Abstract

This paper focuses on theories and practice of mathematical difficulties associated with dyslexia and DLD (developmental language disorders). It discusses definitions, development, comorbidities, and how it can be analyzed within the causal four level model by Morton and Frith (1995). A conclusion is that early identification and intervention of mathematical difficulties is important.

What are Mathematical Difficulties?

When children start school, they not only learn to transform the sound of the language into letters that together form words and meaning they also learn mathematics. In many ways, learning numbers and mathematics can be compared to learning a new language, and the implicit computational skill is formalized and made explicit. Although mathematics assignments may contain words the child already knows, they are used in a new way in mathematics: “Sally had two apples, and then she got one more apple. Then she had three apples.” This is a concrete statement, which can be abstracted to “two plus one becomes three” or to 2 + 1 = 3. The child should not only understand the use of the word “plus”, but also abstract this term into a concise and precise numerical expression, perform the calculation and translate the expression back into the context in which it was presented. Table 3.1 shows the variation in linguistic and symbolic expressions related to elementary mathematics.

Table 3.1. Math vocabulary

When a child does not catch basic arithmetic, it is important to find out what this may be due to. In an international context, two diagnostic manuals are often referred. According to ICD-11 (International Statistical Classification of Diseases and Related Health Problems 2018), specific disorders of computation include the inability to master basic computational skills such as addition, subtraction, multiplication, and division, rather than more abstract mathematical skills needed in algebra, trigonometry, geometry, or complex calculations. As for the diagnostic and statistical manual DSM (Diagnostic and Statistical Manual of Mental Disorders), one must note that the latest edition, DSM-5 (American Psychiatric Association (APA), 2013, has faced criticism for being diffuse. DSM-IV (APA, 1994) has a longer and more specified list of math difficulties. According to the manual, a person has math problems if:

1. the mathematical skills, measured with individually administered standard tests, are significantly lower than expected based on the person’s chronological age, measured cognitive level and age-appropriate teaching
2. the difficulties in criterion A interfere with academic achievements or day-to-day activities that require computational skills
3. it is in combination with a sensory failure, and the difficulties are greater than the difficulties usually associated with this problem.

It is further explained in the APA manual that the difficulties may be of a linguistic nature (i.e. it is difficult to understand or use mathematical terms, operations or concepts and transcode written problems into mathematical symbols), of a perceptual nature (i.e. that it is difficult to recognize or read numbers or arithmetic characters and categorize different things into groups), that it can have to do with attention (i.e. it is difficult to copy numbers and figures correctly, to remember to add numbers and observe mathematical signs), or that the difficulties can be of a mathematical nature (i.e. it is difficult to follow mathematical sequences step by step, count objects and learn the multiplication table).

This way of explaining mathematics difficulties points to problems within two different cognitive domains: a domain-specific or linguistic domain, and a domain-general or neurocognitive domain (O’Brien et al., 2014). Distinguishing between these cognitive areas is important for understanding the difficulties, for interventions and for further research. The French neuropsychologist Dehaene has studied the neural basis of numeric abilities (Dehaene, 2001). According to him, the “the number sense” is short for our ability to quickly understand, calculate and manipulate numerical sizes, and it rests on networks in the brain that have evolved specifically for the purpose of representing basic arithmetic knowledge. Dehaene points to four pieces of evidence for this hypothesis. First, signs of this ability are found in animals; second, an early emergence of arithmetic abilities is seen in infants regardless of other abilities including language; third, there is a similarity between the ability of animals, infants, and adults to process numbers; and fourth, an area of the brain is specific for the number sense. There are human cases of strokes where the number sense is lost, but linguistic skills remain intact.

How Early Can Mathematical Difficulties be Detected?

When talking about early mathematics, the term “numeracy” is often used. In this term, the child’s understanding of and approach to mathematics is reflected. As mentioned earlier, it has been found that the ability to recognize and manipulate small units of numbers mentally is found in very young children. A Norwegian study concluded that although difficulties with numbers can be detected as early as the age of two, it is very unclear whether these observations have predictive value (Reikerås & Salomonsen, 2017). Nevertheless, several studies indicate that early identification and intervention have an effect. According to a study from England, children’s computational skills at school start (age 5 years) predicted not only numeracy skills in 5th grade, but also literacy (Claessens, Duncan, & Engel, 2009). In a U.S. review study of typical and deviant numerical comprehension, the researchers concluded that learning to count, identify, compare, and manipulate sizes is the key to early math skills. Understanding the development of these skills and the underlying neurocognitive skills form the basis for early intervention in children with difficulties (Raghubar & Barnes, 2017).

Today’s children start school with significant skills in the areas of addition and subtraction, but the numbers must be low and the math tasks concrete so that the children immediately understand the problem. In a longitudinal study, the mathematics skills of children from different kindergartens were mapped six times from preschool to the end of third grade. The relationship between early numeracy and mathematical skills was strong throughout the period the children were examined, and the degree of progress predicted the level of mathematics in third grade. The researchers concluded that early numeracy is important for creating a basis for developing skills in mathematics in the early years of school (Jordan, Kaplan, Ramineni, & Locuniak, 2009).

Another study also followed a group of children’s math skills from preschool until they were in third grade. The children were divided into four different groups according to the skills they had in basic mathematics. Interestingly, the researchers found a group of children who had specific difficulties despite average good verbal and non-verbal intelligence and good literacy. Their difficulties could not be uncovered using standard math tests but were associated with weak skills in suppressing irrelevant information while calculating (Geary, Hoard, & Bailey, 2011). This corresponds to what is called “heavy performances” or “inefficient problem-solving strategies”, indicating that the child brings with him or her irrelevant or unnecessary information when solving the tasks (Ostad, 1997). Ostad found that pupils with mathematics difficulties have a limited supply of strategies for how to solve a task compared to pupils with no math difficulties, and that they are far less able to change strategies (Ostad, 1998, 2015).

Comorbidities

In a study by Landerl, Bevana, & Butterworth (2004) a selected group of 8–9-year-olds with diagnosed dyscalculia, reading difficulties or both conditions, were compared to a control group in terms of basic number processing. They found that the children with dyscalculia had difficulties with number processing, including access to verbal number information, counting dots, reading number sequences, and writing numbers. Apart from these areas, the children had no difficulties with tasks related to phonological working memory, non-numeric verbal information, non-verbal intelligence, language skills and psychomotor skills. The children who only had reading difficulties, on the other hand, showed difficulties within all these areas. The researchers concluded that dyscalculia can be explained by an innate difficulty in understanding the basic concept of numbers, and that this is a condition that is independent of other skills.

Later, Landerl & Moll (2010) assessed the comorbidity between dyslexia, specific language impairment, and math difficulties in a large population-based sample of primary school children and a subset of children with at least one learning disability. They found that difficulties with arithmetic, reading or spelling were four to five times higher in groups that already had problems in one academic area compared to the rest of the sample. They also found a stronger correlation between math and spelling difficulties than between math and reading difficulties. The researchers concluded that the comorbidity between these different learning difficulties is probably the result of a complicated interaction between both general and specific etiological factors.

Another study examined the mathematics skills of a group of children with comprehension difficulties without word decoding difficulties. They showed no difference to a control group when it came to arithmetic, but they showed significantly worse results on reasoning in mathematic tasks. The researchers concluded that the understanding of such problems has consequences not only for reading, but also for some components within mathematics (Pimperton & Nation, 2010). From these studies several comorbid combinations of math problems, dyslexia and DLD can be sorted out as proposed in Table 3.2.

Table 3.2. Combinations Math problems, dyslexia, and DLD

Notes. DLD: developmental language disorders (Bishop, Snowling, Thompson, Greenhalgh, & Catalise-consortium, 2017)

Mathematic Difficulties at Four Different Levels

According to Butterworth (2018), math difficulties are provable at each of the four levels of the “causal modelling framework” by Morton & Frith (1995) as depicted in Table 3.3.

Table 3.3. The four-level model

The symptomatic level is where facts can be observed but cannot be explained. Theories to explain poor test performance are placed at the cognitive level, which has to do with information-processing mechanisms including difficulties in visual, auditory, or temporal processing. This again should be anchored within the biological level, concerning gender, heredity, genetics, and current knowledge of brain function. At the same time these three levels have systematically to consider their interaction with the environmental level. Environmental factors are for instance socio-economic, cultural, or pedagogical. According to Frith (1999) ideas about cognitive causes should act as a vital bridge in causal models and should lead to ideas for remediation.

Symptomatic Level

Mathematics difficulties are shown by very poor performance of simple tasks such as comparing numbers and counting a small number of dots. As we have seen, traditional definitions of mathematics difficulties such as the DSM-V definition, require that the child must have scored far below the age norm on standardized tests. A pitfall of this requirement is that such tests assess a range of skills before the scores are merged into a combined score. This poses a great risk of making Type I errors, i.e., the worded hypothesis is true, but the empirical-statistical trial says that it is false. While the tests are diverse, the understanding of what is meant by “mathematics difficulties” can vary between tests. This is probably one of the reasons why scientists have not been consistent about what are the key difficulties in dyscalculia, nor have they been sure how dyscalculia can be defined in a research project (Landerl, Bevan, &Butterworth, 2004).

Thus, the problem for those who are going to investigate mathematics difficulties is that most tests have a ready-made setup in terms of the four basic categories of calculations. Therefore, how the child goes about finding the answer will not be demonstrated or observed. This was one of the conclusions drawn by some researchers after a group of children’s mathematic skills had been followed from preschool until third grade. They found that the difficulties could not be disclosed by using standard mathematics tests, and that the difficulties persisted despite the average good verbal and non-verbal intelligence, literacy and norm scores on many standard tests on working memory (Geary et al., 2011). In a clinical situation, it may therefore be an option to use a qualitative approach.

As shown in the introduction of this chapter, there are several linguistic challenges in mathematics. Consequently, a language difficulty can cause difficulties with this subject. Here we will take a closer look at what characterizes two imagined ten-year old children, Tom and Joan, when asked to solve two math tasks:

Task 1 orally presented: “Dad has got twenty-one apples. He says you are going to give thirteen of them to your friend Peter. How many apples do you have left?”

Task 2 written on a paper: 21 – 13 =?

On Task 1 Tom answered incorrectly, while on Task 2 he answered correctly. With Joan, the opposite happened. She responded correctly on Task 1, but not on Task 2. Tom has a history of DLD, and he has dyslexia. He has difficulties catching all the linguistic elements in a longer, oral message as in task 1. Conversely, Joan had no history of DLD. She comprehends and learns spoken language in line with most children of her own age, but she is diagnosed with dyslexia. The calculations she sets up on paper are not easy to interpret, and the teacher says that she does not master the basic arithmetic skills, addition, subtraction, multiplication, and division.

It is natural to ask how Tom’s and Joan’s math difficulties can be understood. To help arrange the various factors, table 3.2 shows that mathematic difficulties can be observed in four different variants when it comes to comorbid difficulties. The task of a clinician is to differentiate to find the right method of intervention. Questions that need to be asked are the following: Do the math difficulties of a person with dyslexia differ between one who has DLD, and one who does not have DLD? Are there common cognitive traits when it comes to dyslexia and language difficulties, which do not show up in pure mathematics difficulties? Are there common cognitive traits when it comes to dyslexia and mathematics difficulties, which do not show up in pure language difficulties? These questions should be analyzed at the cognitive level within the multiple deficit framework for neurodevelopmental disorders, (Pennington, 2006; Peterson et al., 2016).

Cognitive Level

Table 3.1 shows examples of terminology used at the different levels of mathematics. We know that a child with DLD usually will have difficulties learning new words and concepts, and that they often have difficulty with working memory, such as entering an eight-digit phone number that is read aloud to them. Difficulties of this nature can be related to the phonological loop and storage capacity of the working memory. Many math assignments are given in a verbal-language form, such as the following example that should be read by the child:

“Bonny had 5 candies. Then she lost one. How many did she have left?”

It takes about 5 seconds to read the assignment clearly. The child is to comprehend and store the question, at the same time perform a calculation of subtracting 1 from 5, and then come up with the answer. This requires both well-functioning working memory and numerical knowledge. For a child with language comprehension problems, this is not easy. Nevertheless, the task is short, the child completed it successfully, ready to move on to the next task, which the child should read:

A book that usually costs 12 euro was put down in price to half of this price. When no one bought the book anyway, the store manager decided that the book could be sold for half of this price. How much did the book cost after the second price reduction?

At the symptom level, this task requires literacy. If the child has dyslexia and does not have automatic reading skills, attention will be directed towards the decoding of the text, which may compromise the understanding of content. At the cognitive level, the child should be able to recode the message into mathematical symbol, as follows: [12 – (12 – 6)]/2, apply the subtraction to refute the first prize, remember it, and then divide by two to find the end price and solution of the task. A wrong answer renders three possible interpretation: the child has 1) difficulty reading the text; 2) problems with the calculation, or 3) difficulties with both.

Table 3.4. Six tentative cognitive profiles

Central Executive

Notes. From the Working Memory Model (Baddeley, 2003; Baddeley & Hitch, 1974)

Neither Tom nor Joan managed this task. How can one go about to find out what made the answer wrong? Studies of comorbidities of math problems, dyslexia and DLD point to difficulties within executive function and working memory, including visuo-spatial and number memory tests (Helland & Asbjørnsen, 2000, 2003, 2004; Toll, Van der Ven, Kroesbergen, & Van Luit, 2011). Using the model of Baddeley & Hitch on working memory (1974), six different profiles emerge, as shown in table 3.4. The first profile has no problems as to mathematics, dyslexia or DLD showing a balance between the phonological loop and the visuo-spatial sketchpad. The second profile depicts math problems without comorbidities showing scores above norm on tests related to the phonological loop, but with low scores within the visuo-spatial sketchpad. The third profile depicts the language related impairments dyslexia and DLD with scores above norm on tests related to the visuo-spatial sketchblock, and low scores within the phonological loop, defined by for example rapid naming (RAN) and digit span tasks. A fourth profile with both math problems and dyslexia shows low scores on tests within the visual sketchpad, but variable scores in the lower end on tests related to the phonological loop, especially on serial recall on a digit span tasks (cp Joan). The fifth profile defined by math problems and DLD shows norm scores on tests related to the visuo-spatial sketchpad, but consistently low scores on all tests within the phonological loop (cp Tom). The sixth profile shows low scores on both the sketchpad and the phonological loop most probably leading to massive problems within the three conditions math problems, dyslexia and DLD.

Biological Level

According to several scientists, developmental dyscalculia is a condition that has about the same incidence as dyslexia (Butterworth, 2005; S Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). A study comparing incidence of AD/HD and dyscalculia found a high degree of heredity in both conditions, but that there was no etiological overlapping between the conditions (Monuteaux, Faraone, Herzig, Navsaria, & Biederman, 2005). On the other hand, the significant overlap between math difficulties, dyslexia and DLD is well documented, and heredity plays a significant role in the development of these difficulties when they appear together (Snowling, Moll, & Hulme, 2021; Tercan & Bçakci, 2019).

The previously cited study of Landerl and Moll (2010) examined the relationship between gender and heredity by recording comorbidity between math difficulties, reading difficulties and spelling difficulties. Heredity was much higher in all these groups compared to a control group, and the correlation between spelling and math difficulties was particularly high. Deficits in arithmetic were more typical for girls than for boys. However, another study assessed the common understanding that girls generally do worse in mathematics than boys. School children were followed from first through third grades with emphasis on development in mathematics and visuo-spatial skills. Gender differences were not found in any of these areas (Lachance & Mazzocco, 2006).

As to brain functions Dehaene and Cohen (1995) developed a «triple code» model for how digits are represented in the brain: 1) there is a visual based code for Arab tall (visual digit shapes) in fusiform gyrus; 2) there is a linguistic/language based system to save «number facts», such as multiplication tables and automatized arithmetic knowledge as «2 + 2 = 4», which seem to be stored in the language area of the left hemisphere on line with automatized verbal sequences as week days, the months of the ear etc.; 3) the general number sense in the parietal lobe, specialized for spatial cognition. From this model Dehaene has argued that there is a specific math area in the brain in the horizontal intra-parietal sulcus (Dehaene, 1997). There are large individual variations in how this area interacts with the linguistic network in the brain, and research also shows a change with experience and age (Chinello, Cattani, Bonfiglioli, Dehaene, & Piazza, 2013).

Later studies have both confirmed and expanded the “triple code” model. Brain imaging from many studies have showed that different mathematical tasks activate different areas of the brain (Arsalidou & Taylor, 2011). Nevertheless, there is no clear and unambiguous conclusion that mathematics has a specific neural basis in the brain. Studies of brain strokes that have hit the language areas, show that also the sense of numbers has weakened (Shaywitz et al., 2007).

Thus, it is evident that the biological basis for what can be called specific mathematics difficulties is unclear. Based on a data from brain imaging, cognitive and learning data from 479 children it was concluded that different brain structures are associated with learning disorders as math problems, dyslexia and DLD, but that the results are inconsistent (Siugzdaite, Bathelt, Holmes, & Astle, 2020).

Environmental Level

Weak mathematics skills can be explained by environmental factors, such as curricula, methodology, teacher competence and caretakers’ insight into the school’s way of teaching mathematics.

Mathematics is called the school’s “silent subject”, i.e., that students very often work individually while the teacher walks around the class to explain and help. This means that there is little time for each student, and many students are left to work in their own way and at their own pace. According to Vygotskij, action is controlled by the “inner” language, and this must be corrected by the surroundings, – in this case the language action of the teacher (Veraksa & Sheridan, 2018; Vygotsky, 1964). Since the student is much left to his or her own inner language to solve an arithmetic task, it becomes extra important to gain insight into what strategies he or she uses to solve a basic calculation task.

Since there are many reasons why an individual may be weak in mathematics, it is difficult in a school situation to identify this specific disadvantage. Therefore, mathematics difficulties have often been explained with the symptoms typical of dyslexia. In any case, language difficulties and dyslexia affect all academic writing, including mathematics, and it is therefore important that such difficulties are detected as early as possible.

Dyslexia does not necessarily lead to mathematics difficulties. Indeed, some people with dyslexia are doing consistently well in mathematics (Simmons & Singleton, 2009). However, if there are reports of math difficulties from the pupil, parents, caretakers, or school, this should be investigated further as part of the assessment of dyslexia. Then the pupil’s numerical understanding, numerical knowledge, and basic skills in the four calculation categories should be assessed in relation to age, grade level and teaching methods. Since research shows that low basic skills are likely to be behind the difficulties, emphasis should be placed on assessing these skills.

This paper has focused on different patterns of math problems. One pattern showed mathematics difficulties related to domain-specific skills like language comprehensions and processing. Another pattern showed problems with domain-general information-processing mechanisms including difficulties in visual, auditory, or temporal processing. These different profiles point to different intervention methods. The first group will need linguistic adaptation in mathematics teaching, both in the form of reading help and adapted language on the part of the teacher. The second group will need to learn how to set up the arithmetic tasks, by repeating and internalizing the teacher’s verbal instructions until the child uses it as self-instruction. In Vygotsky’s terminology the child will then be able to use his or her inner language to control and perform each separate calculation operation (Vygotsky, 1964).

The experienced teacher will know that math arises anxiety in some pupils (Ramirez, Gunderson, Levine, & Beilock, 2013). Furthermore, pre-produced assignments usually have a setup that prevents the teacher from observing the strategies or work style of the student. For the interventions to be effective, they must be based on evidence-based assessment, but qualitative testing methods must be used to find out what is the individual student’s difficulties. Therefore, it is recommended that the student work on a blank sheet of paper without lines, and preferably that the tester starts by asking the student to show which calculation assignment he or she masters and feels confident with. The tester then uses this as a basic level where the child feels safe to gradually go on with further mapping.

Here is an example of how to proceed: The student, a fifth grader, choses to start like this: 2 + 3 = 5. The tester approves and says: “Nice! Now it is my turn to write a new calculation task for you”, and writes the following on the paper: 4 + 3 =? The pupil solves the task, and gradually the tester increases the task demands until the student has reached a level that he or she does not master, or the “zone of proximal development” (Vygotsky, 1964). The tester can then note not only the proximal level of the student, but also the strategies applied as to finger counting, ability to align an arithmetic task and fine motor skills. It should be kept in mind that addition, subtraction, multiplication, and division are all demanding on visuo-spatial skills, since the operations change between being horizontal and vertical, some from left to right and some from right to left.

Tom and Joan are examples of the two groups of dyslexia who also have mathematics difficulties, but of different nature. Primarily, Tom has difficulties with the linguistic part, that is, text assignments. But he also needs more time compared to his classmates to learn the multiplication table. This can be related to the relatively weak scores within the phonological loop. He has no trouble with the lineup and calculation of addition pieces and subtraction pieces with multi-digit numbers. He needs help with the linguistic part of mathematics: work with the vocabulary, help to read pieces of text, learn the multiplication table, and get novel material explained in a language adapted to his linguistic level.

Joan belongs in the other group. She does not have difficulties with text assignments or the linguistic part in mathematics, but she needs help to organize her learning. This can be done by taking advantage of her strong skills, which are oral language, to strengthen her weak skills. In other words, every calculation operation should be described in words so many times it becomes internalized and a tool for self-instructions.

Summary

Weak mathematics skills can be explained by environmental factors, such as curricula, methodology, teacher competence and the degree of parental insight into the school’s way of teaching mathematics. There is a high degree of comorbidity between mathematics difficulties, dyslexia and DLD, but there is also evidence that these three conditions can occur independently of each other. All three conditions are explained by an innate disposition and have a significant hereditary component. Brain studies refer to specific areas in the parietal part of the brain that is the center of the sense of numbers. Since there are many reasons why an individual may be weak in mathematics, it is difficult in a school situation to identify this specific disadvantage. Therefore, mathematics difficulties have often been explained with the symptoms typical of dyslexia. In any case, language difficulties and dyslexia affect all academic learning, including mathematics, and it is therefore important that such difficulties are detected as early as possible. Intervention must be based on evidence-based assessments, but qualitative testing methods should also be used to find out what is the individual student’s difficulties. In conclusion, when it comes to investigating and understanding mathematics difficulties, the extended model from Morton &Frith, (1995) as shown in Table 3.5 is very useful summary of how to assess mathematics difficulties. According to this model, the cognitive level is seen as the “bridge” between the levels and is therefore essential in how mathematics difficulties can be understood.

Table 3.5. The four-level model, extended*