 Why do you still have to count on your fingers?
 I'm terrible at math
 I hate math (and math hates me)
Summing Up the Evidence on Math and LD
Let's be honest...math is a subject area that has a terrible reputation. No matter how old (or young) you are, it's likely that you can recall hearing phrases like:
And for students who struggle with learning disabilities, math (both computation and problem solving) can be a source of enormous frustration.
At long last, a number of spotlights are being directed at the area of math and in particular, the science underlying math learning. Some very exciting work is now underway to discover the nature of math learning, the concepts that are important (and perhaps essential) for skill development in math, and the types of experiences and instructional strategies that characterize effective teaching and that have a positive impact on student learning.
Our featured guests, Drs. Daniel B. Berch and Michele Mazzocco are among the national leading experts in the area of math difficulties and disabilities.
Dr. Sheldon Horowitz (Moderator):
Before we begin, please note that the assertions and opinions expressed here, along with any recommended resources (books, articles, websites, other reading materials, curricula, tests, etc.) are those of the authors and should not be taken as representing either official policies of or endorsements by the NICHD, the NIH, the U.S. Department of Health and Human Services, the Johns Hopkins University, or the Johns Hopkins Hospital and Health System.
Let's begin with our first question.
Question from Blair Chewning, Lower School Math Chair, 4th gr. teacher, Collegiate School, Richmond, VA:
While you may not wish to endorse a specific elementary mathematics program, can you speak to the criteria we should look for in a program that is beneficial to both the LD child, as well as to other students? (Our LD students are mainstreamed in our independent school.)
Dr. Michele Mazzocco and Dr. Daniel Berch:
If a program includes attention to the basic skills, opportunities to rehearse, revisit, and practice prerequisite skills, and does not assume that everyone is either on board or at the same level of understanding, it is more likely to be of benefit to both typically achieving children and those with math difficulties. At the very least, one should avoid using programs that coerce a child to move on to the next level without having demonstrated proficiency at the prior level.
Your question makes me think of some work we recently completed in my research lab (MM's) looking at children's ability to rank order fractions or decimals according to quantity, such as stating that 1/2 is bigger than 1/4 which is bigger than 1/9. We found that many children who could not accurately rank order sets of 8 to 10 fractions or decimals in 6th grade continued to fail on this simple task in 8th grade. A task as simple as asking children to read the decimals aloud helped determine who was at risk for this continued failure  that is, children who could not explain what "ohpointfive" meant, and would not produce "five tenths" as an alternative label even with prompting. Our results will appear in a future issue of Developmental Science, but in the mean time, think of ways to see if your child understands the meanings of the specific mathematical notations and symbols.
I’ll add that many of the children in our study who could not rank order fractions and decimals could obtain accurate solutions when adding or subtracting fractions, by using procedures they had mastered. This is an example of the inappropriateness of having children learn skills beyond the foundations that they have not mastered, if all the children are doing are carrying out meaningless steps.
What kinds of learning strategies can be taught to students who struggle, especially in the problemsolving aspect of mathematics?
Dr. Michele Mazzocco and Dr. Daniel Berch:
This depends in large part on the type of difficulty the student is experiencing. These could include, but are not limited to difficulties with: basic sense of number, understanding operations or procedures, with memory span or shortterm memory, attention and organizational skills more broadly defined, or even difficulty understanding the vocabulary of mathematics. These are all factors that have been demonstrated to influence problem solving. Moreover, the answer to your question depends further on whether the child also has reading difficulties.
The place to start is to figure out what the child already knows about math and what possible sources of interference present obstacles to the individual child's problem solving. Lynn Fuchs and Douglas Fuchs have done the leading research on enhancing problem solving, especially among primary school age children with math LD. They address how one major difference between problem solving and calculations has to do with the linguistic components of the former, but that other influences are also involved. To maximize effectiveness, the goal is for the intervention used to generalize to other problems, thereby helping children develop and use strategies.
Selfregulated learning strategies are the topic of one of their 2003 publications in volume 95 of the Journal of Educational Psychology, pages 306315; the benefits of schemabased instruction is the topic of their work appearing in volume 96 (2004) of that journal, pages 635647. For children with math LD, they report the need to foster mastery of problemsolution rules for these benefits to generalize to independent problem solving later on; moreover, they demonstrate the effectiveness of using worked examples as one method to boost problemsolution rules.
My son has dyslexia which effects his reading, writing and math abilities. We have been able to use an OrtonGillingham based structured langauage program to address his language deficits, but no one seems to know a proven intervention for math. Has the research bore out a proven program for math similar to the structure and proven effectiveness of an OG remediation for language?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Although there are a variety of programs available commercially, few programs for math intervention have been tested empirically to date, and none have been evaluated as thoroughly as many of the principal programs for dyslexia. The work of Drs. Ginsburg, Fuchs, Dehaene, and Griffin demonstrate some very different approaches to assisting children who have difficulties with mathematics, depending on the nature of the difficulty and the age of the child. Remember that when selecting a remedial approach, it is necessary to identify the problem  and that mathematical difficulties are not necessarily all likely to be linked to one cause. For instance, research by Bryon Rourke, David Geary, and, more recently Nancy Jordan have demonstrated some differences in children depending on whether the children are experiencing difficulties in math per se, or in both math and reading. Also, see the earlier answer to Anne Osowksi.
As reported in Preschool Matters (Oct. '06), a research team from Northwestern University led by Greg Duncan found that children's math skill at school entry is the most important predictor of later school success and achievement. This rather surprising information has left early childhood educators scrambling for more definitive direction: exactly what math skills should we teach beyond numeral recognition and 1:1 counting? What's the best handson approach to teach math skills to 3 and 4year olds? Can we identify LDMath in children so young? What insights re: math skills would you offer specifically to early childhood educators?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Smokie, you raise the important issues of early identification and early intervention, and also prevention  factors to which all educators want to attend in order to facilitate overcoming obstacles to achievement as soon as possible. To do so prevents children from getting even further behind, and avoids the "waiting for failure" alternative. Keeping in mind that the best predictors of math difficulties are not firmly established, you are correct in pointing out the link between kindergarten performance and later math achievement. Awareness of this association enhances efforts by early childhood educators to provide an environment rich in mathematical learning opportunities. Such an approach serves to nurture mathematical thinking in all children, not only those "at risk" for math LD. Let's talk about what that would look like:
Nurturing math in early childhood doesn't necessarily mean using explicit paperandpencil math lessons. Instead, it includes mathematicsrelated activities and the use of mathematicslanguage in all centers of the room (dramatic play, blocks, art, etc.). For instance, if the dramatic play area is set up as a restaurant, you could have menus with pictures of labeled quantities (two pieces of toast, one tortilla with cheese, three pickles with a sandwich, etc.). You could use number words and comparisons ("the tortilla and cheese costs a lot more than the side of pickles!") But remember that math is not just about numbers; it is about relationships between numbers; concepts such as more or less, equivalence, symmetry in block design patterns, sequences, and so forth. Thus, early childhood educators can provide children with opportunities to learn and express all of those aspects of math, and should use language to reflect those concepts themselves. Susan Levine and her colleagues have demonstrated that more mathtalk by preschool teachers leads to more gains in math knowledge among their preschool students. You can see more examples of this math talk in their article, which appeared in Developmental Psychology in 2006, volume 42, pages 5969.
Using math talk is not only effective for teaching math, learning to talk about math with children is effective for assessing math skills in young children (and older children, too!). This is addressed in Ginsburg, Jacob, and Lopez's book, The Teacher's Guide to Flexible Interviewing in the Classroom: Learning what Children Know about Math.
In terms of the "handson approaches" to which you refer, we want to point out that the use of manipulatives per se does not make an approach effective. Manipulatives can make learning math fun and engaging, and can provide a concrete example of an abstract concept like addition  their use clearly has some benefits. However, as Utall, Scudder, and De Loache have demonstrated, manipulatives need to be treated as symbols, if their use is to generalize to math concepts. Their work reminds me of a 9yearold I worked with who insisted that the drawing I produced of a rectangle was "ten" and not "one" because of its size. Indeed, it was the size of a 10unit rod  the child equated number with size, not quantity, and resisted otherwise until I pointed out to her that my very large computer monitor was indeed only one computer!
One of the few researchbased mathematics curricula for prekindergarten and kindergarten children is Big Math for Little Kids, developed by Herbert Ginsburg, Carole Greenes and Robert Balfanz.
I find many of my LD students have trouble with organizing the information for multiple step problem solving. Do you have any suggestions for how to assist them in organizing the steps and understanding what is being asked?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Kathy, what you are describing are possible "executive function" difficulties. Executive functions are very deliberate skills such as planning, inhibiting a response, shifting attention, or monitoring the use of strategies when solving a problem. In our research, we've demonstrated that children who have very efficient performance on executive function skills are more likely to do well at mathematics, at least during 1st to 5th grades (as reported in a paper I (MM) published with a former student of mine, Sara Kover, earlier this year). Others, including Rebecca Bull and Kimberly Espy have reported not only an association between executive functions and math performance but even an association between specific executive function skills (especially inhibition) and early, emerging math skills. In terms of suggestions for organizing materials, your question is relevant to the organizational components stressed by Lynn and Douglas Fuchs (see our response to Anne Osowksi). However, executive functions do not come into play only with problem solving. One way to tap a child’s thinking processes is to use the "flexible interviewing" techniques we mentioned in response to Smokie's question.
What are your recommendations for parents in developing numeracy skills for preschool children who are having early intervention services for processing deficits and/or learning disabilities?
Dr. Michele Mazzocco and Dr. Daniel Berch:
The recommendations for parents will vary quite a bit depending on one's theoretical framework regarding the presumed core difficulties underlying the development of the dyscalculia. Whereas some researchers place emphasis on number sense or the basic processing of numerosity (such as Brian Butterworth), others emphasize the supporting skills of working memory, language, and spatial reasoning (such as David Geary). In reality, there are probably some children whose difficulties are aligned with one or the other perspective, and some with difficulties in both kinds of skills. If one maintains a broad concept of early math and provides the kind of learning environment we described in our response to Smokie Brawley, one would be addressing both numeracy and other kinds of early math skills.
It is not unusual for parents (and educators) to ask whether dyscalculia (or LD in math) can be "prevented." What are your thoughts about this question?
Your question about preventing dyscalculia would also lead to very different responses depending on one's theoretical perspective, because there is still much ongoing research and dialogue regarding both the nature and origins of dyscalculia. For example, some contend that it is a biologically based disorder. If so, then while it may not be entirely preventable, the level of mathematics proficiency a child achieves could vary depending on the type and degree of instructional and parental support, despite having an inherent difficulty with some aspects of mathematics.
The important point we are emphasizing here is that a biological basis does not mean that ability level is "fixed." If dyscalculia is interpreted to refer to any kind of mathematics difficulty, then it can surely be prevented in those children for whom poor math performance results from inadequate or absent instruction. Can poor achievement be due to other factors, such as anxiety or lack of motivation? If so, then the influences on achievement outcome will differ yet again. In reality, it is likely that some children with dyscalculia do have an inherent difficulty, and it has been argued by some researchers that the use of the term "dyscalculia" be limited to children with a biologically based mathematical learning disability. These same researchers also contend that the term "mathematical difficulties" should be used to represent the broader group of children whose difficulties in math result from a wide range of possible factors.
So, back to prevention: from the answers given thus far, we've mentioned that factors ranging from teachers' mathtalk to schemabased intervention can improve math knowledge or problem solving, but the research base is still too thin to conclusively pinpoint any 'best' programs or any single curriculum that addresses all of the potential causes of math LD or other math difficulties.
Keep in mind that prevention relies at least in part on being able to predict who is at risk for math LD. There is good reason to believe that we can predict math outcome for at least some children. In a paper published in the journal, Learning Disabilities Research and Practice (Volume 20, pages 142155), we reported that children with persistent poor math achievement from kindergarten through third grade could be differentiated from their peers based on test performance in kindergarten. These children were less accurate at reading numbers, number constancy (recognizing that a quantity does not change as a result of rearranging the items in a set), and magnitude judgments (choosing the larger of two numbers). But, we don't know if there are other predictors that are better, or as effective. And, the accuracy of prediction depends in large part on how we define math LD.
What are the "must read" professional text books on learning difficulties in Math? What measures should be included in an evaluation where math is a primary area of concern? What evidencebased math interventions are currently available to teachers? For what specific functional deficits? In the area of reading, there is great research to guide instruction (i.e. where to start, what works, etc.). Is there any similar research in the area of math instruction?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Mathematics does not reflect a single construct, so it is difficult to give a short answer to any parts of this excellent question!
First, I'm not sure what you mean by professional textbooks, so I'll give you a few examples (this is not an exhaustive list!). Although some books address mathematical cognition in general rather than math difficulties specifically, they can provide valuable insights into how children learn math. One such book is an edited volume by Jamie Campbell, The Handbook of Mathematical Cognition (it includes several chapters on mathematical difficulties). Dehaene's book, The Number Sense, helps readers appreciate the complexity involved in understanding quantity. Ann Dowker's book on Individual Differences in Arithmetic provides a comprehensive, interdisciplinary treatment of the multiple components of arithmetic learning. Baroody and Dowker also have an edited volume entitled The Development of Arithmetic Concepts and Skills: Constructive Adaptive Expertise.
Unlike intervention programs for children with dyslexia, there are no widely used programs that have been demonstrated by many researchers to have a clear advantage over others. Ann Dowker describes some of the existing programs in a chapter of her book; also, Baroody and Coslik's teacher's guide, entitled Fostering Children's Mathematical Power: An Investigative Approach To K8 Mathematics Instruction, highlights the issues involved in quality math instruction, and thus addresses good instruction for all children.
For a truly outstanding resource, I would suggest that you look for a soontobe published text by Brookes Publishing called Why Is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities.
Visit the transcript for this LD Talk for a link to excerpts from this terrific new volume.
My son, severely dyslexic, does well in math, is in Phase II (upper division math) in middle school. His strength is grasping the concepts, logic, patterns of mathbut he does not "see" that 10 less 1 is 9, he must count, same with times tables, etc. Of course, this is thus very time consuming and fraught with error. Why is this so and what is the best way to address it? Thank you
Dr. Michele Mazzocco and Dr. Daniel Berch:
Ms. Haynie, I can certainly appreciate the dilemma your son finds himself in, especially as he obviously exhibits a number of strengths in mathematics. That a student can function quite effectively with respect to certain facets of math, yet exhibit some comparatively immature solution strategies in others can frequently be indicative of a learning disability in mathematics. Having said this, I must caution you that I am not in a position to make such a diagnosis with respect to your son, given that this would require extensive, facetoface assessment of his strengths as well as his weaknesses, and usually over a period of time (preferably more than one year) in order to chart the stability or consistency of his performance.
You note that while his conceptual knowledge of math appears to be in good shape, his factual knowledge and procedural skills seem to lacking. Although the development of conceptual, factual, and procedural knowledge and skills are usually interdependent, there is some evidence that these comprise separate components of arithmetic learning. Some of the evidence in support of the distinctiveness of these components comes from studies demonstrating socalled "double dissociations" in adults with known brain damage (note that I am NOT suggesting your son has experienced brain damage). For example, while the conceptual knowledge for some of these individuals is much better than their factual and procedural knowledge, the converse is true for others. Of course, such work does not tell us directly why this has occurred, but does provide us with some indication that different regions of the brain may subserve these different components, even the in the typically achieving individual.
As to possible remedial strategies for your son, I suggest that you read some of the work of Ann Dowker, who has developed what she calls the Numeracy Recovery Program, an intervention program which was designed to target difficulties that children are experiencing with specific components of arithmetic processes and procedures. A summary of this work is described in chapter 12 of her book, Individual Differences in Arithmetic (see also the Mathematics Recovery Program described in this same chapter).
Which are the characteristics that an effective program must have to address difficulties experienced by young children in learning mathematics in their early education years?.
Dr. Michele Mazzocco and Dr. Daniel Berch:
This is a topic of ongoing research, and is related to our answers to the questions from Smokie Brawley and Beth Lusby. For young children, one program that have been tested by the researchers developing them is Big Math for Little Kids (Ginsburg, Greenes, and Balfanz). Other programs include Griffin's Number Worlds (formerly called, Rightstart), The Berkeley Math Readiness Curriculum, 'Round the Rug Math, and Building Blocks, some of which are reviewed in Dowker's chapter 12 (as mentioned earlier). For slightly older primary school age children, Lynn Fuchs and Doug Fuchs have tested interventions for problem solving among elementary school age children, including primary school.
Some of the characteristics of these programs include attending to more than number (such as strategies, or math language, or relationships among numbers)  although a focus on numeracy is also included. Another important point at all ages, not just the early years, is to remember that assessment is part of instruction. Especially for math, which builds on the foundation of earlier established skills, it is essential to master and revisit the earlier concepts and skills before moving forward. I can't tell you how often I've seen children struggle with fractions when they still had not mastered place value concepts; it is no wonder that these children fail to recognize the similarity between 0.10 and 0.1, or why they fail to see the difference between 0.10 and 10.0. Likewise, I've seen children struggle with regrouping when they still have not mastered single digit addition.
What do you foresee emerging from the National Mathematics Advisory Panel?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Dr. Witzel,
As described in the President's Executive Order, the National Mathematics Advisory Panel is charged with advising the President and the Secretary of Education on the best use of scientifically based research to advance the teaching and learning of mathematics. The Panel will examine and summarize the scientific evidence related to the teaching and learning of mathematics, focusing in particular on preparation for and success in learning algebra. Their findings and recommendations will appear in a final report, which must be submitted no later than February 28, 2008. As to my foreseeing what may emerge with respect to the ultimate conclusions and recommendations of the Panel, neither I (as a panel member, albeit ex officio) nor any other member is a position to speak for the Panel itself. However, examination of the progress reports issued by the four task groups at the recent meeting held in New Orleans (see http://www.ed.gov/about/bdscomm/list/mathpanel/5thmeeting/pr.html) should provide some indication of the topics being tackled by the Panel in an effort to fulfill its charge. Furthermore, a preliminary report has just been released, which can also be found on the Panel's website.
Question from Ping Collis, Orton Gillingham ed. therapist, Kailua, Hawai'i:
 Has a specific neuroanatomical region been identified and imaged? [as with Shaywitz et.al.]
 Has an approach such as Slingerland/Orton Gillingham been developed, wherein educators can impart math to students in K12?
Dr. Michele Mazzocco and Dr. Daniel Berch:
To date, the vast majority of research concerning the neuroanatomical correlates of basic numerical cognition has been carried out with young adults whose mathematical knowledge and skills fall within a typically achieving range. Briefly, by making use of various neuroimaging techniques, such as functional magnetic resonance imaging (fMRI), cognitive neuroscientists such as Stanislaus Dehaene, among others, have begun to explore regions of the brain that appear to be recruited during the processing of comparatively simple numerical judgment tasks, such as "Which is larger, 5 or 7?" When "approximate" numerical tasks are administered (i.e., that do not require calculation), the evidence suggests that several different brain regions are activated"most notably in the posterior area of the parietal cortex. Evidence pertaining to "exact" calculation tasks, such as those involving simple arithmetic computation, also seem to activate a fairly widespread cortical network. However, unlike approximate tasks, those that require exact calculation seem to particularly activate the angular gyrus in the left hemisphere.
One of the first imaging studies of mental arithmetic to date using typically achieving children (ages 9 to 18 years) by Dr. Susan Rivera and colleagues suggests that with development, there appears to be a decreasing dependence on working memory and attentional resources associated with the prefrontal cortex, and an increasing functional specialization of the left posterior parietal cortex.
Finally, a recently published article by a team led by Dr. Karin Kucian at the University of Zurich describes one of the first studies that have tested children with developmental dyscalculia (DD) using fMRI, administering both approximate and exact arithmetic tasks. These investigators found that while the neuronal activation pattern of children with DD did not differ from that of a typically achieving group on a measure of exact calculation, they did exhibit weaker activation in almost the entire neuronal network for approximate calculation.
However, as Drs. Tony Simon and Susan Rivera argue, it is important to remember that even for typically achieving adults we should not assume that the neuronal circuits which have been shown to be recruited during numerical processing are necessarily hardwired into the brain. Similarly, these authors point out how we cannot be certain that math LD necessarily arises from lesions to or dysfunctions in those specific circuits. Moreover, bear in mind that while this kind of research is very exciting, it is still so new that definitive conclusions as well as implications for remediation are probably a long way off.
My 14 year daughter was identified as dyslexic when she was 8 years old. While modifications have been in place in her IEP and her reading is at grade level, her math skills have always been below grade level. My experience over the years is that the teachers in the public schools do not have a full understanding on learning disabilities regarding math, much more emphasis is placed on reading skills. What needs to be in place to better educate our teachers with regards to the realities of math and learning disabilities? Is there hope?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Your observation about teachers having more knowledge about reading disabilities than about math LD is a reflection of the research field. The truth is that we have 30 years worth of solid, replicated research on reading that has been applied to classroom practices and intervention programs, whereas the same is not true for mathematics. We all wish we had the answers for what the most effective classroom and intervention strategies for mathematics might be. Keep in mind how multifaceted math is relative to reading. When we say that a child is below grade level in math, do we mean with respect to fact retrieval, or efficient calculation, remembering problemsolving rules, grasping concepts of arithmetic operations, or some other facet of this complex domain? What needs to be in place to educate our teachers are validated screening assessments that help teachers not only spot those children who struggle with math, but also identify why the child is struggling. This is what leads to effective intervention  that is, by first determining what it is that needs intervening. There is much hope, as exemplified by the numerous studies and researchers we have referred to in our earlier responses.
Crosslanguage studies (English Vs. Kannada, one of the Indian languages)suggests that memory span for numbers depends on the number of phonological units in the number lexicon. Do you suggest that we need to sensitize the teachers on this aspect?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Your question is very relevant to the whole issue of the relationship between language and mathematics. Although we are not sure that it is necessary to sensitize teachers to only the very specific aspects in which confusion results from math and general vocabulary, we agree that in general the language  mathematics links should be on teachers' radar screen. For instance, the work of Chris Donlan and others have demonstrated an interaction between linguistic and visuospatial cognition that have implications for mathematical performance and understanding in children with specific language impairments. Yet for all children, there is a need to learn the language of number words, operation terms, concepts like "more" or "greater than," and so forth. Sensitizing teachers to this influence is a good idea, along with the potential cultural influences of what "more" or "add" may mean when translated into the child's primary language. Francis "Skip" Fennell has addressed the confusion that may occur in even typically developing children.
Nationally teachers and/or school district have acknowledged that teachers need assistance with teaching and in some cases understanding math concepts. If this is true, is the diagnosis of LD (specific to math) appropriate for most children or are children being misdiagnosed? If the teachers can not explain, demonstrate or model math concepts, how can they teach math to students?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Carla, what you are really addressing is how to define learning disabilities versus difficulties due to other factors. Teacher education is an important component of child achievement in general, so you are correct in recognizing its role in mathematics education. Along those lines, the answers to some of the other questions we’ve addressed here concern math instruction. Current alternatives for defining LD place emphasis on children's response to intervention  if a child continues to struggle with math (in this case) despite standard instruction, the child is more likely to receive a diagnosis of LD than if the child makes gains once instructional intervention is used. What your question reflects is the need to enhance quality math education for all children, and to be aware of classrooms or schools where the quality of education needs to be enhanced. For instance, many researchers have demonstrated for decades now the association between academic achievement and poverty (including the work of Duncan, Eccles, and Jordan), so the kind of schoolwide enrichment that schools in low socioeconomic regions may benefit from would differ from the kind of intervention needed by a child in an alreadyenriched mathematical learning environment who has an inherent difficulty with the concept of quantity.
When you ask if children are being misdiagnosed, it depends on who is doing the diagnosis. If it is recognized that a child needs help with math, that does not necessarily mean that the child has math LD, and teachers do not make diagnoses. Regardless, if the child needs help with math, help is warranted. And you are right that aspects of instructions (or the lack of instruction) can lead to incomplete mastery of mathematics achievement. Still, we believe that there are children whose difficulties with math are inherent  a notion supported by the work on the genetics of mathematics disabilities by Steven Petrill and Robert Plomin. But, just because a problem is inherent does not mean it will not respond to intervention. So, efforts to help children achieve in math require attention to teacher education, but also to children whose difficulties occur despite high quality instruction.
Is there any evidence for a developmental course for math difficulty? I'm thinking of myself at schoolage (even into young adulthood) who appeared to have significant difficulty grasping/organizing/holding/applying procedural math concepts (e.g., geometry, algebra) who later seemed to "mature" into forming (through subsequent classes and life experiences) a moderate level of facility in classroom math. I am wondering if there is any basis to a neurologicallybased developmental course in math learning.
Dr. Michele Mazzocco and Dr. Daniel Berch:
To date, the evidence does not suggest a single course for the typical development of mathematical knowledge and skills, let alone for their atypical development. For example, according to Dr. Robert Siegler (see Developmental Science, January 2007 special issue), children's thinking (of any kind) is highly variable at every level of analysis, from neural and basic associative levels to higherlevel facets of cognition. Moreover, Dr. Siegler points out that this variability exists not only between individuals, but within them as well.
With respect to math difficulties and disabilities, Dr. Mazzocco's own longitudinal research clearly demonstrates the rather uneven course of development that a math learning disability may take, or at the very least the complexities associated with assessing such development as accurately as possible over successive years. Additional longitudinal research of this type is necessary if we are going to be able to not only make more accurate predictions about children who are at risk for developing math difficulties, but also to be able to assess how appropriate interventions at key points in development can modify developmental trajectories.
As to the neurological basis (or more appropriately, bases) for the development of mathrelated knowledge and skills, there is little doubt that the development of brain structures and neural circuitry are inextricably linked to the acquisition of mathematical knowledge and skills. Nevertheless, it is important to bear in mind that the developmental courses are not likely to show a single pattern, but rather multiple patterns and trajectories which are influenced in part by environmental factors. (For additional information see answer to Ping Collis.)
I have read that the academic progress of students with LD, like dyscalculia, may plateau around the secondary school years. Have you seen research on this, and if so, what do you believe is the general reason or cause of it?
Dr. Michele Mazzocco and Dr. Daniel Berch:
Your question has two general categories of "reasons" that may underlie the slowing of academic progress in math during secondary school. The first concerns the fact that children with LD need to expend effort to complete processes that may be automatic for their peers  such as sounding out words, or calculating rather than retrieving "math fact" problems, which essentially leaves less effort available for learning or more advanced skills. This is similar to the observations that children with dyslexia have difficulty with reading comprehension due to exerting effort to more basic reading problems. However, just as we know that other cognitive underpinnings can lead to poor reading comprehension beyond word level reading disability; there are additional cognitive influences on mastering "higher" mathematics (beyond elementary arithmetic)  including working memory, short term memory, attention, and the other factors we've mentioned in our response to Susan Skok's, you and others today.
The cognitive underpinnings represent one category of potential reasons for the finding that you've raised. Other influences are related to motivation or may even result from mathematics anxiety. In high school in particular, students not only select which math courses to take from an array of options, they can even choose to avoid math classes altogether once they have completed the high school graduation requirements. In other words, there is no single reason or cause for poor math achievement, or leveling off of math achievement, after elementary school.
Can the Test of Math Abilities be used for diagnosing a math disability? If not, what can we use? Currently we are using the WJIII and TOMA2.
Dr. Michele Mazzocco and Dr. Daniel Berch:
We feel that the instrument one decides to use is not as important as is the need to:

attend to several aspects of math,

acknowledge the limitations of any given test or subtest,

evaluate the child's conceptual understanding of math, and

observe the child's strategies and problem solving approaches while engaged in math tasks, rather than solely his or her answers.
Moreover, look beyond whether answers are 'right' or 'wrong.' If correct, how are the answers obtained  efficiently, and with understanding of what the answer or problem means, or routinely, with little evidence of comprehension? If the answer is wrong, does it reflect a slight miscalculation or other common type of error (e.g., 7 X 6 = 49), or does it reflect conceptual gaps (such as 100 + 100 = 2000)?
With reading development, I feel like I have a good understanding of the developmental progression in phonologic, linguistic and other contributory factors and how they intertwine in an individual. I don't have as strong a feeling about how size concept, abstract numeric knowledge, calculation automatization, etc. contribute to the severity and development of LD. Is there a concensus on developmental hierarchies of math subskills and how they affect the classroom performance?
Dr. Michele Mazzocco and Dr. Daniel Berch:
The quick answer is "no." There is no consensus per se. There is consensus on the developmental hierarchies within certain math subskills, such as demonstrated by Robert Siegler's work. For example, he has demonstrated clear developmental progressions in how children use counting strategies during addition, and has demonstrated a shift from logarithmic to linear trends in number line and estimation skills.
I have a student who has asked me, "Why don't numbers make sense? They aren't any different than words! Why doesn't my brain understand 1/.3 = 10/3? Why can I figure out how to work out a formula but I am unable to put a word problem into a formula?" What is going on in this student's brain with her having number sense in some areas but not others?
Dr. Michele Mazzocco and Dr. Daniel Berch:
These are interesting and important questions, Cindy. First off, numbers are both similar to and different from words. As you know, the socalled number words (that is, "one," "two," "five hundred and twentyseven," etc.) are one of the many formats in which numeric information can be presented. However, as you also know, such information can be represented by Arabic numerals, not to mention, Roman numerals, placevalue blocks, etc. What complicates matters here is that "numerosity" is essentially abstract  for example "threeness," which could stand for "three" peanuts or "three" elephants. So, part of the difficulty for children is learning to translate or "transcode" from one representation of numeric information to another. According to Dr. Stanislas Dehaene, the difficulties associated with learning mathematics has to do in part with the developing brain needing to coordinate and orchestrate communication among these various representational codes, which is largely a function of the prefrontal cortex  an area of the brain that we now know continues to develop throughout adolescence and even into young adulthood.
Regarding your specific question about success with formulas and failures translating word problems into a formula, keep in mind that a child may succeed when carrying out steps and procedures due to strong, rote, procedural learning, which  if unaccompanied by conceptual understanding  will not transfer to word problems. An alternative obstacle to success with word problems is difficulties with the language requirements associated with word problems. These are some of the issues explored by Fuchs and others, as mentioned in our response to Dr. Rao's and Anne Osowski's questions.
How can a child, who can remember great details of stories and things he was told years before, lack the memory capacity to retain basic math. He has no difficulty with geometric forms and problems dealing with angles and formulas. He excels in English...and has an unusually high vocabulary.
Dr. Michele Mazzocco and Dr. Daniel Berch:
Your question demonstrates how dissociable aspects of mathematics can be. There are some very unusual cases of individuals who can add successfully, but not multiply; or vice versa, or can add and multiply but not subtract. There are cases of individuals who can calculate responses but not estimate, or vice versa. Delazer and McCloskey have each reported on case studies of individuals whose acquired (rather than developmental) mathematical difficulties (typically resulting from brain injury) demonstrate these dissociations rather dramatically. Your question reminds me (MM) of a child I've worked with who has excellent reading and expressive language skills, and whose teachers insist is not exerting sufficient effort in math. As one teacher put it, "she must be able to do the math, because she is one of the strongest readers in the class! I am sure she is just not trying." By pointing out to the teacher that this was like expecting a great snowboarder to necessarily be an outstanding violinist, I believe I helped the teacher realize that math and other academic domains involve different skill sets, different strengths, and different sources of difficulty. Perhaps it is the case that, for the child to whom you are referring, the math facts simply don't make as much sense to him as do the stories he remembers, and the shapes he is able to reason with.
This leads me to a concluding point: years ago, I had a professor of mathematics education who regularly insisted that if we simply explained to children that math always "makes sense," our students would be able to learn from our instruction. The limitation with that thinking is the assumption that math will make sense to all children as readily as it does for others. But, we know that this is not the case! Just as some children don't "hear" the rhyme of words as easily as do others, some children to not grasp "mathematical sense" as readily as do other children. In other words, math sense is not just common sense!
That concludes our discussion for today. Thanks to everyone for the thoughtful questions and thanks to our experts, Dr. Daniel Berch and Dr. Michele Mazzocco, for their time today.
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