acquisition (many kids with NLD ‘do fine’ in math through third grade or so). • The procedural and visual/spatial subtypes become more obvious as algorithmic and conceptual complexity increases! • Bottom line: As procedural and conceptual complexity increase, the demands on the frontal and parietal lobes increase (Hale & Fiorello, 2004) Student Profiling to Inform Instruction and Learning Plan Student’s Name: _______________ Attention/EF Language Memory Neuromotor Emotional Neuro Profile Math Fact Skill Academic Profile Strategies Algorithm Skill Math Concepts Problem Solving LUNCH TIME!!! 64 Operators Shameless Brookes Standing self-promotion Publishing brookespublishing.com $34.95 By! Company slide!!!! 65 Learning to Remember: December 7, 2010 Augusta Civic Center Essential Brain-Based Strategies for Improving Students’ Memory & Learning Christopher Kaufman, Ph.D. Implications for Instruction BRINGING THE NEUROPSYCHOLOGY OF MATH INTO THE CLASSROOM Firstly: The state of affairs . . . (An empty glass) There has been relatively little in the way of high quality math instruction research! Reading studies outnumber math studies at a ratio of 6:1 Conceptual and Procedural Knowledge Conceptual knowledge has a greater influence on procedural knowledge than the reverse Strong Conceptual Knowledge Procedural Knowledge Weak Sousa, 2004 Key Research Finding Adults often underestimate the time it takes a child to use a newly learned mathematical strategy consistently (Shrager & Siegler, 1998, as cited by Gersten et al., 2005) Step One: Understand a Child’s Specific Problem(s) • Look for deviations for normal development (re: the acquisition of counting and early arithmetic skills) • Look for error patterns that are suggestive of weakness in the semantic/memory, procedural/algorithmic, and visual-spatial domains An Important First Intervention Step: Look for Error Patterns (Hale & Fiorello, 2004, p. 211) • Math fact error (FE) – Child has not learned math fact, or does not automatically retrieve it from LTM (Teacher: Michael, what’s 4 X 4? Michael: Um, 44?) • Operand error (OE) – Child performs one operation instead of another (e.g., 6 + 3 for a 6 X 3 problem) • Algorithm error (AE) – Child performs steps out of sequence, or follows idiosyncratic algorithm (i.e., attempts to subtract larger from smaller number) • Place value error (PE) – Child carries out the steps in order, but makes a place value error (common among kids with executive functioning and visual/spatial deficits) • Regrouping errors (RE) – Child regroups when not required, forgets to subtract from regrouped column during subtraction, or adds regrouped number before multiplication Example of an Algorithm Error (revealed via a ‘think aloud’ examination) (Hale & Fiorello, 2004, p. 211) 64 +13 14 “First I look to see if it’s addition or subtraction. Okay, it’s addition, so you always go top to bottom and left to right. So I add 6 + 4, and that equals 10, and then 1 + 3 equals 4. And then I add them together, top to bottom, and so 10 + 4 equals 14.” A Great Calvin and Hobbs Example ‘John has a problem with multiplication’ • What kind of problem? How broad is the scope? • Kids who can’t (despite adequate instruction and chances to practice) seem to recall the product of 8 X 7 have a fact recall difficulty (LTM deficiency – temporal lobe) • Kids who have no difficulty recalling the product of 8 X 7, but can’t solve 16 X 7 on paper may have an algorithm process difficulty (working memory or arithmetic reasoning deficiency – frontal lobe or parietal lobe) THE CORE STRATEGIES 1. Emphasize the development of an internal number line (in grades K and 1) to build number sense 2. Teach the concept and the algorithm (not just the algorithm in isolation), and keep teaching the algorithm until mastery 3. Distributed practice works better than massed practice (smaller doses of practice over time is better than a lot all at once) 4. Emphasize the verbalization of strategies/algorithms as kids problem solve (and after they’ve arrived at a solution) 5. Build automaticity of fact retrieval 6. Minimize demands on working memory/simultaneous processing (encourage kids to download info from working memory to paper by encouraging thinking on paper) 7. Enhance the explicit structure of math problems (using multiple colors, graph paper, boxing techniques, etc.) 8. Body-involved, ‘kinesthetic learning’ is good! Strategies to Build Number Sense Meet Caleb Caleb’s a ‘feisty little guy’ (to quote his mother) who’s just entered kindergarten. He wore sandals to school, but took them off somewhere in the classroom and now can’t seem to find them. He’s knows his primary colors and all basic shapes, but his letter/number ID and formation skills seem low. He can count to 20 in a rote manner, but seems unsure as to what the numbers mean (e.g., yesterday said that 4 was more than 6). Also, his ability to count with 1:1 correspondence is still shaky (can only do it with direct adult support). He gets frustrated very easily in task contexts and is apt to cry and throw things when stressed. What, exactly, is number sense? • Definitions abound in the literature . . . Berch, 1998: Number sense is an emerging construct that refers to a child’s fluidity and flexibility with numbers, sense of what numbers mean, ability to perform mental mathematics, and ability (in real life contexts) ‘to look at the world and make magnitude comparisons.’ Number Sense and Environmental Factors • Most kids acquire number sense informally through interactions with parents and sibs before they enter kindergarten • Well-replicated research finding: Kids of moderate to high SES enter kindergarten with much greater number sense than kids of low SES status • Griffin (1994) found that 96% of high SES kids knew the correct answer to the question, “Which is bigger, 5 or 4?” entering K. Only 18% of low SES kids could answer the question correctly (this study controlled for IQ level) • Number sense skill in K and 1st grade is critical, as it leads to automatic use/retrieval of math info and is necessary to the solution of even the most basic arithmetic problems (Gersten, 2001) Building Number Sense • It’s critical that parents, during the preschool years, really talk to kids about numbers and amounts and magnitude (“Let’s count these stairs as we climb them!”) • Head Start and other preschool programs for low SES kids should really push number concept games and related activities (just as they should push phonological awareness activities as a precursor reading skill) • During the K and 1st grade years, it’s essential for children to develop a mental (internalized) number line and to ‘play’ with this line in various ways • Without strong number sense, kids often are unable to determine when a numeric response makes no sense (i.e., 5 + 12 = 512) 10 9 8 7 6 5 4 3 2 1 Building Number Sense: Some Concrete Strategies (Bley & Thorton, 2001) 1. More or less than 10? • 8+4: Is this more than 10 or less than 10? (kids should check with manipulatives and number line work) What’s 5+5? Is 5 + 9 more or less than that? How do you know? • Variations for older grades More or less than ½? Ask students to circle in green all fractions on a sheet that are more than ½. Closer to 50 or 100? Have students circle in green those numbers that are closer to 50 than 100, using both visual and ‘mental’ number lines Over or under? Provide repeated instance in which students are asked to decide which of two given estimates is better and explain their reasoning. E.g., 652 – 298 =? A. Over 400 B. Under 400 Building Number Sense: More Strategies (Bley & Thorton, 2001) 2. What can’t it be? Provide computational problems and a choice of two (or more) possible answers. Ask the children to predict which of the choices couldn’t be possible and to state why. Example: A. 28 + 37 = 65 B. 28 + 37 = 515 Verbalized response: The answer can’t be 515. It’s not even 100, because 50 + 50 is 100, and both numbers are less than 50. 3. What’s closest? Ask the children to predict which of the answer choices is closest to the exact answer? How do you know? Example: 92 – 49 = ? A. 28 B. 48 C. 88 It’s B. The problem is sort like 100 – 50, and the answer to that is 50, and so 48 is closest. Digi-Blocks Strategies Targeting Semantic/Memory Weakness Meet Katie . . . Katie is a generally shy and sweet-natured 7th grader with a longstanding speech/language impairment. Although her once profound articulation difficulties have abated in response to years of SL therapy, she continues to have a hard time with receptive language tasks of all sorts. She’s of basically average intelligence, but has gotten numerous accommodations over the years related to literacy tasks. Although math computation had been her area of relative strength, she’s had a much harder time in middle school now that the technical math vocabulary demands have really increased. Her father reports that she now “hates math” and says things like, “If they’d just show me what to do and make it clear, I could do it – I wish they’d just show me what they mean!” When language comprehension is the problem Carefully teach math vocabulary, with all the possible forms related to the different operations posted clearly in the classroom Addition Subtraction Multiplication Division Sum Add Plus Combine Increased by More than Total Take away Remaining Less than Fewer than Reduced by Difference of Product Multiplied by Times Of 3 X 3 = 3(3) Quotient Per A (as in gas is $3 a gallon) Percent (divide by 100) Operations Language Chart in a Simpler Form Add = Plus = + Subtract = Take Away Multiply Divide = = Times = Divided By = Minus = – X = Per = ÷ When language comprehension is the problem • Link language to the concrete (have a clear visual and kinesthetic examples of all concepts readily available) • Teach math facts and basic vocabulary in a variety of ways (brains love multi-modal instruction!) • Use lots of manipulatives to clearly demonstrate ‘taking away,’ ‘total,’ ‘divisor.’ • Make liberal use of kinesthetic/multisensory demonstrations • Have kids put math vocabulary into their own words (and then check for the accuracy of these words!) Illustrating the Pythagorean Theorem c a Teacher: John, can you remind us what an hypotenuse is? b John: Um, nope – I haven’t got a clue . . . Teacher: John, we’ve spent the last two days talking about this stuff. John: So?! I don’t remember, All right?! What’s your problem?! Geez!! 13 5 12 Other language targeted strategies • Trying to always present a concrete visual (‘draw it out’) whenever you present the oral/verbal form of math concept (kids who have significant language deficiencies should have quick ‘cheat sheets’ available) • Keep verbal instructions short and to the point • Having kids read instructions into a tape recorder and then play them back When factual (declarative) memory is the problem • Ensure that the child clearly grasps the concept (i.e., that 3 X 4 mean ‘3 four times’) • If the child doesn’t grasp the concept, then teach the concept in multiple ways until he does (kids grasp/recall math facts much better when they ‘get’ the concepts behind them) • Drills (i.e., flashcards) really work (kids retain rote information best when it’s acquired/practice right before sleep) • Fact family sorts (e.g,. Sorting flash cards by into ‘families’) • Use games (e.g,. ‘Multiplication War’ - see supplemental handout) • Graph progress with the kid (kids often love to see their improvement, and the graphing, by itself, is a worthwhile math activity) Three Kinds of Math Facts Autofacts – Math facts a student knows automatically Stratofacts – Math facts a student can figure out using an an idiosyncratic strategy (i.e,. counting on fingers and using hashmarks) “Clueless” Facts – Math facts a student cannot recall or access at all Meltzer et al., 2006 Gimme the facts, Madam, just the facts . . “Terrific Tens” Strategy + 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 10 10 10 10 10 10 10 10 10 Meltzer et al., 2006 And then there’s good ‘ol ‘Touch Math’ Developers and it’s proponents claim that it ‘bridges manipulation and memorization’ Also often called a ‘mental manipulative’ technique Multi-sensory, in that kids simultaneously see, say, hear, and (most importantly) touch numbers As they learn to count and perform an array Of computational algorithms Published by Innovative Learning Concepts Curriculum now extends into secondary grades Multiplication Fact Strategies 0 Rule: 0 times any number is 0 1’s Rule: 1 times any number is the number itself 2’ Rule: Counting by two’s 5’s Rule: The answer must end in a 5 or 0 (e.g., 35 or 60) 10’s Rule: The answer must end in a 0 (10, 40, 80, etc.) 9’s Rule: Two-hands counting rule 2 hands ‘Rule’ when it comes To solving the tricky 9’s! Meltzer et al., 2006 A key developmental asset in teaching kids division and division facts . . . Greed (balanced by an insistence on fairness) “How many do we each get?” Strategies Targeting Executive Functioning (Procedural/Algorithmic) Weakness Meet Andrew . . Andrew, a fourth grader, knows his multiplication and division facts cold, but has had gobs of difficulty ‘getting’ double/multiple digit multiplication and has had even more difficulty performing even the most basic aspects of long division (to quote his teacher: “He’s just so all over the place with it!”). Although Andrew is a reasonably wellmotivated youngster who’s attended some extra help sessions with his teacher (and will seemingly ‘get’ the multiplication and division algorithms in these sessions), he seemingly ‘forgets’ the procedures by the time he gets home or to school the next day (Mom: “It’s like I’m always at square one with him on this stuff”). Completing assignments of all kinds is also a big issue for this kid. The most important thing to remember in helping ADHD (“EFD”) kids with math It’s all about . . . Diminishing demands on working memory Mastery of algorithms is important in the end, but . . Go slowly, in a very stepwise manner, and scaffold, scaffold, scaffold!! Download as much as possible into the child’s instructional environment, with emphasis given to presentation of algorithm steps in easy to follow formats A key distinction: Factual Memory vs. Procedural Memory Factual memory . . Procedural memory Refers to an individual’s ability to recall discrete bits/units of information Refers to an individual’s ability to remembers processes; that is, procedural steps (e.g,.7 X 7 = 49, the capital of France is Paris, my mother’s middle name is Dorothy, ‘sh’ makes the /sh/ sound) e.g., How to bisect an angle, how to swing a golf club, how to bake blueberry muffins, how to divide 495 by 15 Working memory demand: Working memory demand: Fairly minimal Moderate to marked, depending upon the process being recalled Helping EFD (‘ADHD’) Kids with Math: First Steps • To the extent possible, avoid multiple step directions (and good luck with that . . .) • Have the kids do one thing (and only thing) at a time (e.g., “Let’s just first circle all the signs on the page” or “let’s just highlight the key words in this word problem”) • Mel Levine: Break algorithms down into their most basic sub-steps and carefully, slowly teach each sub-step. Thus, in teaching two digit by one digit multiplication (47 X 6) • First ensure the child’s single digit multiplication facts are solid (or that he is at least facile in the use of the chart/grid) • Second, achieve mastery of single by double digit multiplication without regrouping (24 X 2) (will likely need lots of massed practice at this stage) • Third, introduce the concept of ‘carrying’ in double digit multiplication, but do so in a manner that makes use of the parts of the times tables a kid has mastered (e.g., 24 X 5) (again, lots of massed practice here) • Fourth, bring in more challenging multiplication elements from the higher, ‘scarier’ end of the times table (e.g., 87 X 9) • Than move, after mastery, by adding a third digit to the top number, and then a fourth, always building in plenty of time for massed practice, and distributed practice in the form of reviews of earlier, easier stuff. Helpful Strategies to Aid Algorithm Acquisition and Practice • Graph paper rocks! • Box templates are even better • Box templates that include written reminders are even better • Box templates that include written reminders and include color coordination are even better A good multiple digit multiplication ‘box template’ X + (Adapted from Bley & Thorton, 2001) A better multiple digit multiplication ‘box template’ + 7 2 3 X 3 2 1 4 4 6 2 1 6 9 2 3 1 3 6 (Adapted from Bley & Thorton, 2001) Long Division Algorithm Box Template 7 — 0 4 9 3 4 8 2 8 – 6 8 6 3 R5 5 4 3 = 4 X 4 X 4 = 64 Pneumonics/Heuristics: Excellent Ways to Help EFD Kids Learn and Retain Arithmetic Algorithms Does McDonalds Sell Burgers Done Rare? 1.Divide 2.Multiply 3.Subtract 4.Bring Down 5.Repeat (if necessary) Improving Error Checking Top Three Hits The 3 most common errors a kid exhibits in math Example: Steven’s Top 3 Hits: 1. Misreading directions 2. Misreading signs 3. Arriving at errors that can’t possibly make sense. P.O.U.N.C.E P – Change to a different color pen or pencil to change your mindset from that of a student to a teacher O – Check Operations (Order right?) U – Underline the question (in a word problem) or the directions. Did you check the question and follow the directions? N – Check the numbers. Did you copy them down correctly. In the right order? Columns straight? C – Check you calculations. Check for the types of calculation errors you tend to make. E – Does your answer agree with your estimate? Does your answer make sense? Strategies Targeting Visual/Spatial Weakness For Kids with NLD: Emphasize the Verbal 1. Kids with pronounced visuo-spatial comprehension/integration deficits often struggle with forming in LTM visual images of objects and particularly struggle with visual representations of concepts (i.e., an isosceles triangle) 2. Emphasize the verbal (simple, direct, concrete) over the visual whenever possible 3. The goal for these students is to construct a strong verbal model for quantities and their relationships in place of the visual-spatial mental representation that most people develop. 4. Descriptive verbalizations also need to become firmly established in regard to when to apply math procedures and how to carry out the steps of written computation. 5. Complex visuals can really freak out kids with visual/spatial weakness (avoid busy graphs, maps, and charts) Other Strategies Targeting Visual-Spatial Weakness • Fewer items on a page • Avoid flashcards (too visual – better to do rote learning via auditory exercises – e.g., via rhymes) • Use blocks to isolate problems on the page (see next slide) • Emphasize the use of concrete manipulatives in the teaching of abstract concepts (being able pick up, feel, and talk about manipulatives helps these kids) • Encourage these kids to ‘think on paper’ (help them draw very simple pictures – stick figures -- to represent what is going on in a math problem (Levine) • Kinesthetic learning experiences may be particularly helpful for this population, providing clear verbal explanations accompany the demonstrations Addition (‘plus’): Do these first Subtraction (‘minus’): Do these next 47 +56 88 -45 83 +31 45 -24 29 +93 62 -39 68 +55 96 -48 Division Cards – A Great Device for NLD Kids Problem: 5 255 Question 1: Is there a number which can be multiplied by 5, and be equal to or less than 2? Answer: No, and so zero is placed above the 2 and the card is shifted to the right to get a bigger number. Question 2: Is there a number which can be multiplied by 5, and be equal to or less than 25? Answer: Yes, and the number is 5, so a 5 is placed above the dividend. Etc. 05 5 25 John’s Division Card MAKING THE ABSTRACT CONCRETE A) The problem: What’s 5/8 of 16? ___ ___ ___ ___ ___ ___ ___ ___ B) Concrete illustration of 5/8: C) Concrete illustration of 5/8 of 16 ___ ___ ___ ___ ___ ___ ___ ___ * * * * * * * * * * * * * * * * (Adapted from Bley & Thorton, 2001) D) Answer is 10 “Buy Out”: A great technique for kids who are ‘motivationally challenged’ Operates from the perspective That few things are as motivating As the chance to get out of work Thus, kids are motivated to work By the opportunity to ‘work their way Out of work’ E.g.: For every two problems you do, you get to cross out one! 34 X 45 56 X 13 89 X 64 92 X 35 76 X 56 83 X 83 69 X 31 78 X 64 27 X 59 39 X 37 71 X 82 90 X 90 Two Effective (Evidence-Based) Remedial Programs Case Studies/Student Profiling