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A toolkit designed to help teachers and parents to identify dyscalculia.

This toolkit provides information on how to get support for children with dyscalculia. It is designed to raise awareness of dyscalculia and mathematical learning difficulties.

It also signposts opportunities to learn more about dyscalculia through our short courses, presentations and blogs.

Background

Good mathematical abilities have been related to successful financial, physical, and mental health outcomes. Yet, a significant proportion of individuals (up to 22%) have mathematical learning difficulties and up to 6% of children in primary schools have dyscalculia1. Although this suggests dyscalculia is as prevalent as dyslexia (1-2 children per class of 30), dyscalculia is less well known, and it has been less well researched in comparison to other learning difficulties.

Individuals with developmental dyscalculia (or dyscalculia for short) may experience a number of delays in acquiring foundational mathematical skills and concepts and may experience later difficulties with developing more advanced mathematical facts and procedures. These difficulties cannot be explained by general underlying ability in the below average range or other developmental disorders.

As dyscalculia is heterogeneous and has a high co-morbidity rate with other developmental disorders, such as dyslexia and attention deficit hyperactivity disorder (ADHD), the cause of dyscalculia is still unclear. However, recently there has been further research in the field and a number of new trials have evaluated mathematical interventions, which have demonstrated benefits for children with dyscalculia and mathematical difficulties. These studies have the potential to provide further insight into the causes of dyscalculia and how we can best provide support for these children.

References

Morsanyi, K., van Bers, B.M.C.W, McCormack, T. and McGourty, J. (2018). The prevalence of specific learning disorder in mathematics and comorbidity with other developmental disorders in primary school-age children, British Journal of Psychology, 109(4): 917-40.

This toolkit was co-created with teachers and UCL staff with expertise in mathematical development and dyscalculia:

Special thanks go to all of the teachers who have helped us put this toolkit together.

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Awareness of Developmental Dyscalculia and Mathematical Difficulties toolkit (ADD UP)

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A toolkit designed to help teachers and parents to identify dyscalculia.

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Dyscalculia App Wheel (Webinar from CALL Scotland, 2021) 

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Dyscalculia App Wheel (Webinar from CALL Scotland, 2021) 

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Presenter(s): Shirley Lawson, CALL Scotland

Curriculum for Excellence Benchmarks for Numeracy and Mathematics.

Do you have iPads in your class / school and have pupils who could benefit from extra support with their maths? This webinar will point you in the right direction of free and low cost apps that are engaging, educational and can boost the confidence of pupils when they experience success within the app and can use this as a transferable skill in a real life scenario or separate mathematical setting.

Newsletter: join thousands of other people

Once a month we'll send you an email with news, research and thoughts, as well as training courses and free webinars you may wish to attend.

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Dyscalculia App Wheel - apps to support pupils with numeracy difficulties

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Presented by CALL Scotland (University of Edinburgh) and focussing on assistive technology to support learners with additional support needs.

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November 2023 article: Developmental dyscalculia is not associated with atypical brain activation

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November 2023 article: Developmental dyscalculia is not associated with atypical brain activation

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It Just Doesn't Add Up: Explaining Dyscalculia and Overcoming Number Problems for Children and Adults

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It Just Doesn't Add Up: Explaining Dyscalculia and Overcoming Number Problems for Children and Adults [Moorcraft, Paul] on Amazon.com. *FREE* shipping on qualifying offers. It Just Doesn't Add Up: Explaining Dyscalculia and Overcoming Number Problems for Children and Adults

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I follow and learn from a lot of fantastic experts via Twitter - be sure to check them out!

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Discover Mathigon, the Mathematical Playground. Learning mathematics has never been so interactive and fun!

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The progression videos are a series of videos that show how math concepts build and grow through the grade levels. Here is a post that gives a "behind the scenes" of what it takes to make the videos.

Click on the image below to watch the progression video and download the resources.

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Dyscalculia is a specific and persistent difficulty in understanding numbers which can lead to a diverse range of difficulties with mathematics. It will be unexpected in relation to age, level of education and experience and occurs across all ages and abilities.

Mathematics difficulties are best thought of as a continuum, not a distinct category, and they have many causal factors. Dyscalculia falls at one end of the spectrum and will be distinguishable from other maths issues due to the severity of difficulties with number sense, including subitising, symbolic and non-symbolic magnitude comparison, and ordering. It can occur singly but often co-occurs with other specific learning difficulties, mathematics anxiety and medical conditions.

(BDA Definition)

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Dyscalculia - British Dyslexia Association

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What is Dyscalculia? Dyscalculia is a specific and persistent difficulty in understanding numbers which can lead to a diverse range of difficulties with...

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The aim of this new SASC guidance on dyscalculia, SpLDs and maths learning difficulties is to provide assessors with updated, evidence based, operationally effective definitions and procedures which will enable them to differentiate between three categories of mathematics learning difficulties:

1. Dyscalculia - a SpLD whose core feature is a problem with sense of number

2. Other SpLDs which do not include a problem with sense of number, but which may have an impact upon mathematics learning

3. Maths learning difficulties arising from lack of appropriate teaching environmental factors or other medical conditions.

The guidance can be accessed from the Downloads page of this website or click the link below.
Dyscalculia guidance

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The aim of this new SASC guidance on dyscalculia, SpLDs and maths learning difficulties is to provide assessors with updated, evidence based, operationally effective definitions and procedures which will enable them to differentiate between three categories of mathematics learning difficulties: 1. Dyscalculia - a SpLD whose core feature is a problem with sense of number 2.

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DYSCALCULIA is a specific learning difficulty with mathematics, primarily arithmetic. It was defined in a UK Government document in 2001 as: 'Dyscalculia is a condition that affects the ability to acquire mathematical skills. Dyscalculic learners may have a difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence.'

It is defined in the USA (Diagnostic and Statistical Manual of Mental Disorders, 2013) as:
'Difficulties in production or comprehension of quantities, numerical symbols, or basic arithmetic operations that are not consistent with the person's chronological age, educational opportunities, or intellectual abilities.'

Research into dyscalculia has lagged far behind research into dyslexia, but this is beginning to even out. The new powerful neurological tools are shedding new light on this difficulty. Despite this, our understanding of dyscalculia lags behind our knowledge of dyslexia. However, researchers agree that there is no single profile. There are many contributing factors. It is described as a heterogeneous problem which is compounded by the constellation of demands made by maths.

Bearing this in mind, the estimate of people who are dyscalculic is around 5%. There will be a spectrum of maths difficulties. A UK survey into young adults suggests that about 22% are functionally innumerate. My own research suggests some startlingly depressing statistics of success rates for basic problems in a wide range of topics, for example, multiplication and fractions. Again, the research is not definitive, but it some research suggests that the occurrence of dyscalculia is the same for girls and boys, and may even be more prevalent in girls.

Impact

Whilst it is socially acceptable to say, 'I'm hopeless at maths' and for people to accept that statement without criticism, it doesn't mean that there are no everyday problems that are a consequence of those difficulties with maths. Obviously, there may be problems with finances and money, time management, remembering sequences of numbers (for example, phone numbers, pin numbers), reading timetables, correctly diluting medicines, speed of calculating and achieving the maths qualifications needed to enter many professions and jobs.

Maths, more than any other subject, causes anxiety and that anxiety usually amplifies problems. I think one of the key contributors to this situation is the fear of being wrong, the fear of negative evaluation. Much of maths is very judgemental, for example, whilst saying 7 plus 8 is fourteen is close to the right answer, the answer is wrong. Children rarely get credit for being almost right.

There is now neurological evidence to confirm a link maths anxiety to a fear of physical harm.

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DYSCALCULIA is a specific learning difficulty with mathematics, primarily arithmetic. It was defined in a UK Government document in 2001 as: 'Dyscalculia is a condition that affects the ability to acquire mathematical skills.

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all.

Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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Build Math Minds | Learn to LOVE Teaching Math

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With the implementation of new standards, assessments, and (for some) new curriculum, teachers are feeling bogged down and yet left out on their own to make sense of it all. Gain knowledge of how kids learn math so that you can confidently close your textbook and do fun lessons that meet your students where they are.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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About dyscalculia

Dyscalculia is a specific learning difficulty that affects roughly 5% of the population. It is often described as "like dyslexia, but for maths". I find this a helpful analogy because we have all, nowadays, heard of dyslexia but not everyone has, yet, heard of dyscalculia.

Dyscalculia is a relatively new field. In the UK, the first official government recognition of its existence came in 2001 when a definition was published by the DfES (as it was then), a definition that has not since been added to or amended; compare this date with the UK's earliest government definition of dyslexia in 1970.

Recent and current research will gradually increase our understanding of dyscalculia so that, in time, teachers and other practitioners will have enough information to underpin a consensus about the best way to help dyscalculic learners. In the meantime, there is a general feeling that the best approach is to build on what has proved to be successful with dyslexic learners, namely multisensory teaching methods combined with an awareness on the part of teachers and schools of the precise nature of the students' difficulties.

Dyscalculia is often called a 'maths' difficulty but it actually only affects those aspects of mathematics to do with number. For example, a dyscalculic learner may have no problems at all with geometry or algebra but be unable to remember a simple multiplication fact. It would be more accurate, therefore, to think of dyscalculia as a difficulty with arithmetic, or perhaps as a specific arithmetic difficulty, rather than a difficulty with maths as a whole. Dyscalculia need not be a bar to understanding mathematical concepts or to taking a degree in a scientific or mathematical subject.

I have met and worked with many learners with developmental dyscalculia over the years and the most striking thing they have in common is their lack of 'number sense'. For those of us who are more numerate, it can be difficult to imagine what it means to have absolutely no feel for numbers. To give you some idea of what it might be like to live in a world in which everything to do with numbers feels alien and challenging, here are three examples from my own experience of working with learners with dyscalculia. Note that these pupils were all of average, or above average, intelligence.

Learners with dyscalculia have no sense of the size of the quantity that is represented by a number. For example, immediately after establishing that there were 13 glass nuggets in a group, by touching and counting them, an 8-year old was presented with a group of roughly 30 of the same nuggets. The challenge was to estimate how many. After much thought (but no opportunity to count them) the pupil guessed that there were 11, without noticing that this guess was less than 13, despite seeing me add to the previous quantity. Another example comes from a highly intelligent 9-year old who was trying to work out what to add to 4 to make 9, by reasoning from a known doubles fact. The pupil said, "Could it be 5? No, because, well, 4 + 4 is 8 but 5 is a much bigger number than 4, so 4 + 5 would be much bigger than 9."

Learners with dyscalculia may have no idea that there are repeating patterns within our decimal number system. For example, a 9-year old trying to add 14 + 10 used fingers to count up 10 steps, one at a time, starting from 14. Another child of the same age, who produced a fast and correct answer to 14 + 10 was unable to work out 26 - 6 without using fingers to count back, one by one.

Learners with dyscalculia may have very little ability to conjure up an accurate mental number line in their heads. I have lost count (no pun intended!) of the number of children who have counted out aloud for me as follows: "... 95, 96, 97, 98, 99, 100, 200, 300 ...".

Indicators for dyscalculia are:

No 'feel' for numbers and relative quantities

an inability to subitise (see without counting) even very small quantities

an inability to estimate whether a numerical answer is reasonable

weaknesses in both short-term and long-term memory

an inability to count backwards reliably

a weakness in visual and spatial orientation

directional (left/right) confusion

slow processing speeds when engaged in maths activities

trouble with sequencing

a tendency not to notice patterns

a problem with all aspects of money

a marked delay in learning to read an analogue clock to tell the time

an inability to manage time in their daily lives.

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About dyscalculia Dyscalculia is a specific learning difficulty that affects roughly 5% of the population. It is often described as "like dyslexia, but for maths". I find this a helpful analogy because we have all, nowadays, heard of dyslexia but not everyone has, yet, heard of dyscalculia. Dyscalculia is a relatively new field.

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Note: For general tips on what Numeracy Routines (e.g. Number Talks, WODB, Estimation 180, Notice/Wonder, Clotheslines, etc.) are, why they are important, and how to begin please click on the Numeracy Routines pages. The following page is specific to Number Talks.

To begin, I'd recommend you attend a session in your District on Number Routines where you can engage in the activities as a learner and see them modelled. If that isn't available to you, I would dig into one of the following books and/or the following websites.

Number Talks
Author Sherry Parrish

Recommend for Intermediate Teachers

​Making Number Talks Matter
Authors Cathy Humphreys and Ruth Parker

Number Sense Routines
Author Jessica Shumway

High Yield Routines K - 8
Authors Ann McCoy, Joann Barnett, Emily Combs

Dot cards
Dice
Dominoes
Five and Ten Frames
Rekenreks
Hundreds Frames
Pictures of items
Images of real life arrays - this PPT has many great images - Greg Tang books also have great images that can be used.
Number Strings - Video example of a Grade Four class
Any addition, subtraction, multiplication, or division question

The teacher will present a problem on the board. Students will be given thinking time to solve the problem mental. I typically introduce "Thinking Thumbs" and explain to the students that I do want them to rob their friends of their thinking by blurting out answers. Sherry Parrish feels that this strategy is better than having students raise their hands because she when several hands are in the air, students often stop thinking because they are used to teachers calling on the students with raised hands for the answer. When students are ready, they will indicate by holding up one thumb up to show they have a solution. If the students have more than one way to find the solution they will raise a finger by their thumb. When most students show that they have a solution (e.g., majority of thumbs are up) that is when I ask students to share their thinking. Try not to rush the thinking time as Parrish states "the aim is not to reward speed, but to focus on the thinking about mathematical relationships (Parrish, 2010, Number Talks, pg. 18).

Next I ask students to share their answers. I record all of these, even the incorrect answers. In Number Talks, we use incorrect answers to discover misconceptions students have. Next students are asked to communicate their thinking. What strategies did you use? The teacher will record what the students says and ask if anyone used a different strategy. The rest of the class is actively listening and thinking about the strategies being shared. They are considering and thinking about other students' strategies and checking to see if they make sense. Through this, they are building their own repertoire of strategies. The teacher facilitates the discussion and is responsible for asking guiding questions, such as "How is this strategy similar or different to _______'s strategy?", helping students to make connections.

Before you provide a question for a Number Talk, you will want to think through all the potential responses your students might provide. You will need to consider how you will capture each student's thinking.

Some potential strategies for addition include:

counting all

counting on

doubles/near doubles

making tens

friendly numbers (e.g., 23 + 48 - Student adds 48 + 2 to get 23 +53 = 73 and then takes away 2 from 71

compensation (e.g., 8 + 6 The student takes away one from 8 and adds one to 6 to get 7 + 7)

decomposing numbers by place value (e.g., 12 + 43 becomes (10 + 40 and 2 + 3)

adding in chunks (e.g., 45 + 28 The student begins by adding 45 + 20 = 65 and then 65 +8 = 73)

Some potential strategies for subtraction include:

adding up

decomposing numbers into easy to remove parts e.g., 65 - 32 The student thinks of it as 65 - (10 + 10 + 10 + 2)

decomposing and subtracting by place value components

Teacher Education By Design (tedd.org/about/TEDD.org) - Created by the University of Washington designed to support teachers
You will need to register to access this free site full of amazing resources. Nested under the "Content Area" of Mathematics you will find links to Quick Image, Number Strings, Mental Math and True/False number talks. In part one, the introduction, you will find everything you need to get you started, such as videos from various grades, and teacher info sheets. In part two, the planning section you will find templates and suggestions. Part three, enact, provides key resources you can use and part four, provides valuable reflection prompts.

Number Talk Images - ntimages.weebly.com/ntimages.weebly.com - Curated by Pierre Tranche
This website has a collection of images, including dots, number strings, arrays that can be used for Number Talks.

Janice Novakowski - Richmond School District's Math Consultant Teacher.
Janice has done a lot of work in classrooms with Number Talks and has written about these experiences on her blog. I highly recommend you click on this link to her blogs.sd38.bc.ca/sd38mathandscience/?s=number+talks website to see the many ways Number Talks can be used in various classroom contexts.

Number Talks Build Numerical Reasoning by Sherry Parrish

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Note : For general tips on what Numeracy Routines (e.g. Number Talks, WODB, Estimation 180, Notice/Wonder, Clotheslines, etc.) are, why they are important, and how to begin please click on the...

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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Contact Email 1+ (313) 300-1901 Is it a Learning Disability? Dyscalculia Checklist Learning Disabilities Checklist Expert Diagnosis Learning Disabilities Hub

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About the Author

About the Author

The Understood Team is made up of passionate writers and editors. Many of them have kids who learn and think differently.

Reviewed by

Reviewed by

Daniel Ansari, PhD is a professor in developmental cognitive neuroscience at Western University, Canada.

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The Understood Team is made up of passionate writers and editors. Many of them have kids who learn and think differently. Daniel Ansari, PhD is a professor in developmental cognitive neuroscience at Western University, Canada.

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spld dyslexia dyspraxia dyscalculia education aimhigher university progression support provision

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National Network providing multi-faceted provision for learners with SpLD

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education / special education / learning disabilities

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special education - learning disabilities

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teaching of students with specific learning difficulties

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teaching of students with specific learning difficulties / needs

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education / special education / learning disabilities

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special education - learning disabilities

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teaching of students with specific learning difficulties / needs

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The Dyscalculia Assessment [Emerson, Jane, Babtie, Patricia] on Amazon.com. *FREE* shipping on qualifying offers. The Dyscalculia Assessment

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more trouble with maths: a complete manual to identifying and diagnosing mathematical difficulties (nasen spotlight)

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special education - learning disabilities

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teaching methods & materials - mathematics

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bird; chinn; diagnostic toolkit; dyscalculia; dyspraxia; emerson; mathematical difficulties; mathematics; maths

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education / special education / learning disabilities

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education / teaching methods & materials / mathematics

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education/teaching methods & materials - mathematics

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special education - learning disabilities

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teaching methods & materials - mathematics

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teaching of specific groups & persons with special educational needs

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teaching of students with specific learning difficulties

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teaching of students with specific learning difficulties / needs

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education / special education / learning disabilities

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education / teaching methods & materials / mathematics

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education/teaching methods & materials - mathematics

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teaching of specific groups & persons with special educational needs

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teaching of students with specific learning difficulties / needs

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More Trouble with Maths: A Complete Manual to Identifying and Diagnosing Mathematical Difficulties (nasen spotlight)

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More Trouble with Maths: A Complete Manual to Identifying and Diagnosing Mathematical Difficulties (nasen spotlight) [Chinn, Steve] on Amazon.com. *FREE* shipping on qualifying offers. More Trouble with Maths: A Complete Manual to Identifying and Diagnosing Mathematical Difficulties (nasen spotlight)

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Forefront Education: Number Sense Screeners

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Forefront Education: Number Sense Screeners

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This page has moved.

Please find our page devoted to the K-5 Universal Screeners for Number Sense project now linked here: https://forefront.education/solutions/usns-project/

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Forefront Education - Universal Screeners for Number Sense

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An index of some of our favorite lessons. We recommend starting with openers or games if this is your first time here.

Want more?
The Math for Love curriculum offers enough supplementary lessons to enrich the entire school year, or teach a month-long summer program.

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An index of some of our favorite lessons. We recommend starting with openers or games if this is your first time here. Want more? The Math for Love curriculum offers enough supplementary lessons to enrich the entire school year, or teach a month-long summer program.

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teaching of specific groups & persons with special educational needs

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bird; butterworth; dyscalculia; dyslexia; emerson; maths; numeracy; numerical difficulties

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teaching of specific groups & persons with special educational needs

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Dyscalculia: from Science to Education [Butterworth, Brian] on Amazon.com. *FREE* shipping on qualifying offers. Dyscalculia: from Science to Education

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Understanding Maths Learning Difficulties by Judy Hornigold

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Understanding Maths Learning Difficulties by Judy Hornigold

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​Whether you are an experienced teacher, a teacher trainee, a teacher educator, a parent, a researcher or a policy maker, we hope you will find this website interesting and useful. Specifically, we want to share with you the benefits of te﻿aching and learning mathematics through stories and creative writing, as well as to offer practical ideas for teachers and parents on how to embrace this approach.

This website has been designed and continuously developed based on the feedback we receive from the key stakeholders as noted above. There are plenty for you to do on this website, including, but not limited to:
Please take your time to explore the website using either the menu bar at the top of the page or from the menu below.

Finally, consider s ubscribing to our free quarterly newsletters, and please help spread the word about our project and its website :-)

Best wishes,

Dr. Natthapoj Vincent Trakulphadetkrai B.A. (Hons), M.Sc. (Oxford), M.Sc. (London, UCL), Ph.D. (Cambridge), QTS

@NatthapojVinceT Natthapoj.org

Associate Professor of Mathematics Education
​University of Reading (UK)

Founder & Principal Investigator
​MathsThroughStories.org
​

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"A book by Steve Chinn is welcomed and a real pleasure to read but it also makes us feel uneasy when there is still so much to do. His wealth of experience relating to maths in schools cannot be surpassed globally. His insight and perspectives in this book make it a 'go to' book, I believe this is not only for teachers of maths (and parents) but also for any educator who want to gain a deeper understanding of how we need to take a learner's perspectives. While it is focused on maths, we can also learn about the emotions that are evoked when it is harder for learners to engage in the subject. This is a reflective book honed from extensive experience and practice but it also remains a really practical book." - Professor Amanda Kirby

"In his latest book Steve Chinn is reflecting upon his extensive experience of teaching mathematics in the classroom and provides a fresh view on how to support learners that are left behind by the current system. He is taking what he has learnt from teaching mathematics to pupils with learning difficulties and distills it into a seasoned vision of how to teach mathematics to all learners. This book is highly readable with many examples and illustrations from his teaching practice. Chapters cover a wide range of important areas of teaching mathematics ranging from an overview of mathematical topics that cause the most difficulties, mathematics anxiety and other learner characteristics that might affect learning to a captivating chapter on the use of materials and visual images. The book is thought-provoking and provides a good overview of the obstacles a successful learner has to overcome." - Dr Silke Goebel, Associate Professor, University of York, UK

"In his book, Steve Chinn uses his vast experience to strip back maths learning and teaching. He focuses on some fundamental questions relating to how numbers are considered and how this can inform teaching practices. A variety of insightful, practical teaching strategies are provided, and a series of anecdotes are used to bring these to life. The book fosters an inclusive approach to maths learning and teaching. In particular, consideration is given to ways in which learners may approach maths, emphasising the need to tailor presentation and communication to effectively support the development of conceptual understanding. Moreover, the book crosses boundaries and considers how emotional and cognitive barriers may impede maths learning. A critical and common-sense approach is taken when discussing assessment of maths learning difficulties. The book is written in the author's typical accessible style and should appeal to any audience interested in supporting maths learners, especially those who struggle." - Dr Thomas Hunt, Associate Professor in Psychology, University of Derby, UK

"This book is a must-have for any teacher training programme and maths staff room. Drawing upon Steve's vast background from research, training and the classroom, this book will make both new and experienced teachers reflect upon their own practice. What I particularly admire about Steve is that he doesn't prescribe what we should do, but offers us a critical yet friendly eye into what he has seen over many decades. This book raises the bar in terms of books that educate teachers about how to teach maths." - Bobby Seagull, school maths teacher, author of The Life-Changing Magic of Numbers and BBC presenter for the Monkman & Seagull Genius Guides

" How to Teach Maths is a stimulating and accessible book, presenting a learner-centred approach to mathematics teaching. The book sets out the highly ambitious aim to introduce methods that work for all students, regardless of their ability levels or disability status. It covers a range of topics that are of great relevance to everyday classroom practice, from the concepts students struggle with the most to how visual aids can support understanding. The book also introduces the concepts of metacognition, mathematics learning difficulties and mathematics anxiety, and their relevance to teaching practice. Full of memorable examples, useful tips and entertaining anecdotes, this thoroughly researched, passionately argued, insightful and original book will be an extremely valuable resource to anybody interested in mathematics teaching and related research." - Dr. Kinga Morsanyi, Senior Lecturer, Centre for Mathematical Cognition, Loughborough University, UK

"Steve Chinn is a remarkable man. His insight into the ways that children think and understand maths is extraordinary. Many children have difficulties with maths; they can develop anxieties, put up barriers or just believe they can't do. Often these problems can become deep-seated and may even stem from the way the subject has been approached and taught, with misconceptions being seeded in early foundations. There are pressures on teachers to push on through topics: teach with expediencies, teach rules, teach towards tests. Problems are compounded. Anxieties grow. Teachers may be frustrated with pupils' difficulties in a topic. Why don't they get it? The problem may be much lower down, way back, say, to the simplest understanding of number, the meaning of a fraction or the concept of division. Steve Chinn understands the psychology of maths for the developing child's brain like none other. The approaches he has developed are so well thought out. They carefully knit together into a jigsaw of interrelated topics to kindle real depth of understanding, switching on light-bulbs and lifting the clouds. How to Teach Maths: a bold title. Sit up; sit straight; listen! We might learn something." - Patrick Davis. A national examiner, Head of maths, and teacher with 32 years' experience of teaching maths to 8 to 13 year-olds

"This is a fascinating book by an author who always produces excellent reads. I have read Steve Chinn's works since The Trouble with Maths and have recommended them to teachers, parents and colleagues - I wish I knew a few education ministers to whom I could recommend them as well. Steve Chinn's books, and How to Teach Maths is no exception, have the wonderful style of being knowledgeable and readable so that we can all follow the ideas and strategies presented. As with previous publications, there are some great ideas/strategies in this book, but presented within the context of a general curriculum and a perspective on the development of mathematics ability. The general ethos that if you can teach someone who is struggling with maths, you can teach everyone, is very apt - and let's face it, who doesn't struggle with at least some aspects of mathematics. Steve Chinn can teach us all, and in an enjoyable and accessible way. A lovely book!" - John Everatt, PhD, is a Professor of Education in the College of Education, Health and Human Development at the University of Canterbury, New Zealand.

"In this thoroughly engaging book, Chinn elegantly combines his decades of professional experience teaching students maths with what we know from research to provide an invaluable resource for maths educators" - Professor Daniel Ansari, Canada Research Chair in Developmental Cognitive Neuroscience & Learning, Department of Psychology & Faculty of Education Brain and Mind Institute, Western University

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teaching of specific groups & persons with special educational needs

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arithmetic; basic maths; developmental maths; dyscalculia; maths anxiety; maths learning difficulties; number sense; teaching maths; visual images for learning maths

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teaching of specific groups & persons with special educational needs

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Amazon.com: How to Teach Maths (9780367862718): Chinn, Steve: Books

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We are very pleased to feature a guest post from PD Dr. Karin Kucian, associate professor at the Centre for MR-Research of the University Children's Hospital of Zurich. In her article for the Dyscalculia Blog, Dr. Kucian explores changes in brain function and brain anatomy and how these relate to developmental dyscalculia.
Evidence is growing that Developmental Dyscalculia (DD) is associated with various alterations in brain function and brain structure. Recent work in the field of DD has used brain-imaging techniques to study the brains of people performing number tasks. These techniques have allowed researchers to generate high-resolution images of participants' brains, making it possible to observe brain activation patterns during number processing.

Number calculation is a demanding cognitive ability and is processed by a complex neuronal network.
Kucian, K. (2016). Developmental Dyscalculia and the Brain. In D. B. Berch, D. Geary, & K. Mann Koepke (Eds.), Development of mathematical cognition: Neural substrates and genetic influences (Vol. 2, pp. 165-193). Amsterdam: Elsevier.

The last few years have given us a clearer picture of what happens in our brains during number processing and calculation. Brain imaging reveals that these operations require an entire neuronal network to carry out. Within this network, an area of the brain called the intraparietal sulcus (IPS) is the centre for numerical processing. You can see the area marked on the illustration.

During research studies, the IPS activated whenever subjects had to deal with numerical magnitude. This included not just math tasks but even situations such as simply looking at changing amounts of dots.

Numerical processing also involves memory, as well as attentional, perceptual, motor, and spatial functions.

Add to this that the prefrontal cortex in general is associated with cognitive processes involved in calculation tasks and we start to get an idea of just how complex is number processing.

Altered brain function in dyscalculia and hope for remediation

Research findings conclude that brain activation pattern during number tasks is less precise in children with DD (developmental dyscalculia). These children may also exhibit deficits in core brain regions used for number processing. To make up for this, dyscalculic children may rely more heavily on supporting brain areas associated with memory, attention, or finger counting. It's not that dyscalculics use other brain regions to calculate. Rather they need to recruit other regions to make up for reduced activation in areas generally associated with numerical processing.

There is an urgent need to transform this basic knowledge into evidence-based, practical applications for special needs education and remediation. First steps in this direction have already been taken and provide promising results for affected children. Our brain is a highly plastic organ and adapts constantly-not only to typical development, but also to special stimulation such as remediation.

What were some of the results when children with dyscalculia completed a specific intervention? Brain imaging depicted a decrease in activation for those supporting areas we talked about. Furthermore, increased activity was observed in areas such as the IPS (see the illustration above) that are pivotal for number processing.

Brain anatomical changes in developmental dyscalculia

There is a growing body of knowledge showing DD is not only associated with alterations in brain activation, but also with changes in brain anatomy. Simply put, our brain can be divided into gray and white matter. Gray matter includes the outer layer of our brain with all the neuronal cell bodies and white matter the inner part with the axons of the neurons.

There is an urgent need to take research findings and apply these towards practical special education resources for dyscalculia.
Photo by Joshua Hoehne on Unsplash

When it comes to gray matter, a reduced volume in areas that are thought to play a key role in numerical abilities characterises DD. Basic number processing and calculation depend on a complicated network that includes different and widespread brain areas. In dyscalculics, anatomical alteration in gray matter has been reported across almost the entire network.

These brain regions are linked and communicate by complicated connections among different cortical and subcortical regions. Fast and accurate connections between the different brain regions are crucial for efficient transfer and adjustment of information.

There is still little known about impairments of brain connections and how these relate to DD. But the evidence to date supports the hypothesis that developmental dyscalculia is associated with a disconnection or reduced connection between brain areas important for number processing. Affected brain regions appear to vary, which could be explained by the diverse nature of dyscalculia.

Key takeaways:

Number processing and calculation are highly demanding cognitive skills

Number skills require a complicated neuronal network

Deficits in brain function, brain structure, or brain area interconnections have been reported across almost the entire neuronal network of people with dyscalculia

Brain activation pattern in children with DD is less precise than in those without dyscalculia

Brain areas affected in dyscalculics are mainly the parietal areas, but other cortical and subcortical regions can also be affected

The stronger recruitment of supporting areas associated with working memory, attention, monitoring, updating, or finger representation are supposed to reflect compensatory mechanisms in dyscalculic children

The use of supporting areas for number tasks could also reflect deficits in these domain-general skills that might contribute to the development of DD

Intervention and remediation for dyscalculic children can improve brain activation in areas associated with number processing

There is an urgent need to take research findings and apply these towards practical special education resources for dyscalculia

About the author

Karin Kucian studied Neurobiology and Educational Science at the Swiss Federal Institute of Technology. After her PhD on the development of cerebral representations of numbers in typically achieving children and children with developmental dyscalculia, she focused further on the investigation of neuronal correlates of number processing in children with developmental dyscalculia and interventions to improve numerical understanding. In 2015, she obtained the venia legendi of the medical faculty of medicine of the University of Zurich and is now working as associate professor at the Centre for MR-Research of the University Children's Hospital Zurich.

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We are very pleased to feature a guest post from PD Dr. Karin Kucian, associate professor at the Centre for MR-Research of the University Children's Hospital of Zurich. In her article for the Dyscalculia Blog, Dr. Kucian explores changes in brain function and brain anatomy and how these relate to developmental dyscalculia.

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Brain areas associated with numbers and calculations (Arsalidou et. al., 2018)

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Brain areas associated with numbers and calculations (Arsalidou et. al., 2018)

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Abstract

Children use numbers every day and typically receive formal mathematical training from an early age, as it is a main subject in school curricula. Despite an increase in children neuroimaging studies, a comprehensive neuropsychological model of mathematical functions in children is lacking. Using quantitative meta-analyses of functional magnetic resonance imaging (fMRI) studies, we identify concordant brain areas across articles that adhere to a set of selection criteria (e.g., whole-brain analysis, coordinate reports) and report brain activity to tasks that involve processing symbolic and non-symbolic numbers with and without formal mathematical operations, which we called respectively number tasks and calculation tasks. We present data on children 14 years and younger, who solved these tasks. Results show activity in parietal (e.g., inferior parietal lobule and precuneus) and frontal (e.g., superior and medial frontal gyri) cortices, core areas related to mental-arithmetic, as well as brain regions such as the insula and claustrum, which are not typically discussed as part of mathematical problem solving models. We propose a topographical atlas of mathematical processes in children, discuss findings within a developmental constructivist theoretical model, and suggest practical methodological considerations for future studies.

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Brain areas associated with numbers and calculations in children: Meta-analyses of fMRI studies

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Children use numbers every day and typically receive formal mathematical training from an early age, as it is a main subject in school curricula. Despite an increase in children neuroimaging studies, a comprehensive neuropsychological model of mathematical functions in children is lacking.

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Over the past 5 years, this blog has seen nearly 3.1 million views. The most popular strategies from the blog all have one thing in common: They include downloadable resources and strategies which can be immediately used in the classroom. They are also free! I'm sometimes asked these two questions:

"Which are the most popular strategies and downloads?"

"Is there a way you could put them in one place?"

Here they are.

Part 1: The Quick List for Easy Access

(See Part 2 for a Deeper Dive.)

Cube Conversations - and the follow-up post 80 Printable Cube Conversations Pages and 4 Instructional Strategies

The Maze Hundreds Chart - and also the post that followed, Strategies for Using the Maze Hundreds Chart

Tiled Area Questions - and the accompanying 3 Powerful Tile Strategies (and 40 New Downloadable Pages).Also note: Primary Tile Questions is part of this series.

More detailed information - AND PICTURES - can be found in Part 2.
Splat!
Two years ago - nearly to the day - I posted the first Splat! lesson. Since that time, I've posted 21 levels on the blog through various posts. The sequence of lessons ranging from very small numbers through algebraic reasoning with multiple variables has proven to be the single most popular concept. There are other versions of Splat! that I have yet to share. Among them, Splat! in Motion and the Rare, Rogue, Red Splat! are in the wings and I look forward to sharing them!
Esti-Mysteries: Estimation Meets Math Mysteries
This concept has been really fun to share. Estimation and a wide range of math concepts are fused together in a very invitational format. The first time you use Esti-Mysteries, you may be amazed at how deep the conversation goes - and at how loud the cheering becomes when you reach The Reveal at the very end of the journey. Your students will ask you for another Esti-Mystery right away.
51 Esti-Mysteries
If it looks like I posted the same thing twice, that's because the original Esti-Mystery post only had 12 Esti-Mysteries. I received so many requests for more Esti-Mysteries, that I decided to write 51 more during the first few months of this school year. I planned to post 3 per week for 17 weeks in a row. The result is that there are a LOT of Esti-Mysteries suitable for many grade levels that can be immediately downloaded and used in class.
The Fraction Splat! Series
The Fraction Splat! Series is one of my favorite resources. Following Splat! the fraction series dives into some much deeper territory and invites students to think about numbers in many different ways.
20 Days of Number Sense and Rich Math Talk (K-12)
The purpose of this series was to create a set of resources that any teacher - at any grade level - could use for 20 days in a row. It's mean to provide rich experiences to students and a rich journey for teachers. While the series contains entirely exclusive content, it draws upon 4 routines found in my blog. Days 1-5 are Estimation Clipboards. Days 6-10 are Splats! Days 11-15 are Esti-Mysteries. Days 16-20 are Cube Conversations. Each of the 20 days is labeled so that it is very easy to find which grade level to use. Any teacher at any grade level can use this single post to have access to 20 days of rich math discourse - and may even discover a new part of their teaching journey just by listening to the insights of the students.
Splat for Google Slides - 40 Lessons
After writing Splat! I heard from many educators who do not have access to PowerPoint. So, I rewrote the series from scratch with a new combination of animations that would make it work in Google Slides. This set is for everyone who has Google Slides and wants to use Splat!
100 Subitizing Slides and 10 Challenge Patterns
This is an all-time favorite resource. The concept is a set of 10 lessons. Each lessons shows a series of quick subitizing images (which also energizes the classroom) then quickly transitions to a challenge dot pattern. It's more than subitizing, and it's more than a dot talk. It's the combination of the two that make this work so well. The 100 slides are set in 10 lessons and there are printables to match the challenge patterns.

Cube Conversations - and the follow-up post 80 Printable Cube Conversations Pages and 4 Instructional Strategies

Cube Conversations is a series in which I feature a 3-D cube structure and then (with a click) allow you to show your students how it could be decomposed usefully. The shape changes colors and surprising groupings appear. Then, with two more clicks, two more examples of how the shape can be decomposed appear before the students. Many of the recognize similarities to how they have decomposed it already. The animations serve as an example as 1 more shape appears and allows the students to decompose it 4 different ways. This is one that you really have to see and experience to detect the full power of how it propels student insights.
The Estimation Clipboard
Estimation has always been near and dear to me. Yet, I've seen so many, many times when students are given ONE thing to estimate (1 jar, 1 image, 1 length of rope) and then they move on to something completely different. The Estimation Clipboard offers something new in that it provides 4 highly similar images in a row. Students estimate, share their reasoning with each other, and eventually see the answer... And then they are given a 2nd image that is similar to the first one. They have a rich context in which to estimate. Then they see a third image - and then a fourth. After you've taught your class the Estimation Clipboard routine, you are very likely to hear even more cheering in the classroom.

The Maze Hundreds Chart - and also the post that followed, Strategies for Using the Maze Hundreds Chart

This interactive hundreds chart allows you to create mazes on the screen in front of your students. Better yet, it allows them to amaze you with their thinking. It's hard to describe exactly how this chart works without showing it in person - or guiding you to the video. I encourage you to take a look at the post, watch the video, and see just how this highly versatile chart works.
The Animated Multiplication Table
This interactive PPT file allows you to see the multiplication table - and many of the relationships that sometimes stay hidden - in many different, vivid ways. Take a look at the video, and you will see what I mean.

Tiled Area Questions - and the accompanying 3 Powerful Tile Strategies (and 40 New Downloadable Pages).Also note: Primary Tile Questions is part of this series.

I was going to make this list show the top 10 pages - but when I saw that one of my all-time favorite strategies wasn't within the list of the top 10 page views on my blog, I knew it needed to be shared again. This strategy can travel into so very deep, very rich, and very colorful territory. I love this strategy so much that I actually wrote 3 blog posts about it.

I am certain that this list will change over time. Some of these posts (like Splat!) have been around for two years, while others (like 51 Esti-Mysteries) have only been around for a handful of months.

Thank you for all of your questions and support. I hope this helps to answer these two questions:

"Which are the most popular strategies and downloads?"

and

"Is there a way you could put them in one place?"

I look forward to reflecting on this post in the future and seeing how it changes. There are many new resources which I'm excited to share very soon!

All my best,

Steve

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The 12 Most Popular Math Strategies and Downloads - Steve Wyborney's Blog: I'm on a Learning Mission.

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Over the past 5 years, this blog has seen nearly 3.1 million views. The most popular strategies from the blog all have one thing in common: They include

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Differences in brain structure (McCaskey et. al., 2020)

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Differences in brain structure (McCaskey et. al., 2020)

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Ursina McCaskey 1,2*, Michael von Aster 1,2,3,4, Ruth O'Gorman 1,2,5 and Karin Kucian 1,2,4

Center for MR-Research, University Children's Hospital, Zurich, Switzerland

Children's Research Center, University Children's Hospital, Zurich, Switzerland

Clinic for Child and Adolescent Psychiatry, German Red Cross Hospitals, Berlin, Germany

Neuroscience Center Zurich, University of Zurich and ETH Zurich, Zurich, Switzerland

Zurich Center for Integrative Human Physiology, University of Zurich, Zurich, Switzerland

Developmental dyscalculia (DD) is a learning disability affecting the acquisition of numerical-arithmetical skills. Affected people show persistent deficits in number processing, which are associated with aberrant brain activation and structure. Reduced gray matter has been reported in DD for the parietal cortex including the intraparietal sulcus (IPS), but also the frontal and occipito-temporal cortex. Furthermore, dyscalculics show white matter differences for instance in the inferior (ILF) and superior longitudinal fasciculus (SLF). However, the longitudinal development of these structural differences is unknown. Therefore, our goal was to investigate the developmental trajectory of gray and white matter in children with and without DD. In this longitudinal study, neuropsychological measures and T1-weighted structural images were collected twice with an interval of 4 years from 13 children with DD (8.2-10.4 years) and 10 typically developing (TD) children (8.0-10.4 years). Voxel-wise estimation of gray and white matter volumes was assessed using voxel-based morphometry for longitudinal data. The present findings reveal for the first time that DD children show persistently reduced gray and white matter volumes over development. Reduced gray matter was found in the bilateral inferior parietal lobes including the IPS, supramarginal gyri, left precuneus, cuneus, right superior occipital gyrus, bilateral inferior and middle temporal gyri, and insula. White matter volumes were reduced in the bilateral ILF and SLF, inferior fronto-occipital fasciculus (IFOF), corticospinal tracts, and right anterior thalamic radiation (ATR). Behaviorally, children with DD performed significantly worse in various numerical tasks at baseline and follow-up, corroborating persistent deficits in number processing. The present results are in line with the literature showing that children with DD have reduced gray and white matter volumes in the numerical network. Our study further sheds light on the trajectory of brain development, revealing that these known structural differences in the long association fibers and the adjacent regions of the temporal- and frontoparietal cortex persist in dyscalculic children from childhood into adolescence. In conclusion, our results underscore that DD is a persistent learning disorder accompanied by deficits in number processing and reduced gray and white matter volumes in number related brain areas.

Introduction

Currently, there is one study that investigated children between 8 and 14 years and aimed to describe how the differences between dyscalculic and control children vary during cortical development ( Ranpura et al., 2013). Relative to TD children, gray matter volume in dyscalculics increases with age in the left dorsolateral prefrontal cortex and the right superior occipital lobe, but decreases slightly in the left primary motor cortex. White matter development of children with DD showed notable delays relative to the control group. Whilst TD children showed an age-related increase in frontal and parietal areas, the white matter volumes of DD children remained stable over time ( Ranpura et al., 2013).

The gray and white matter regions and their relationship with numerical or arithmetical skills were also investigated in TD children. Gray matter volume is positively correlated with arithmetic scores or performance gains specifically in the left IPS and angular gyrus ( Li et al., 2013; Supekar et al., 2013; Evans et al., 2015). Price et al. (2016) showed that gray matter volume in the left IPS measured in first grade predicted the math score at the end of the second grade. This result is not confined to the parietal cortex. Increased gray matter volume in the frontal (dorsal and ventral prefrontal cortices, IFG), occipito-temporal (cuneus, fusiform gyrus) and in the hippocampus also relates to better math performance ( Li et al., 2013; Supekar et al., 2013; Evans et al., 2015; Wilkey et al., 2018). Moreover, associations between brain volume abnormalities and math performance have been reported for the gray matter volume of parietal regions in other populations prone to math difficulties (prematurely born children, very low birth weight, Turner syndrome; Isaacs et al., 2001; Starke et al., 2013; Zhao et al., 2013).

To summarize, gray matter volume in various regions of the frontoparietal network, but specifically in the IPS-which is known as a key area for number processing-has been associated with better performance in numerical processing and arithmetical skills. In line with that, children with DD show reduced gray matter volumes in parietal, but also frontal, occipito-temporal and hippocampal areas. Furthermore, white matter differences have been reported in the main white matter tracts connecting the parietal with the frontal and the temporal cortex in subjects with DD. However, to the author's best knowledge, there is no study investigating the development of these structural differences in children with DD using a longitudinal study design. Therefore, the present work aims to elucidate the developmental trajectory of gray and white brain matter volume in children with and without DD from childhood to adolescence.

Materials and Methods

Participants and Procedure

A total of 35 (23 DD, 12 TD) children between 8 and 11 years were recruited into this longitudinal study, of which 27 took part in a previous study ( Kucian et al., 2011). This longitudinal study included structural and functional MRI data (for the results of fMRI data please see McCaskey et al., 2018). Children were evaluated by neuropsychological tests and MRI at baseline and returned after 4.2 ( SD = 0.46) years for a follow-up measurement. We approached the subjects of our study through the school setting or School psychological Services (DD subjects). The children visited us twice at the Center for MR-Research of the University Children's Hospital Zurich. On both occasions, they first completed a neuropsychological session and then underwent the MRI measurement.

Inclusion criteria for all children were no history of neurological or psychiatric disorders and an IQ ≥ 85, measured by the third edition of the WISC ( Tewes et al., 1999; Similarities, Block Design, Vocabulary, Picture Arrangement). Furthermore, DD children had to score below the 10th percentile in the total score or three subtests of a standardized numerical test battery (ZAREKI-R) at baseline. These criteria are in line with the diagnostic criteria for DD of the ICD-11 (6A03.2 Developmental learning disorder with impairment in mathematics; WHO, 2018). TD children had to perform above the cut-off of value in the test batteries for numerical abilities at baseline and follow-up (10th percentile in the ZAREKI-R and 67 points in the BASIS-MATH 4-8, respectively). Following these criteria, six DD children and one TD child were excluded from the study. For the MRI analysis, two children were excluded due to missing imaging data at baseline or follow-up and three for image quality reasons, resulting in 13 DD and 10 TD complete data sets for the whole study. Groups did not differ in age, gender, handedness, and pubertal status determined by the Edinburgh Handedness Inventory ( Oldfield, 1971) and an adapted version of the Self-administered Rating Scale for Pubertal Development ( Carskadon and Acebo, 1993).

Parental consent and child assent were obtained at the beginning of the study. The study was approved by the Ethics committee of Zurich, Switzerland based on guidelines from the World Medical Association's Declaration of Helsinki ( WMA, 2013).

Neuropsychological Testing

During the neuropsychological assessment, we acquired numerical abilities, general cognitive abilities as well as measures for the most common comorbid disorders such as developmental dyslexia, attention deficit and hyperactivity disorder, and working memory deficits.

Numerical Abilities

At baseline, numerical abilities were assessed using the revised version of the Neuropsychological Test Battery for Number Processing and Calculation in Children (ZAREKI-R; von Aster et al., 2006). The Zareki-R is a multidimensional test, measuring basic numerical skills as well as calculation, and widely used for the diagnosis of DD in the German-speaking area (see Supplementary Material for detailed information about the subtests). Based on this test battery children scoring below the 10th percentile in the total score or three subtests were identified with DD.

Also, the Arithmetic subtest of the Wechsler Intelligence Scale for Children (WISC-III; Tewes et al., 1999) was performed. In this subtest, children had to solve story problems of increasing difficulty within a set time limit (reported test values are IQ scores).

At follow-up, the numerical achievement was assessed with the test for Basic Diagnosis in Mathematics Education for Grades 4-8 (BASIS-MATH 4-8; Moser Opitz et al., 2010). The Basis-Math is a criterion-based test battery measuring various arithmetical abilities such as counting, decimal system, and calculation. Criteria for numerical deficiencies are met if the performance is under a threshold value of 67 points (maximum score of 83 points). This is interpreted as not reaching mastery of basic mathematical concepts (see Supplementary Material for detailed information).

The curriculum-based subtest Quantity Comparison of the Cognitive Abilities Test (KFT 4-12+R; Heller and Perleth, 2000) was performed to assess the arithmetic performance at a peer level. In this subtest, subjects had 10 min to solve as many quantity comparisons as possible of increasing difficulty (reported test values are T scores).

The spatial representation of numbers was measured using a computerized number line task adopted from Kucian et al. ( 2011; for a detailed description see McCaskey et al., 2018). Children had to indicate by mouse-click the position of 20 Arabic digits on a number line with the labeled endpoints 0 and 100. Accuracy was measured by calculating the percentage distance from the marked to the correct position of the given number (reported measures are raw values).

Children also solved 40 basic arithmetic problems (20 addition and 20 subtraction) in the number range 0-1,000 (for a detailed description see McCaskey et al., 2018). Each problem was presented visually on the computer screen and solutions were given via the keyboard. The number of correctly solved items was quantified (reported test values are raw scores, maximum value 20).

Domain General Cognitive Abilities

At baseline, intelligence was measured with the third edition of the WISC (Similarities, Block Design, Vocabulary, Picture Arrangement; Tewes et al., 1999). At follow-up, the fourth edition of the WISC was used (Similarities, Block Design, Matrix Reasoning; Petermann and Petermann, 2007). Table 1 shows the estimated general IQ.

Table 1. Demographic characteristics and scores on numerical abilities, domain-general cognitive abilities, working memory, attention, and reading.

Working Memory

Visuospatial working memory was measured with the Block-Suppression-Test ( Beblo et al., 2004). The task required subjects to reproduce every second block of a previously presented sequence on a board with nine cubes. The sequences had a length of 3-9 cubes. Three items per sequence were presented. The longest sequence which was reproduced correctly twice was quantified (reported test values are raw scores, maximum value 9). Verbal working memory was measured with the subtest Digit Span of the WISC-IV ( Petermann and Petermann, 2007). In this task, subjects had to repeat an auditorily presented sequence of numerals forward or backward. The sequences had a length of 2-9 numerals (reported test values are IQ scores).

Attention

Levels of attention and inhibition were measured using the subtests Alertness and Go-Nogo of the computerized Test battery for Attentional Performance (TAP; Zimmermann and Fimm, 1993). In the Alertness subtest, subjects had to react as quickly as possible when the target stimulus "x" appeared (intrinsic alertness). Half of the trials were preceded by an acoustic cue stimulus (phasic alertness). The test has four runs and a total of 80 target items. For each subject, the percentile rank of the median RT was quantified (reported test values are percentile ranks). In the Go-Nogo subtest, subjects had to react as quickly as possible to a target stimulus ("x," go condition), but inhibit reactions on a second presented stimulus ("+," nogo condition). The test has a total of 40 items (20 go and 20 nogo items). For each subject, the percentile rank of the median RT was quantified (reported test values are percentile ranks).

Reading Abilities

The 1-Minute-Reading-Task from the Salzburg Reading and Orthography Test (SLRT-II; Moll and Landerl, 2010) assessing word and pseudoword reading fluency was used to estimate the reading performance. Two sheets of paper with either 156 words or 156 pseudowords of increasing length and difficulty were presented. Subjects had 1 min per sheet to read as many words as possible. The amount of correctly read items was quantified (reported test values are percentile ranks). Because of lacking test norms in grades 7 and 8, we interpolated the norms from the test manual (grade 6) and Kronschnabel et al. (2013; grade 9).

Behavioral Data Analysis

Behavioral data were statistically analyzed with SPSS (Version 22). To account for the difficulties regarding the performance of statistical tests of normality in small samples, we performed nonparametric tests (Mann-Whitney U Test) to assess group differences. Effect sizes are reported as Pearson's correlation coefficient r and are interpreted as small ( r = 0.10), medium ( r = 0.30) or large ( r = 0.50).

Brain Imaging

Image Acquisition

MRI data were acquired on a 3T General Electric Signa Scanner (GE Medical Systems, USA) using an 8-channel head coil. T1-weighted structural images (voxel size = 0.94 × 0.94 × 1.00 mm ³) were acquired with a fast spoiled gradient echo sequence (3D FSPGR, slice thickness = 1 mm, no interslice skip, matrix size = 256 × 256, field of view = 240 mm, flip angle = 20°, echo time = 3 ms, repetition time = 10 ms). Participants were carefully instructed and supplied with hearing protection before entering the scanner. To minimize head motion, the head was stabilized with padding.

MRI Data Preprocessing

The data were preprocessed using the Computational Anatomy Toolbox (CAT12, Structural Brain Mapping Group, University of Jena, Germany), which is an extension to Statistical Parametric Mapping (SPM 12, Wellcome Trust Centre for Neuroimaging, University College London, UK) running on Matlab (Release 2012b, The MathWorks Inc., USA).

In a first step, the longitudinal data pair (baseline and follow-up image) of each subject was registered to the mean image for each subject by an inverse-consistent realignment, which also includes a bias correction between the different time points. The mean image of each subject is then segmented and the spatial normalization parameters are estimated with the help of a Dartel Normalization. These spatial normalization parameters (Dartel deformations) are then applied to the gray and the white matter segmentations of the baseline and follow-up image. The white and gray matter segmented images were smoothed with a Gaussian kernel of 8 mm FWHM (full width half maximum). As the last step, the total intracranial volume (TIV) was estimated for the baseline and follow-up image of each subject (please see Supplementary Material for the Matlab scripts of the preprocessing steps).

Quality Control

The CAT12 Toolbox provides image quality measures describing the properties of the images before the preprocessing. The image quality rating (IQR) is a weighted average of the noise contrast ratio, the inhomogeneity contrast ratio and the resolution of the input image. The images all reached ratings above 79 at both measurement time points (baseline range 79-87, follow-up range 81-87). Note that typical scientific (clinical) data is expected to get good to satisfactory ratings (70-90; Gaser and Dahnke, 2016, see also www.neuro.uni-jena.de/cat/). The groups did not differ in their IQR at baseline (DD Mdn = 85.0, TD Mdn = 85.3, U = 46.5, z = −1.15, p = 0.263) or follow-up (DD Mdn = 85.0, TD Mdn = 85.5, U = 48.0, z = −1.05, p = 0.313), and there was no significant difference in the quality measures between the time points (baseline Mdn = 85.1, follow-up Mdn = 85.3, z = −0.69, p = 0.494).

Also, the segmented and normalized gray and white matter images were visually inspected and the sample was checked for homogeneity (mean correlation). Based on the visual inspection and the Mahalanobis distance, which combines a measure of image quality before (weighted overall image quality) and after preprocessing (mean correlation), we excluded two DD and one TD data sets resulting in the final group size of 13 DD and 10 TD.

Statistical Model

For the statistical analyses of the gray and white matter volumes, two separate flexible factorial models with the factors subject, group (DD, TD) and time (baseline, follow-up) including TIV and puberty score as covariates were defined. Statistical results are shown with a threshold of p < 0.05 family-wise error (FWE) correction (see Supplementary Material for the design matrix and the Matlab scripts of the statistical model as well as the defined contrasts). Anatomical localization of the gray matter volume results was attained trough the SPM Anatomy Toolbox v2.0 ( Eickhoff et al., 2005, 2007). White matter regions were labeled according to the JHU (Johns Hopkins University) white-matter tractography atlas ( Hua et al., 2008).

Results

Behavioral Data

The neuropsychological results and the demographic data for all subjects are summarized in Table 1. All participants scored in the normal range of IQ (DD IQ = 93-111, TD IQ = 101-125). However, groups differed in the estimated general IQ at baseline (WISC-III p < 0.001, r = 0.69) and follow-up (WISC-IV p < 0.001, r = 0.72). IQ measures are known not to be fully independent of measures of math ability, and the present sample, therefore, reflects the cognitive pattern typically observed in DD.

In the attention, working memory, and reading task no differences between DD and TD children were found (all p > 0.05).

Numerical Achievement

As expected, numerical abilities, assessed by the Zareki-R at baseline, differed significantly between the TD and the DD groups ( p < 0.001, r = 0.84). The groups also differed in the subtest Arithmetic (WISC-II), with the DD subjects scoring significantly lower than the TD group ( p < 0.01, r = 0.55; Table 1).

At follow-up, adolescents of the DD group still performed worse in comparison to their peers (Basis-Math p < 0.001, r = 0.84). In fact, all the subjects identified with DD at baseline still fulfilled the diagnostic criteria for DD at the follow-up measurement. Furthermore, they also scored significantly lower in the curriculum based test Quantity Comparison (KFT 4-12+R p < 0.001, r = 0.83), the number line task ( p < 0.01, r = 0.61) and the basic arithmetic operations (addition p < 0.05, r = 0.52, subtraction p < 0.01, r = 0.62; Table 1).

Pearson's correlations of the whole brain gray and white matter volume with behavioral measures were calculated for baseline and follow-up. At the follow-up, gray matter volume was positively correlated with the Basis-Math ( r = 0.64, p < 0.05) in TD children. However, we found a negative relationship in TD adolescents between performance in addition and gray ( r = −0.69, both p < 0.05) and white matter volume ( r = −0.60, p < 0.05), respectively. For DD children, no significant correlations between numerical abilities and volume of the brain structure were revealed (see Supplementary Material for a complete table of correlations).

Structural Results

Gray Matter

For the gray matter volume, the flexible factorial analysis revealed a significant effect of group ( Figure 1A, Table 2). DD children showed decreased gray matter volumes in the bilateral inferior parietal lobe assigned to the IPS, the bilateral supramarginal gyri, the left precuneus, the left postcentral gyrus, and the right paracentral lobule compared to TD children. In the occipital lobe, differences were found in the left calcarine gyrus/cuneus, the left middle occipital gyrus (MOG), and the right superior occipital gyrus (SOG). Decreased gray matter was found in the bilateral inferior (ITG) and middle temporal gyri (MTG), the left rolandic operculum, and the bilateral insula over the examined time of 4 years.

Figure 1. Group differences in gray and white matter. Results are shown on a pediatric template ( Fonov et al., 2009) with a significance level of p-cluster < 0.05, FWE corrected. (A) Decreased gray matter volumes in dyscalculics compared to typically developing children over both time points. (B) Decreased white matter volumes in dyscalculics compared to typically developing children over both time points. Abbreviations: CST, corticospinal tract; IFOF, inferior fronto-occipital fasciculus; ILF, inferior longitudinal fasciculus; INS, insula; IPS, intraparietal sulcus; ITG, inferior temporal gyrus; MOG, middle occipital gyrus; PCUN, precuneus; SLF, superior longitudinal fasciculus; SMG, supramarginal gyrus.

Table 2. Gray matter.

The main effect of time and the interaction group by time were not significant.

White Matter

For the white matter volume, the flexible factorial analysis revealed a significant effect of the group ( Figure 1B, Table 3). DD children showed reduced white matter volumes in a widespread set of brain regions including the bilateral CST, the bilateral superior and ILF, the bilateral IFOF, and the right ATR. Compared to TD children the DD children further showed increased white matter volumes in the right CST.

Table 3. White matter.

The main effect of time revealed an increase in the left SLF adjacent to the precentral gyrus over both groups. The interaction group by time was not significant.

Discussion

Our study aimed to investigate the neural structural development of children with DD and TD peers using a longitudinal study. Until now, there have only been a handful of studies investigating the structural differences between children with and without DD, and only one study examining changes of regional differences in cortical development ( Rotzer et al., 2008; Rykhlevskaia et al., 2009; Ranpura et al., 2013). However, all of these study results are based on cross-sectional data. To our knowledge, this is the first study investigating the neural developmental trajectory using longitudinal data in children with DD. On the behavioral level, we found that the children of the DD group performed significantly worse in all the numerical and arithmetical tasks. This result remained stable over time. All children that were identified with DD at the beginning of the study still fulfilled the diagnostic criteria of DD 4 years later. On the neural level, children with DD showed reduced gray and white matter volumes in various regions and prominent tracts of the frontoparietal numerical network. These differences do not vanish over time, but persist from childhood into adolescence.

The dyscalculics showed reduced gray matter volumes in the parietal lobes specifically, but also in the occipital, temporal, and frontal parts of the brain, consistent with the results of previous studies. Less gray matter volume in the IPL including the IPS and the SPL have been reported in all studies with dyscalculic children ( Rotzer et al., 2008; Rykhlevskaia et al., 2009; Ranpura et al., 2013). These regions are known from functional studies to be the key areas for number processing and quantity representation (for a meta-analysis see Sokolowski et al., 2017; Arsalidou et al., 2018). Furthermore, we also found reduced gray matter volumes in the bilateral supramarginal gyri, which are thought to play a crucial role in the retrieval of arithmetical facts ( Menon, 2016). Similar to Rykhlevskaia et al. (2009), our results revealed lower gray matter volumes in the MOG/SOG, the cuneus/precuneus, and the temporal gyrus, although our results include bilateral MTG/ITG. Unlike the studies conducted before with DD children ( Rotzer et al., 2008; Rykhlevskaia et al., 2009; Ranpura et al., 2013), we found reduced gray matter volumes in the insula. However, the insula has a high likelihood to be activated when children solve number and calculation tasks and have been proposed to play a role in intrinsic motivation about learning and training ( Arsalidou et al., 2018). Also, our study did not find any volumetric differences in the parahippocampal areas, which was reported in Rykhlevskaia et al. (2009).

In terms of white matter, our study revealed reduced white matter volume in dyscalculics in the bilateral superior and ILF, the CST, the IFOF, and the ATR. Reduced white matter volumes were also reported in the same tracts by Rykhlevskaia et al. (2009), except that they found additionally reduced volumes in the forceps major and the splenium of the corpus callosum. Moreover, the CST, the ILF and SLF, and the corona radiata (of which the ATR is part) have all been associated with numerical and mathematical processing by numerous studies ( Kucian et al., 2013; for a review see also Matejko and Ansari, 2015). The SLF connects frontal and parietal regions of the brain, which are known to be the main areas activated when solving number and arithmetic related tasks. It has further been proposed that the ILF is involved in visual processing related to numerical or mathematical problem solving ( van Eimeren et al., 2008; Matejko and Ansari, 2015).

Children with DD revealed increased volumes in the right CST. This tract connects the cortex with the brainstem and is typically associated with motor functions. Research has further demonstrated that there is a link between finger and number representation ( Noël, 2005; Matejko and Ansari, 2015). As children with DD often rely on finger counting strategies to compensate for the deficits in fact retrieval, the increased volume in the CST might be related to the frequent finger use during calculation in DD.

However, it is important to note that these observed structural differences in gray and white matter are related to underlying microstructural mechanisms of development and learning. Among the candidate mechanisms explaining gray and white matter plasticity are morphometric changes in the neuron (e.g., axon sprouting, dendritic branching, synaptogenesis, neurogenesis), in fiber organization (e.g., axon branching, sprouting, axon diameter or the number of axons) as well as in the myelination (for detailed information see Zatorre et al., 2012). But also vascular changes (angiogenesis) or changes in morphology and number of glia and astrocytes could explain the increase in gray and white matter volume ( Zatorre et al., 2012). Regarding the structural differences in DD children, one could speculate that DD children show reduced white and gray matter volumes as the underlying microstructural process does not take place to the same extent as in TD peers. This would also be in line with results of functional and DTI studies reporting decreased activation and lower FA values in number related areas and tracks, respectively ( Davis et al., 2009; Kucian et al., 2013; for an overview see Peters and De Smedt, 2018).

Over development, we did not observe a gray matter decrease or prominent white matter increases. The lack of a gray matter decrease can be due to the age of our subjects. We examined children between the ages of 9 and 14. The developmental trajectory of the gray matter volume follows an inverted u-shape and depending on the study the peak of gray matter volume has been reported at age 8 ( Mills et al., 2016), age 12 ( Groeschel et al., 2010), or between ages 11-14 ( Gogtay and Thompson, 2010). The time point of the peak further varies depending on the brain region ( Giedd et al., 1999) and the sex and/or pubertal stage, with females reaching gray matter peaks 1-2 years earlier than males ( Gogtay and Thompson, 2010; Mills and Tamnes, 2014). Therefore, it could be the case that we did not detect developmental changes in the gray matter, because the children and adolescents we studied are around one of the reported gray matter peaks. If we look at the individual trajectories of our study participants, some children still show a gray matter increase, whilst other children show a gray matter decrease or almost no change in the gray matter volume over the 4 years ( Figure 2, upper panel). Moreover, the fact that we controlled for pubertal status and therefore indirectly for sex might be an additional reason why we do not find developmental changes. In the white matter volume, we found a significant increase during development in the left SLF (MNI x = −40, y = −20, z = 28), located right next to the reported developmental changes in the study of Giorgio et al. (2010; MNI x = −42, y = −22, z = 28). However, our results do not show the prominent white matter changes as reported in the literature ( Giorgio et al., 2010). A possible explanation is that our study does not look at the correlations between white matter volume and age. On the other hand, the individual trajectories of our subjects show a clear increase in the total white matter volume ( Figure 2, lower panel), which is in line with previous research ( Aubert-Broche et al., 2013; Mills et al., 2016).

Figure 2. Spaghetti plots of whole brain volumes. Gray matter (top panel) and white matter volumes (lower panel) are plotted against age for children with dyscalculia (left, in blue) and typically developing children (right, in gray). The red line depicts the mean of the group.

The main focus of our study was to find out more about the developmental trajectories of gray and white matter volumes in children with DD. We found no significant interactions, but the group differences in gray and white matter volumes remained stable during the examined time window. Although the differences found between DD and TD children are well in line with the literature ( Rotzer et al., 2008; Rykhlevskaia et al., 2009), and also reflect the known and persistent behavioral differences ( Nelson and Powell, 2018), the developmental trajectories differ from the results reported by Ranpura et al. (2013) in such that we did not find an increase in white matter volume in frontal and parietal areas in TD children only. Studies in adults find no or very subtle differences between the subjects with and without DD ( Cappelletti and Price, 2013; Moreau et al., 2019). Based on these results, one could argue that the volumetric differences should diminish over development. However, Moreau et al. (2019) used a rather lenient criterion for DD, which could also be a reason why volumetric differences were not found. Also, there is an age gap of 10-15 years between our results and the adult studies, in which developmental changes can still take place. Our results, therefore, point towards stable and persistent differences in the dyscalculics gray and white matter volumes from childhood to adolescence. More research, specifically longitudinal studies over a longer time interval, are urgently needed to enable us to conclude the developmental trajectory of the brain structure in DD. However, findings from the present study suggest that development proceeds in a similar manner between DD and TD children.

To better understanding the present findings, it is important that we advance our knowledge about the typical developmental trajectory of the brain structure and its spontaneous variations in numerical cognition. Furthermore, the effects of schooling and specific interventions on brain structure need to be explored more profoundly. This knowledge should build the basis for the investigation and a better understanding of the deviant and/or delayed development as reported in children with DD. In a next step, it would be interesting to investigate if the abnormalities in gray and white matter are a result of a developmental delay or a specific marker of DD. This open question could be tackled by comparing the structural brain development of DD children with TD children that perform on a similar numerical level. Furthermore, the investigation of structural changes caused by a specific intervention could help clarify the questions about the neurobiological cause of dyscalculia.

Limitations

Our results are important for the field since this is to our knowledge the first study investigating the developmental trajectory of structural white and gray matter volumes using longitudinal data in DD. However, several limitations should be considered when interpreting the results of our study. First of all, due to braces and movement artifacts, the drop-out rates in longitudinal MRI studies with children and adolescents are high. Our study includes only small sample sizes and should for this reason be interpreted with caution. However, the data included had good data quality ratings, as assessed by objective criteria ( Gaser and Dahnke, 2016). Moreover, we replicated the main findings of previous studies examining volumetric gray and white matter differences in DD children ( Rotzer et al., 2008; Rykhlevskaia et al., 2009; Ranpura et al., 2013), which were also performed with larger sample sizes (e.g., Rykhlevskaia et al., 2009). Therefore, we are confident that despite the small sample sizes our results contribute valuable knowledge towards an understanding of the developmental trajectory of brain structure in children with and without DD.

Second, the rather large age range within our sample (at each time point) may attenuate the developmental effects. For methodological reasons, a narrower age range would be much better to examine general developmental effects and detect group differences over time. A closer look at our data revealed that the individual developmental trajectories showed similar trends irrespective of the age at the entry of the study. However, future longitudinal studies should investigate limited age ranges to control better for the effect of schooling and the rapid changes in development.

A third restricting point of our study is the significant difference in IQ between the TD and the DD group. Developmental imaging studies have shown that there is a positive relationship between intellectual abilities and white/gray matter volume, especially in the dorsolateral prefrontal cortex, parietal lobe, the anterior cingulate cortex and in temporal and occipital regions ( Wilke et al., 2003; Tamnes et al., 2011; Brancucci, 2012). For this reason, we ran our analyses with IQ as an additional covariate showing that the main results remained unchanged (see Supplementary Material).

Last, it should be noted that some authors argue that the sensitivity to examine the white matter using voxel-based morphometry is limited, as white matter areas are characterized by large homogenous regions with only subtle changes in intensity ( Kurth et al., 2015).

Conclusion

In conclusion, the present study reveals for the first time the gray and white matter trajectories of the dyscalculic brain. The findings confirm the structural differences as reported in earlier research and support the notion that DD is characterized by persistent structural and behavioral abnormalities. There is an urgent need for longitudinal studies examining the typical and atypical neural development, but also the effect of interventions and therapy on numerical and mathematical abilities. Advancing the knowledge about the developmental course of DD and the effects of schooling, therapy, and intervention would enable us to support affected children and adolescents more effectively.

Data Availability Statement

The datasets generated for this study will not be made publicly available. According to the Ethics committee of Zurich, Switzerland the consent was not obtained for sharing data outside the research team. Therefore we do not have the allowance to make the data publicly available. Requests to access the datasets should be directed to the Ethics committee of Zurich in Switzerland ( Info.KEK@kek.zh.ch, project-number: 2011-0384, title of the project: Plasticity after training and development in children with and without dyscalculia).

Ethics Statement

The present study was reviewed and approved by the Ethics committee of Canton of Zurich, Switzerland. Written informed consent to participate in this study was provided by the participants' legal guardian/next of kin.

Author Contributions

All authors have contributed and have approved the final manuscript. UM contributed to the design of the study, the acquisition, analysis, interpretation of the data, and writing the manuscript. MA and RO'G contributed to data interpretation and revised the manuscript. KK contributed to the design of the study, the acquisition, data interpretation, editing, and revision of the manuscript.

Funding

This research was supported by a grant from the NOMIS Foundation.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We would like to thank all subjects and their parents for participating in this study.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnhum.2020.00272/full#supplementary-material.

References

Heller, K. A., and Perleth, C. (2000). Cognitive Abilities Test for Grades 4 to 12 - Revised Version. Göttingen: Beltz.

Kurth, F., Luders, E., and Gaser, C. (2015). "Voxel-based morphometry," in Brain Mapping, ed. A. W. Toga (Waltham, MA: Academic Press), 345-349.

Moll, K., and Landerl, K. (2010). Salzburg Reading and Orthography Test - Revised version. Bern: Huber.

von Aster, M., Weinhold Zulauf, M., and Horn, R. (2006). ZAREKI-R. Neuropsychologische Testbatterie für Zahlenverarbeitung und Rechnen bei Kindern, Revidierte Version [Neuropsychological Test Battery for Number Processing and Calculation in Children]. Frankfurt am Main: Harcourt.

WHO. (2018). International classification of diseases for mortality and morbidity statistics (ICD-11). Available online at: https://icd.who.int/browse11/l-m/en. Accessed November 4, 2019.

# data[].attributes.attachment_link.title
Persistent Differences in Brain Structure in Developmental Dyscalculia: A Longitudinal Morphometry Study

# data[].attributes.attachment_link.description
Developmental dyscalculia (DD) is a learning disability affecting the acquisition of numerical-arithmetical skills. Affected people show persistent deficits in number processing, which are associated with aberrant brain activation and structure. Reduced gray matter has been reported in DD for the parietal cortex including the intraparietal sulcus (IPS), but also the frontal and occipito-temporal cortex.

# data[].attributes.headline
Impaired estimation, intact subitizing (Decarli et. al., 2020)

# data[].attributes.subject
Impaired estimation, intact subitizing (Decarli et. al., 2020)

# data[].attributes.attachment_link.readable_text
Gisella Decarli,

Emanuela Paris,

Chiara Tencati,

Chiara Nardelli,

Massimo Vescovi,

Luca Surian,

Manuela Piazza

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Abstract

It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation. Here we tackled this question by asking 32 DD children and 32 controls to perform a series of tasks on visually presented sets, including exact enumeration of small sets as well as comparison of large, uncountable sets. In children with DD, we found poor sensitivity in processing large numerosities, but we failed to find impairments in the exact enumeration of sets within the subitizing range. We also observed deficits in visual short-term memory skills in children with dyscalculia that, however, did not account for their low ANS acuity. Taken together, these results point to a dissociation between quantification skills in dyscalculia, they highlight a link between DD and low ANS acuity and provide support for the notion that DD is a multifaceted disability that covers multiple cognitive skills.

Citation: Decarli G, Paris E, Tencati C, Nardelli C, Vescovi M, Surian L, et al. (2020) Impaired large numerosity estimation and intact subitizing in developmental dyscalculia. PLoS ONE 15(12): e0244578. https://doi.org/10.1371/journal.pone.0244578

Editor: Jérôme Prado, French National Center for Scientific Research (CNRS) & University of Lyon, FRANCE

Received: May 21, 2020; Accepted: December 13, 2020; Published: December 31, 2020

Copyright: © 2020 Decarli et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All data files are available from the Open Science Framework (OSF) database (accession URL: https://osf.io/gjpb3/?view_only=91847e46de8e4cd5a899d75cc1b55af8. The data are not fully available because the name, surname and date of birth of the participants can not be shown. We therefore assigned a code for each participant, specifying the group in which it was inserted (DD or control group).

Funding: The author(s) received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Humans possess two basic non-verbal systems underlying the quantification of the number of objects in sets. One system allows approximate estimation (also called approximate number system, or ANS) and it generally follows the Weber law, according to which the threshold of number discrimination between two stimuli increases proportionally with the intensity of the stimuli, i.e. with the magnitude of the numerosities. More recently, some authors have proposed that the discrimination threshold follows the Weber law only in the case of low-density numerosities, while for high-density and crowded numerosities a texture-like mechanism drives the numerosity comparison [ 1- 3] (i.e., when the dots are too dense, numerosity judgements seem to be performed using a texture-based mechanism, evident from the fact that in those cases number discrimination threshold increases with the square root of numerosity instead of with its log). The other numerical system allows exact and fast quantification but it is limited in capacity, as it only applies to small sets of few objects (also called subitizing, or object tracking system, OTS).

The distinction between these two systems [ 4- 9], evident in human adults [ 10, 11], is traceable from the very first days of life. Young infants, able to discriminate large sets differing by at least of 200% (e.g., 10 vs. 30 items), are not able to distinguish sets differing by the same amount in the small number range (e.g., 1 vs. 3), as their still immature multiple object tracking system only extends to sets of 1-2 objects [ 6]. According to some authors, young infants present an immature OTS, that would develop during the first year of life [ 12, 13]. As demonstrated by some studies that have assessed the visual short-term memory (the ability at the base of the tracking system), infants present a span of one object at 6 months, and it develops up to 4 elements at 12 months.

Both systems improve during the lifespan (due to maturation and experience), but their developmental trajectories appear rather different: while the multiple object individuation ability reaches the adult level during early childhood, the approximate number system continues to improve up to adulthood. Moreover, there is some evidence that small sets individuation and large sets quantification are dealt with by separate cognitive and neuronal mechanisms through the entire life-span [ 8, 14, 15]. The qualitative difference between these two systems appear evident when looking at neuroimaging data, since each system appears associated to specific alteration of the EEG signal, in both adults and preverbal infant: the OTS corresponds to the modulation of an early posterior brain wave, which amplitude varies proportionally with number, but only for small sets of 1-3 items; the ANS, on the contrary, gives rise to a ratio-dependent modulation of a late posterior wave only during processing of large numerosities [ 14]. This suggests that the ANS and OTS may emerge from at least partially segregated cortical circuits. More recently, Fornaciai and Park [ 15] identified distinct polarity as well as topographic distributions of electrodes sensitive to small and large numerosities specifically.

One aspect that further differentiates these two systems is the extent to which they correlate and longitudinally predict formal mathematical skills. On one side there is now solid evidence for a reliable and bidirectional link between the ANS acuity (the precision in assessing the number of items in large sets) and formal math skills: the ANS correlates and longitudinally predicts mathematical skills [ 16- 19], and is in turn modified by the acquisition of formal math skills [ 20, 21]. One potential interpretation of these results is that when children learn the meaning of symbols they initially do so by relating them to the pre-existing representations of non-symbolic magnitudes and their transformations, and that the two systems remain connected during the life-span [ 22; however see 23 for a different view]. For example, Gilmore and colleagues showed that 5-to-6-year-old children during the first stages of acquisition of symbolic numbers treat them as symbolic referents to approximate quantities [ 24]. Traces of this link can also be found in older children and in adults who, when asked to compare symbolic numbers, show the same distance and magnitude effects (i.e., speed and accuracy decrease as the ratio across numbers increases) that also characterize non-symbolic magnitude processing. However, it should also be noted that other authors failed to find an association between symbolic and non-symbolic formats. For example, some authors have found higher RTs and error rates in comparing mixed format of dots-numerals, than single format of dots-dots, concluding that if the symbolic and non-symbolic formats (one highly accurate, while the other highly inaccurate) are linked then the performances should not be worse in the mixed format compared to the single one [ 25]. Moreover, some neuroimaging findings failed to provide evidence for a common neuronal representation between the two formats [e.g., 26]. In sum, the issue is still heavily debated.

On the other hand, as for the OTS, the rare studies focusing on the inter-individual differences in exact small number processing (typically indexed by the range of numerosities enumerated with no errors and in a very short time, also referred to as "subitizing range") fail to find a reliable association with math abilities in typically developing populations [ 27].

These observations suggest that large number estimation and small number quantification might also be differentially involved in developmental dyscalculia (DD), a neurodevelopmental specific learning disability in number and calculation skills. Children with DD show a broad range of deficits in math that primarily include mental and written calculation but also extend to more basic skills such as number comparison or numbers to magnitudes association. Previous studies also reported a low ANS acuity in dyscalculia, indexed by impairments in dot comparison tasks, where DD seem to have a 5 year-delay along the typical developmental trajectory [ 28- 32]. These findings, however, are not undisputed, since using similar (but not identical) dots comparison tasks, some researchers did not find differences between DD and controls [ 33- 35]. Moreover, there is an ongoing debate on the factors underlying the DD poor performance in this task: some reported evidence that it is determined by a non-specific deficit in a general inhibitory system [ 36], while others failed at replicating it [ 37].

The literature on small set quantification (subitizing) in DD appears at least as equally controversial, if not more. Again, some authors found impairments in DD [ 38- 42] while others did not [ 43- 45].

Of particular interest for the question of whether the estimation and subitizing are or not commonly impaired in dyscalculia are those studies that investigate the two skills in the same group of DD children. These types of studies report divergent results: one did not find any deficit in any kind of set quantification (neither in subitizing nor in estimation) [ 46]; another found impaired estimation but not in subitizing [ 47], and a final one found an impairment in both estimation and subitizing [ 48]. To our knowledge no study to date reported an impairment in subitizing but no impairment in estimation in DD, potentially suggesting that whenever quantification tasks are compromised, subitizing tends to be more resilient to impairments compared to large set estimation.

When considering the results on subitizing, some methodological considerations are worth mentioning: first, many of the aforementioned studies investigated subitizing using a task where the to-be-enumerated dots remained on the screen for an unlimited time until the response of the participant was detected [ 38, 41, 42, 46- 48]. Giving no limits to stimuli presentation could lead children with DD, who tend to be more unsure whenever they have to deal with numerical tasks, to adopt a different strategic choice compared to controls, such that of counting the dots one by one also when presented with sets of 1-3 objects. This would be coherent with the fact that the aforementioned studies reported an impairment in RTs (some in the form of a steeper increase in RTs) but not in accuracy in DD. This possibility reminds us that, while very often response times give important information on the structure of the internal representations, they may also be heavily influenced by strategic choices.

A second methodological concern is the definition of subitizing range. In some previous studies the interval defined as the subitizing range was set not on the bases of the data, but a priori (to either to 3 or 4) [ 41, 45- 47, 49]. However, it is known that the subitizing range is influenced by specific features of the paradigm, such as presentation time, the spatial layout of the stimuli, the presence of a mask [ 50, 51] and also the age range of participants [ 52]. Therefore, the subitizing range should be determined empirically on the data and not a priori. This might seem like a minor aspect. However, we believe it is not. In fact, setting the subitizing range a priori may lead to the inclusion of data that, given the particular protocol used, already fall in the serial counting or estimation regime instead of subitizing properly. This might thus let the researchers conclude that the DD have an impaired subitizing range when in fact they are impaired in serial counting, a process that, contrary to subitizing, involves serial visuo-spatial attentional shifts, working memory, and phonological processing [ 53].

In sum, while there is not full consensus as for whether dyscalculia is associated in deficit in large sets estimation, the question of whether they are impaired in subitizing is even more controversial. Particularly problematic in this respect seems the large variety of protocols used to investigate subitizing together with often arbitrary assumptions related to the extent of the subitizing range. The absence of a clear convergence on the issue of whether children with DD are impaired in both number estimation and subitizing clearly calls for further investigations.

In the present study we assessed numerosity estimation and exact enumeration performances, within the same group of children with developmental dyscalculia, and we compared them with those of non-dyscalculic children of the same age and general intelligence. Because the tasks typically used to assess numerosity perception rely on visually presented sets of items, one may think that potential impairments in these tasks could be directly depending on domain-general impairments in visuo-spatial skills. Several authors have indeed provided evidence that DD is associated with deficits in visuo-spatial skills, especially in visuo-spatial working memory [ 54- 57]. In order to investigate the potential dependency between number perception and visuo-spatial skills we also tested the latter, using both a perceptual change detection task as well as the widely used Corsi working memory test.

The aim of the present study was to investigate whether children with DD are concurrently impaired in numerical judgements of large uncountable sets as well as small countable ones, tapping respectively on the ANS and on the OTS system. In parallel, we also assessed visuo-spatial short-term memory, a domain-general skill that, on the bases of previous research, we hypothesized could have a link especially with the OTS system. Indeed, previous studies, using correlational and dual task interference paradigms on adult subjects, demonstrated not only a dissociation between ANS and subitizing, but also a correlation and a specific interference between subitizing (but not the ANS) and visuo-spatial attention [ 4; 10; see also 11], suggesting that the OTS (but not the ANS) share resources with the visuo-spatial attention system.

We therefore hypothesized that dyscalculics would be impaired in the ANS and that this impairment would not correlate with visuo-spatial working memory skills. On the contrary, based on the aforementioned literature that points towards a link between OTS and visuo-spatial skills, we predicted that if the subitizing is impaired in DD, the impairment would be accompanied with concurrent impairments in visuo-spatial attention.

Materials and methods

The research has been approved by the Human Research Ethics Committee of the University of Trento (approval number 2015-026). Consensus for participation in the study was written and was obtained from the parents of the children.

Participants

The study included thirty-two dyscalculic children and thirty-two typically developing children with no learning disabilities. Parents/legal representatives gave written consent to participate in the study. The experiment was performed in accordance with the ethical standards established by the Declaration of Helsinki.

Children with DD.

Thirty-two children with severe deficits in the mathematical domain were selected from a sample of children referred to a regional reference center for the diagnosis and rehabilitation of learning disabilities, the "Azienda Sanitaria Beato de Tschiderer" (Trento, Italy), because of a specific learning disability in the mathematical domain. Their mathematical abilities were assessed through one of the following widely used age-standardized Italian batteries for assessing mathematical skills: (1) the "Battery for the assessment of Developmental Dyscalculia" (BDE-2) [ 58], a standardized battery for children aged 8 to 13 years which items are divided into 3 areas: number reading/writing, exact oral/written calculation and semantic number understanding (comprising symbolic number comparison, number-to line mapping, and approximate calculation), or (2) the "Test for evaluating Calculation Abilities" (AC-MT) [ 59], a standardized battery for children aged 6 to 11 which produces 8 indices that consider speed and/or accuracy in written and mental calculation, verbal reciting of the number list, number spelling (dictation) and arithmetic facts retrieval. Both batteries provide normative data for every school grade; the raw score of each subtest is compared with the corresponding normative score to obtain the level of the child performance compared to typically developing children with the same age and grade. Therefore, each child performance can be assessed in terms of percentiles for the AC-MT and quotients for BDE-2.

Participants, included in the sample, presented severe difficulties in number processing and calculation corresponding to the strict diagnostic criteria for DD: performances under or equal to the 5th percentile in at least 4 out of 8 indexes of the AC-MT (N = 18) or in at least 2 out of 4 quotients of the BDE-2 (N = 14). We also included 1 subject who did not perform the full AC-MT battery, but only completed 2 out of 5 tasks: the mental calculation and the arithmetic facts retrieval tasks. In both tasks, and for both RTs and accuracy, he performed below the 5th percentile. An additional subject was included even if performance fell below the 5th percentile only in 2 (instead of 3 indices), but was consistently below the 10th percentile in 4 additional ones.

Further selection criteria were: general intelligence, assessed with the Wechsler Intelligence Scale for Children [ 60], within the normal range; normal or corrected-to-normal vision and hearing; normal schooling; no neurological or psychiatric disorders. Co-morbidity with other developmental disorders was not set as an exclusion criterion because our aim was not to generally describe the cognitive profiles of "pure" DD, but rather to contrast performance in two specific quantification tasks in children with DD, irrespective of the presence of other concurrent weaknesses. In our group, 14 children presented a selective deficit in the domain of mathematics (they were "pure" DD), while the remaining ones presented co-occurring deficits in other cognitive domains (written expression (N = 3), reading (N = 3), written expression and reading (N = 7), language production (N = 5), motor coordination (N = 1), or attention-deficit/hyperactivity (N = 1).

The group had a mean age of 9.51 ( SD = 1.55, range 7.4-14) and an average total IQ of 97 ( SD = 9.48) (Verbal IQ = 108, SD = 12; Performance IQ = 102, SD = 11). Children were tested in a quiet room of the "Azienda Sanitaria Beato de Tschiderer", in one or two sessions (separated by a max. of 14 days), depending on their availability. The order of the tests was randomized across participants.

Non-dyscalculic controls.

Thirty-two children (mean age = 9.79, SD = 1.66, range 7.5-14.3) were selected from a sample of eighty-seven children, recommended as normal calculators by their teachers, from a primary school in Malo (province of Vicenza, in the north of Italy) to match the DD group in chronological age (Age: t(62) = -.71, p = .48), and in general intellectual functioning (Similarities, taken as an index of verbal skills, and Matrix Reasoning indexing visual processing. Note that 3 DD children were addressed to the Center after they were already diagnosed elsewhere, and for them we did not have the WISC single scores for Similarities and Matrix Reasoning, but only the total scores) (Similarities: M dysc = 10.55, SD = 2.9; M contr = 10.34, SD = 2.39, t(59) = .31, p = .76; Matrix Reasoning: M dysc = 9.1, SD = 2.51; M contr = 10.03, SD = 2.52, t(59) = -1.44, p = .16; see Table 1). The typically developing children were tested in a quiet room of the school, during school-time, in one test session. Also in this case, the order of the tests was randomized across participants.

Tasks and procedure

Symbolic number comparison task.

Because our control group did not perform the diagnostic tests for dyscalculia, we wanted to have to least one measure of symbolic number processing that would confirm their difference with the DD group. We therefore used a classic symbolic number comparison task. Participants were presented with pairs of stimuli, appearing within white circles on a computer screen. Stimuli consisted in Arabic digits from 1 to 9. Participants were asked to decide as quickly as possible which of the two digits indicated the larger quantity. Stimuli remained on the screen until participants pressed a response-key on the mouse. The task started with 8 training trials where feedback on accuracy was given to participants, and then it comprised 128 trials divided into 8 blocks, where no feedback was provided. It lasted about 10 minutes.

Large numerosity comparison task.

Participants were presented with pairs of arrays of black dots on white background, appearing laterally on a central fixation cross on the screen ( Fig 1).

They were instructed to judge, fast and without counting, the more numerous one, by pressing the response-key on the mouse located on the same side. The arrays remained on the screen until children gave the answer.

Of each pair of sets, one (n1) always contained either of 16 or 32 dots. Stimuli paired with arrays of 16 dots (n2) could contain 12, 13, 14, 15, 17, 18, 19, 20 dots, while stimuli paired with 32 dots could present 24, 26, 28, 30, 34, 36, 38 or 40 dots. In half of the trials the size of the dots remained constant across all the n1 and n2 sets (therefore, as a consequence, the total occupied area increased with number), while for the other half the size of the total occupied area remained constant for the two arrays (thus individual item size was anti-correlated with number). The task started with 8 training trials where feedback on accuracy was given to participants, followed by 128 experimental trials, divided into 8 blocks, where no feedback was provided. It lasted about 10 minutes.

Small exact enumeration task.

Participants were presented with arrays of colored dots varying in number from 1 to 8, and appearing, for 500 ms in a central grey circle ( Fig 2).

Every dot had a specific color that was randomly chosen among 8 highly discriminable colors (red, pink, yellow, orange, blue, white, green, cyan and black) so that two dots could not have the same color in the same image. The stimuli were identical to those used for the visuo-spatial short-term memory task [see 10]. Participants were asked to say aloud and as quickly and accurately as possible the precise number. They were informed that the maximum number of dots could be 8. The responses were recorded via a microphone mounted on headphones to maintain the stability of the mouth-to-microphone distance across trials. In order to calibrate the microphone's sensitivity to the average children's voice pitch, before starting the task participants were instructed to say aloud numbers from one to eight. The microphone level was then calibrated accordingly. Stimuli were generated such that, across numerosities, half were controlled for size and half for total occupied area. The task comprised 10 training trials, followed by 128 trials divided into 4 blocks, and lasted about 30 minutes. No feedback was provided.

Visuo-spatial short-term memory task.

Participants were presented with a sequence of two stimuli, both appearing in the center of the screen. Both stimuli were sets containing the same number of colored dots. In half of the trials, the second set was identical to the first one. In the other half, one of the dots appeared in a different color. The first image appeared for 700 ms, followed by a blank image for 1000 ms. Then, the second image remained on the screen for 2000 ms. Stimuli were identical to those used in the exact enumeration task. The task closely matches those typically used to assess visuo-spatial short-term memory in experimental psychology/psychophysics [ 10, 61].

Participants indicated whether the two arrays were identical or differed by expressing their judgment aloud, and an experimenter pressed the corresponding answer on the keyboard. While typically adult subjects perform this task through button press following simple response mapping rules (e.g., press right if same, left if different), here we used the vocal response to rule out that our results might reflect a potential difficulty in maintaining in memory the mapping rule. The experiment started with 10 training trials, followed by 128 test trials divided into 4 blocks. No feedback was provided.

Corsi block-tapping task "forward" and "backward".

The canonical version of the Corsi test was used [ 62]. The stimulus consisted of nine blocks placed on a board. The blocks were numbered from 1 to 9 only on one side of the block, visible to the experimenter but not to participants. The experimenter, seated in front of the child, taps the blocks with the finger following a sequence. Participants were instructed to repeat the same sequence with their own fingers. The task started with a sequence of two blocks, and then the experimenter increased in the number of blocks when two out of three trials of the same number were performed correctly. Otherwise, the test was terminated. In the "backward" version participants had to repeat the sequences in inverse order.

Results

Symbolic number comparison task

We analyzed mean accuracy and mean RTs using a mixed ANOVA with group (dyscalculics and controls) and numerical distance (1-8: e.g., pair 1-3, numerical distance = 2; pair 2-9, numerical distance = 7) as between and within factor respectively. Results indicated that dyscalculic children made significantly more errors and were slower than controls (main effect of group for accuracy: M contr = .97, SD = .02, M dysc = .95, SD = .03; F(1,62) = 8.94, p = .004, η ² G = .04; for RTs: M contr = 920.04 ms, SD = 238.26 ms; M dysc = 1139.2 ms, SD = 321.49 ms; F(1,62) = 10.07, p = .002, η ² G = .12), and that, for both groups, close numbers were more difficult to compare than distant numbers (main effect of distance for accuracy: F(7,434) = 57.04, p < .001, η ² G = .4; for RTs: F(7,434) = 53.7, p < .001, η ² G = .1). Importantly, however, dyscalculics were more impaired at comparing pairs of numbers characterized by a small numerical distance, indicated by a significant group by distance interaction for RTs and by a marginally significant interaction for accuracy (accuracy: F(7,434) = 1.87, p = .07, η ² G = .02; for RTs: F(7,434) = 2.48, p = .02, η ² G = .005).

Large numerosity comparison task

We first analyzed overall mean accuracy: controls were significantly more accurate than DD ( M contr = 67.77%, SD = 6.59; M dysc = 63.84%, SD = 7.8, t(62) = -2.18, p = .03, d = .54). Importantly, the two groups did not differ in RTs ( M contr = 1595.79 ms, SD = 632.45; M dysc = 1556.38 ms, SD = 624.91, t(62) = -.25, p = .8, d = .06); thus, the poorer performance of DD children was not simply the result of an overall faster decision making.

In order to directly compare the current results with previous reports, we also analyzed the psychometric curves and estimated the internal Weber fraction (hereafter w) for each participant [ 32]. Assuming the hypothesis that numerosities are internally represented by a logarithmic number line with fixed Gaussian variability, w is proportional to the standard deviation of the estimated Gaussian distribution of the internal numerical representation that generates the observed performance. Because this measure is dependent upon model fitting, we excluded the subjects for which the model did not fit well (R2 < 0.2; N = 6, 4 dyscalculics and 2 controls). After this selection the two groups remained balanced for IQ and age (Similarities: p = .69; Matrix Reasoning: p = .32; age: p = .49). The DD group had larger w fractions ( M contr = .24, SD = .10; M dysc = .32, SD = .14; t(55) = 2.55, p = .01, d = .68; see Fig 3).

RTs also did not differ across groups in these reduced samples (RTs: M contr = 1645.39 ms, SD = 644.63 ms; M dysc = 1622.37 ms, SD = 631.22 ms, p = .89, d = .04).

Small exact enumeration task

We first analyzed mean RTs and mean accuracy (Figs 4 and 5, respectively) through a mixed ANOVA with group as between factor and numerosity as within factor.

We reported the analyses including the set sizes up to numerosity 7, because children were informed that the number of dots presented could be from 1 to 8. Therefore, the results of numerosity 8 were influenced by this information given to the participants. Indeed, for numerosity 8 we found a typical guessing end-effect [see 10, 64, 65], showing a drop in RTs and in error rate compared to numerosity 7. The ANOVA revealed that both errors and RTs increased with number (main effect of numerosity accuracy: ( F(6, 372) = 251.7, p < .001, η ² G = .72; RTs: F(6, 372) = 144.84, p < .001, η ² G = .55), and that dyscalculics made more errors and were slower (main effect of group accuracy: F(1, 62) = 4.1, p = .05, η ² G = .02; RTs: F(1, 62) = 4.6, p = .04, η ² G = .03; interaction for accuracy: F(6, 372) = 1.94, p = .07, η ² G = .02); interaction for RTs: F(6, 372) = 4.79, p < .001, η ² G = .04).

In order to specifically investigate whether dyscalculics present a reduced subitizing range we first estimated it by conducting pairwise comparisons among successive numerosities (1 vs. 2; 2 vs. 3; 3 vs. 4) on both accuracy and RTs in our control group. Since this is a study testing whether a clinical group is different from a control group, we estimated the "standard behavior" from the control group and then test if the clinical group differs from it or not. However, while in theory we could have also decided to perform the opposite analysis (set the standard subitizing range in the DD and then test if the controls behave differently), this would have been a much less sensitive approach for detecting a difference across groups: if, as we might expect from previous literature, dyscalculics have a smaller subitizing range than controls [e.g., 39], then the subitizing range computed from DD would include that of the controls. This would lead to no difference across groups, and to an erroneous conclusion. We found that significant differences in both accuracy and RTs appeared only between 3 and 4 (accuracy: 3 vs. 4: p < .01, all previous pairwise comparisons were not significant; RTs: 3 vs. 4: p < .001, all previous pairwise comparisons were not significant), suggesting that on this task our control group displays a subitizing range of 1-3. We thus proceeded performing separate analyses for the trials in the 1-3 (subitizing) and 4-7 (counting) range. Across groups, in the subitizing range there were no significant differences neither in accuracy ( M contr = 99%, M dysc = 98%; t(62) = -1.3, p = .197, d = .33) nor in RTs ( M contr = .42 s, M dysc = .39 s; t(62) = -.85, p = .4, d = .21). However, the groups did perform differently in the counting range, as shown both in their RTs and, only marginally, in accuracy (RTs: M contr = 1.05 s, M dysc = 1.36 s; t(62) = 2.71, p < .01, d = .68; accuracy: M contr = 64%, M dysc = 56%; t(62) = -1.94, p = .06, d = .49).

Another possible approach previously used to estimate the subitizing range is that of fitting a sigmoid function to the data and of taking its flex as its estimate [ 10, 11]. However, the sigmoid function is fitted on the full data set, including responses to both small and large numerosities. Because of this, the estimated flex is determined by how the data changes within the small and the large numerosity ranges: for any fixed performance in the small numerosity range, the steepness of the increase in performance within the counting range influences the estimate of the flex. Because DD and controls differed in both RTs and errors in the large numerosity range, this would have impacted on the estimate of the flex. On the contrary, computing pairwise comparisons between neighboring small numerosities is not influenced by performance with higher numerosities. For this reason, we have decided to use the first index instead of the flex.

We decided to perform two additional analyses to provide further evidence at the claiming of no difference between DD and controls in the subitizing range. We chose two indexes that had already been used in the literature when assessing children's ability in enumerating small numerosities. The first one consists in combining RTs and accuracy in the so-called inverse efficiency measure (EM = mean correct reaction times / accuracy) and computing the steepness of the increase of EM (EM maxN-EM minN / EM maxN) within subitizing and counting ranges separately [ 66]. We compared those indices across groups, and we found no effect in the subitizing range, and a small but not significant difference in the counting range (subitizing range: M contr = .2, SD = .23, M dysc = .26, SD = .24; t(62) = 1.01, p = .32, d = .25; counting range: M contr = .77, SD = .12, M dysc = .65, SD = .35; t(59) = -1.68, p = .097, d = .43).

The second approach we adopted focused on the response distributions, that we assessed computing the coefficient of variation, providing a quantification of the errors' extent [ 29, 40, 67]. This is computed by dividing the standard deviation of the responses by their mean for each numerosity. We found no significant differences between dyscalculics and controls neither in the subitizing range ( M contr = .03, SD = .1; M dysc = .01, SD = .06; p = .42, d = .20) nor in the counting range ( M contr = .13, SD = .06; M dysc = .14, SD = .06; p = .41, d = .21), indicating that for both ranges the size of the errors was similar across groups (see Fig 6).

Visuo-spatial short-term memory

We first analyzed mean accuracy through a mixed ANOVA with group as between factor and numerosity as within factor. In both groups accuracy declined with increased set size (set size 1: .96; 2: .94; 3: .92; 4: .82; 5: .69; 6: .71; 7: .63; 8: .61) (main effect of set size: F(7, 434) = 173.07, p < .001, η ² G = .66), and dyscalculics performed worse than controls (main effect of group: F(1, 62) = 14.3, p < .001, η ² G = .06), especially so for sets of 3 and more items (interaction: F(7, 434) = 2.21, p = .03, η ² G = .02) (See Fig 7).

We then calculated Cowan's K [ 68] (i.e., K = S(H-F), where S is the set size, H is the hit rate and F is the false alarm rate; the score obtained with this formula allows the estimation of the number of objects encoded) for each set size, and then computed the average K across set sizes for each subject as an estimate of the visuo-spatial short-term memory span. The average K was 1.76 for the dyscalculics (range = .68-2.77, SD = .5) and 2.26 for the controls (range = 1.23-3.26, SD = .52), a highly significant difference, t(62) = -3.85, p < .001, d = .96.

Corsi block-tapping test

Separate analyses were performed for Corsi test "forward" and "backward". Controls performed significantly better than DD, showing a higher visuo-spatial span both forward and backward ("forward": M contr = 4.69, SD = .96, M dysc = 4.06, SD = .62, p = .003, d = .77; "backward": M contr = 4.59, SD = 1.29, M dysc = 3.12, SD = 1.29, p < .001, d = 1.14).

Because we found impairments in both the ANS acuity and visuo-spatial short-term memory, we addressed the question of the potential link between the two impaired functions by correlating the w fraction and accuracy in the ANS task with the Cowan's K and the average score between the forward and the backward in the Corsi test in DD and controls. We did not find any significant correlation neither in the children with DD, nor in controls ( w fraction and Cowan's K: r contr = -.17, p = .39; r dysc = -.23, p = .23; w fraction and Corsi test: r contr = -.13, p = .51; r dysc = .13, p = .51; dot comparison accuracy and Cowan's K: r contr = -.13, p = .49.; r dysc = .17, p = .34; dot comparison accuracy and Corsi test: r contr = -.13, p = .49; r dysc = .29, p = .1). Moreover, we performed the same analyses for the OTS and we did not find significant correlations between subitizing and visuo-spatial short-term memory (accuracy 1-3 and Cowan's K: r contr = -.18, p = .33; r dysc = .32, p = .07; accuracy 1-3 and Corsi test: r contr = .26, p = .15; r dysc = .16, p = .37; RTs 1-3 and Cowan's K: r contr = -.15, p = .4.; r dysc = -.09, p = .63; RTs 1-3 and Corsi test: r contr = .0004, p = .99; r dysc = -.19, p = .3).

Discussion

The main question addressed in our study concerns whether DD is associated with impairments both in the quantification of large sets (supported by the ANS) and of small sets (supported by subitizing, or OTS). Although the dissociation between ANS and OTS is well-established in typically developing infants, young children, and adults, there is currently no agreement on whether the two systems are concurrently impaired in DD. Different theories predict either a defective ANS and a non-impaired OTS [ 69], or a deficit in both ANS and OTS [ 70]. Other theories posit that DD results from an impairment in more general cognitive abilities, especially tapping on visuo-spatial skills, such as visuo-spatial working memory [ 55- 57]. In the current study we tested all those three systems (large numerosity comparison, exact enumeration and visuo-spatial short-term memory skills) in the same group of DD children with strong impairments in symbolic number processing and arithmetical calculation (< 2 SD s in a standardized mathematical test), and we compared them with a group of children matched for age and general intelligence without deficits in maths.

Regarding the ANS, our study provides strong evidence for a deficit in DD, tightly replicating previous reports [ 29- 32]. DD children showed worse performances compared to typically developing children, resulting in different estimates of their internal Weber fraction, indexing their ANS acuity (dyscalculics: w = 0.32; controls: w = 0.24). Despite being tested on a different group of DD and controls, who also had a slightly larger age range, it is remarkable that the current findings closely replicate those obtained by Piazza et al. [ 32], who, using the same experimental paradigm and stimuli as used here, reported a w of 0.34 in dyscalculics and of 0.25 in controls. Coherent with previous findings, we also observed that although children with DD showed impaired accuracy in this task, they did not present differences in RTs compared to typically developing children. This suggests that, while spending the same time to take the required number comparison decision, dyscalculics rely on less precise internal representations.

Our second skill of interest was subitizing, the ability to accurately enumerate small sets of items. We found no group differences (neither in RTs, in errors, in their combination, nor in error extent) in the subitizing range (1-3). The two groups diverged only in the counting range and only when considering response times, not when considering accuracy.

These data contradict some previous findings [ 38- 42] that reported the presence of a deficit in DD in the subitizing ability. However, an important difference across our study and previous ones in the experimental paradigm might explain the divergent findings: while in our paradigm the dots remained on the screen for a limited amount of time (500 ms), in some of the aforementioned studies children could look at the images and provide their response without time limits. We speculate that this might have prompted the use of divergent strategies (DD using serial exploration, controls using parallel individuation) and that the difference reported does not necessarily reveal a real difference in the underlying skills across groups. Indeed, when children are not allowed to implement serial visual exploration strategies, both RTs and errors performed by children with DD are identical to those of the controls. Moreover, contrary to some of the previous studies, here we defined subitizing range empirically, on the bases of performance in the control group. This is important, as it is well known that the subitizing range depends upon the specificities of the experimental paradigm and stimuli as well as the age of participants. It is possible that those previous studies that set subitizing range a priori erroneously attributed groups difference to subitizing when in fact they might have been related to counting.

Our results of an intact subitizing and impaired large numerosity comparison in DD supports the separation between ANS and multiple-objects individuation system found in typically developing population. These findings lead to confirm a qualitative difference in the mechanisms underlying the perception and representation of large vs. small numerosities.

Another relevant finding of the current study was the significant impairment observed in DD in the one-digit symbolic comparison task. In this study we introduced this task because our control group did not perform the diagnostic tests for dyscalculia, and therefore we wanted to have at least one measure of symbolic number processing that would confirm their difference from the DD group. In line with previous studies [ 34, 71], we found that children with DD performed overall significantly poorly in the symbolic comparison and the difference between the two groups was particularly evident in the case of small distances (1, 2 and 3). This is in line with the idea that the mental representation of numerical symbols in DD is less precise (such that close numbers are less differentiated from each other) than in controls. The fact that DD children present impairments in symbolic comparison is part of the predictions put forward by many authors, not only those that ascribe DD to an impaired number sense that would later affect the symbolic number processing [ 32, 72] but also those who postulate a first deficit in the exact numerical representation of symbolic numbers with a later and consequent dysfunction in the ANS acuity [ 73]. The contrast between these two positions is still debated in the literature and the question of the possible causal/effect link between symbolic and non-symbolic number processing remains open.

Finally, our results also confirmed previous reports [ 46] that children with DD are also heavily impaired in visuo-spatial short-term memory, which we measured both through a computerized change detection task and the Corsi test, a widely used test in neuropsychological assessment. In the change detection task, we found that children with DD retain less information of a complex visual scene (they have a smaller Cowan's K). Similar results were obtained in the Corsi test where children had to repeat ("forward" or "backward") a sequence of previously shown touched blocks. This task required not only encoding/storage/retrieval of the information but also its mental manipulation, in particular in the "backward" version. In both versions of the tasks, children with DD were dramatically less accurate compared to the controls. Importantly, however, we found that deficits in those visuo-spatial tasks unlikely account for the impairments in the ANS as the two did not correlate across subjects. It is possible that impairments in visuo-spatial skills and in the ANS play a differential role in numeracy: while the former may potentially have a higher impact on learning and using written and mental arithmetical calculation procedures [ 74, 75], the latter may be more fundamental for learning and using the semantics of symbolic numbers.

Previous work [ 10] on adult participants found that subitizing and visuo-spatial short-term memory are correlated as they share common limited attentional resources. From this, one would predict that the two abilities should be commonly impaired (or commonly spared) in DD. However, we found that subitizing was fully spared and visuo-spatial short-term memory deeply impaired. It is hard to speculate on why there is this apparent discrepancy with the adult results. One possibility is that the two functions are unrelated in childhood but they get related during development. This issue needs more studies directly comparing different age ranges using the same experimental paradigms.

In sum, the present study reports a strong and independent weakness of dyscalculics in two different cognitive domains: approximate number processing (ANS) and visuo-spatial processing. The absence of correlations in performance across the two domains suggests that the impairments in the ANS may not be determined by general impairments in visuo-spatial processing, and that the two functions are separable even if they are concurrently impaired in DD.

The present study failed to observe weaknesses in subitizing, thus failing to provide evidence for the subitizing deficit hypothesis in DD. We speculate that the contrasting findings with previous studies could be explained by methodological differences. It would be important that the field converges on set of stable methods used universally to investigate the ANS and subitizing, such that the results will be more directly comparable.

Implications for practice

If further replicated, these results lead to important clinical implications: on one side they suggest that clinicians could increase symbolic abilities since we found strong impairments in the capacity to compare digits, also in line with previous data showing a strong link between symbolic number knowledge and maths skills. Clinicians could also combine exercises that aim at strengthening the ANS acuity especially using non-symbolic approximate calculation [ 76- 78; note however that this issue remains controversial as some authors questioned the effectiveness of the ANS training and others suggested that ANS training studies lack of sufficient statistical power; see 79- 81]. More generally, combining the results of the present study with those in the literature, we think that it is reasonable to define DD as connected to multiple deficits rather than a single core deficit [ 82- 84]. From the point of view of the potential clinical implications, this work suggests that training for dyscalculics should also include exercises aiming at boosting visuo-spatial skills.

Acknowledgments

We would like to thank all children and families who participated in the study. Moreover, we are grateful to the school "Istituto Comprensivo G. Ciscato" of Malo (VI, Italy) for the collaboration to the collecting data.

References

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4. 60. Wechsler D. (2003). WISC-IV: Wechsler intelligence scale for children: Manual. Psychological Corporation.

5. 62. Corsi P.M. (1972). Human memory and the medial temporal region of the brain. Unpublished Thesis. Montreal: McGill University.

6. 72. Wilson A. J., & Dehaene S. (2007). Number sense and developmental dyscalculia. In: Coch D., Dawson G., Fischer K. (Eds.), Human Behavior, Learning, and the Developing Brain: Atypical Development (pp. 212-237). New York: Guilford Press.

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Impaired large numerosity estimation and intact subitizing in developmental dyscalculia

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It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation.

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TEDx Talk: My World Without Numbers, Line Rothmann (2015)

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TEDx Talk: My World Without Numbers, Line Rothmann (2015)

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My world without numbers | Line Rothmann | TEDxVennelystBlvd

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The Exploding Dots Experience - The Global Math Project

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.
Watch the video to learn how sharing works.

# data[].attributes.attachment_link.description
Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.

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Online manipulatives bring all of the learning and fun without the mess. Our math virtual manipulatives help students grasp abstract concepts through modeling. Literacy virtual manipulatives create a fun space for practicing spelling, rhyming, and more!

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Virtual Manipulatives | Mathematical Modelling | Toy Theater

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Original teaching tools aimed at supporting teachers in the elementary school classroom. Two color counters, 3D dice, probability spinners, graph builders, fraction bars, base ten blocks, and more!

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Intergenerational Effects of Parents’ Math Anxiety on Children’s Math Achievement and Anxiety (Maloney et. al., 2015)

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Intergenerational Effects of Parents’ Math Anxiety on Children’s Math Achievement and Anxiety (Maloney et. al., 2015)

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A large field study of children in first and second grade explored how parents' anxiety about math relates to their children's math achievement. The goal of the study was to better understand why some students perform worse in math than others. We tested whether parents' math anxiety predicts their children's math achievement across the school year. We found that when parents are more math anxious, their children learn significantly less math over the school year and have more math anxiety by the school year's end-but only if math-anxious parents report providing frequent help with math homework. Notably, when parents reported helping with math homework less often, children's math achievement and attitudes were not related to parents' math anxiety. Parents' math anxiety did not predict children's reading achievement, which suggests that the effects of parents' math anxiety are specific to children's math achievement. These findings provide evidence of a mechanism for intergenerational transmission of low math achievement and high math anxiety.

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Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and Anxiety - Erin A. Maloney, Gerardo Ramirez, Elizabeth A. Gunderson, Susan C. Levine, Sian L. Beilock, 2015

# data[].attributes.attachment_link.description
A large field study of children in first and second grade explored how parents' anxiety about math relates to their children's math achievement. The goal of the study was to better understand why some students perform worse in math than others. We tested whether parents' math anxiety predicts their children's math achievement across the school year.