Psycholinguistics/Language and Arithmetic

From Wikiversity
Jump to navigation Jump to search
Skull and sagittal brain.svg

Language and Arithmetic[edit | edit source]

Introduction[edit | edit source]

At first it would appear that language and math are two completely separate entities; however, how different are they really? When analyzed, it is evident that both language and math are firmly rooted in a form of syntax. These set rules govern the way in which people can communicate both linguistically and arithmetically. Moreover, it would appear that language and arithmetic not only share governing rules, but also a set of cognitive resources. For instance, Spelke and Tsivkin examined the relationship between language and math in a study on bilingualism (2000) [1]. They studied Russian-English bilinguals in a series of arithmetic equations with a set of geographical facts, or historical facts that either did or did not contain numerical information. It was found that when participants were trained to perform a series of math questions, they performed those skills better in their native language. Moreover, information regarding non-numerical facts and numbers was similarly remembered in both languages. This suggests that language has an impact on the learning of exact numbers in an array of fields. While language and mathematics initially appear to be two starkly different subjects, they do hold some similarities. As these seemingly divergent subjects are examined closely, they become more complimentary to one another. On that basis, it would seem only natural to examine both language and arithmetic in context of one another. After all, aren't both language and arithmetic concerned with relationships, whether communicative or theoretical? Over the course of this chapter several elements of language and arithmetic will be examined, including brain systems and cognitive resources, the number sense, the development of numerosity, the importance of math and language in a social and cultural context, as well as contemporary research and its benefit to society.

Brain Systems and Cognitive Resources[edit | edit source]

Cognitive Neuroscience has given researchers the opportunity to study the specific brain systems that influence how humans operate. In this section, the neural pathways that coincide with language and arithmetic functions will be analyzed through an array of neuroimaging techniques. In analyzing these cognitive resources, the way in which math and language are both similar and different at a cognitive level will become evident.

Language[edit | edit source]

Language meaning in the brain is generally evenly distributed; however, there are some areas that are more known for their influence on language representation. As the major brain areas are covered in the textbook chapter “Language and the Brain,” this section will cover the main points regarding language localization and will act as a brief review before delving into the shared cognitive resources between language and arithmetic.

To begin with, most people have heard at some point about the notion that language is housed in the left hemisphere of the brain. While this idea for the most part holds true, language lateralization can vary within a population. In general, language does tend to be left lateralized, but some aspects, such as metaphor, prosody and language organization, are found in the right hemisphere (Lancker, 2000) [2]. Language functions thus seem to be bilateral, rather than strictly lateralized.

One of the primary brain areas involved in language is Broca’s area. Paul Broca was the first to qualify this area through an autopsy of one of his patients, Leborgne, a man who lost his ability to generate sentences, and could only say “Tan” and “Sacre nom de dieu!” (“Goddamnit!”) (1861) [3]. The postmortem analysis concluded that Leborgne had a deficit in the left frontal lobe, specifically in the area of the inferior frontal gyrus. Because of his research, Broca’s area is now the area of the brain known to have an effect on speech production. Another brain system related to language is Wernicke’s area, a region found in the angular gyrus (Wernicke, 1874) [4]. Lesions in this area of the left cerebral cortex cause deficits in speech comprehension.

Brain limbicsystem.svg

While Broca and Wernicke’s areas are known as the 'language centres' in the brain, there are other neural systems that influence the way in which humans communicate linguistically. The Limbic system is one such area (Lamendella, 1979) [5]. This neocortical system tends to aid children with exchanges of language, while it plays a part in message interpretation in adults. The basal ganglia, an area that facilitates prosody production and comprehension, as well as automatic speech, also has a great effect on language (Lancker, 2000) [2]. The influence of the basal ganglia coincides with its other bodily functions, as it primarily deals with the motor system. One other area known for its role in language is the amygdala (Stuss & Alexander, 2000) [6]. Not only is the amygdala known for emotional awareness, but also for the integration of other information, such as language.

Arithmetic[edit | edit source]

Many studies and reviews have examined the relationship between arithmetic and neuropsychology through neuroimaging. Of these works, it has been found that one of the major brain areas involved with arithmetic is the horizontal segment of the of the bilateral intraparietal sulcus (HIPS) (Dehaene, Molko, Cohen, & Wilson, 2004) [7]. When Stanislas and colleagues reevaluated their theory of the number sense, they found through meta-analysis that the HIPS is the area of the brain that shows the most activity when processing numbers, as well as making judgments and comparisons on numbers. Moreover, Simon, Mangin, Cohen, Le Bihan, and Dehaene’s functional magnetic resonance imaging (fMRI) study found that once accounting for noise, the HIPS was the sole area of the brain active during calculation; thus, it appears to be the primary neural circuit active in regards to arithmetic (2002) [8].

Another area of the brain that is highly involved in arithmetic is the angular gyrus (Dehaene, Spelke, Pinel, Stanescu, Tsivkin, & Science, 1999) [9]. In neuroimaging studies, it has been found that the angular gyrus shows the most activity when retrieving facts from working memory to use in arithmetic operations, such as multiplication. Further evidence for the impact of the angular gyrus is evident in Lee’s case study of a patient with a lesion in the angular gyrus that affected her ability to do simple multiplication tasks (2000) [10]. However, in another case study van Harskamp, Rudge, and Cipolotti analyzed the mathematical skills of another woman with similar damage to her angular gyrus (2002) [11]. In this case, the patient’s ability to subtract was impaired, while her ability to multiply and comprehend was intact. This recent study caused problems with the initial belief of the importance of the angular gyrus in arithmetic. As there is some debate on the function of the angular gyrus in arithmetic functions, it does not appear to have as concrete of a place in the mathematical brain system as the HIPS.

Moreover, meta-analysis of neuroimaging illustrates that the precentral sulcus and inferior frontal gyrus also appear to be triggered when managing a series of numbers in working memory (Stanislas et al., 2004) [7]. This seems to be specifically the case when completing arithmetic tasks, such as mental calculation.

In studying the meta-analysis of neuroimaging in terms of arithmetic, it becomes apparent that several areas of the brain are involved when processing mathematics. However, one must also consider the possibility that not all areas of the brain involved with math are used at once. Based on this idea, Rosenberg-Lee, Lovett and Anderson studied the specific brain loci activated when performing an array of math-related tasks (2009) [12]. They suggested that as individuals use different sets of strategies to solve math problems, there must also be different neural responses to correlate with the arithmetic functions. Their research illustrates that long multiplication calculations cause early activation in the posterior superior parietal lobule, and the posterior parietal cortex, but not in the HIPS, or lateral inferior prefrontal cortex. This suggests that areas, such as the HIPS that tend to be associated with calculation, may relate more to quantity processing and estimation than to certain arithmetic tasks, such as long division (Stanislas et al., 2004) [7]. This study illustrates that distinct areas of the brain devoted to mathematical calculation may be affected differently, depending on the manner in which the arithmetic is processed.

Communal Cognitive Resources[edit | edit source]

Although language and arithmetic appear to be different domains, they do share a set of cognitive recourses. Dehaen, Spelke, Pinel, Stanescu, Tsivkin, and Science explored the idea of communal neural pathways in a study that examined bilinguals and the role of language and visual representations on mathematics (1999) [9]. They found that visuo-spatial pathways were active during approximations (i.e. guesstimating), while language areas were affected with exact calculations (i.e. calculating 258 - 62). The fMRI scans showed activation in the left inferior frontal lobe, left cingulated gyrus, left precuneus, right parieto-occipital sulcus, left and right angular gyri, and middle temporal gyrus for exact calculations. Generally, all activation when doing exact calculations was left lateralized and coincided with language neural-pathways. While the study illustrates the need for multiple neural pathways when it comes to arithmetic, it also demonstrates a set of shared cognitive resources between language and arithmetic.

As Dehaen’s study illustrated, there is an apparent connection between neural responses responsible for language and those for arithmetic (Dehaen, et al., 1999) [9]. Of particular interest in his study is the interconnectedness of language and math through the angular gyrus. Indeed, the angular gyrus is not only activated with arithmetic tasks, but also with language tasks such as phoneme detection (Stanislas, Molko, Cohen, & Wilson, 2004) [7]. As the angular gyrus is often associated with arithmetic operations comparable to multiplication (if based on on the case study by Lee (2000)) [10], then it would appear that there is some language-based retrieval dependency (similar to phoneme detection) for arithmetic operations. Moreover, Dehaen’s study suggests that operations, such as multiplication, are merely viewed as another type of fact to be retrieved at a neural level (Dehaen et al., 1999) [9]. It becomes apparent that even in retrieval from cognitive resources, arithmetic and language are not as different as one might imagine.

Brain Areas Math Skills Language Skills
the HIPS Number Processing Speech Production
The Angular Gyrus Operations (ex: multiplication) Speech Comprehension

Communal Cognitive Resources for Math and Language

Despite the fact that previous studies have shown a relationship between the HIPS, quantity processing and arithmetic estimation, there is also association between the HIPS and linguistic syntax (Carreiras, Carr, Barber & Hernandez, 2010) [13]. Carreiras and colleagues illustrated such a correlation in their study on agreement violations. Participants were exposed to one of four different sets of word pairs, determinant-noun versus non-adjective pairs, and three disagreement conditions (word pair disagreements based on gender, number and baseline. i.e. "math affect language" would be a number disagrement). Words were shown one after the other and consisted of one of the word pairs and a disagreement. In the cases where there were issues with number agreement, the HIPS was activated; this suggests that the HIPS is not only affected by quantity in arithmetic, but in language as well. Moreover, when there was a violation in either gender of number, the left premotor and left interior frontal areas were activated. Once again, there is a distinct connection between math and language neural circuits.

Brain Activation Word Pair Disagreements
Left Premotor Area Syntactic Number and Gender Violations
HIPS Syntactic Number Violations
Left Interior Frontal Area Syntactic Number and Gender Violations

The activation of brain areas when syntactic rules are violated

The Number Sense[edit | edit source]

The Triple-Code Model[edit | edit source]

This diagram depicts the bilateral properties associated with math

The number sense is one of the three components in Dehaen and Cowen’s triple-code model (1997) [14]. In order to understand the significance of the number sense and its importance in terms of arithmetic, one must first have some insight into Dehaen’s model. To begin with, the model states that digit identification, as well as quantitative representation, occurs in both the right and left hemispheres. As the triple-code model suggests bilateral properties for mathematic ability, it makes sense that the three codes that are used to judge various math functions come from an array of brain areas. The verbal area associated with math, found in the Broca’s area, facilitates the understanding and learning of arithmetic that is tied to language. For instance, the learning of one’s 'multiplication table" is attributed to the function of Broca’s area. This mathematic function correlates with the other roles of Broca’s area, that is, speech production and verbal working memory. The visual area, found in the inferior temporal cortex, aids in the understanding of written mathematics; thus it is the visual-symbolic code that makes it possible to read and understand an arithmetic problem written on a page. Finally, the quantity area, found in the parietal lobe (The HIPS), deals with the idea of basic math functions (i.e. counting), numerosity and making judgments regarding a number set. It is this quantity code that we will focus on in our explanation of the number sense.

The Number Sense and Numerosity[edit | edit source]

As previously mentioned, the number sense is interrelated with the quantity code in the triple-code model (Dehaen & Cowen, 1997) [14]. Essentially, numerosity is the ability to approximate and manipulate smaller math problems almost automatically, without the use of other cognitive resources, such as language devices (Binet, Simon & Kite, 1916) [15]. Thus, the skill of numerosity is the ability to identify and manipulate small number sets without having to count, as counting requires the use of language centers in the brain. As a result, the number sense and numerosity are mathematically defined as one’s ability to use intuition to approximate. This innate ability not only allows us to approximate a number of items almost automatically, as Jevons did in his study of iconic memory, but to estimate whether one group of items is larger than another group (Jevons, 1871) [16]. Furthermore, an individual’s capacity to quickly perform these tasks while making judgments is known as one’s ability to subitize (Kaufman, Lord, Reese, & Volkmann, 1949) [17]. This, of course, is related to numerosity, as it is the skill associated with the number sense.

Another property associated with the number sense is the linear representation of quantities in the human brain, also known as the number line (Binet, Simon & Kite, 1916) [15]. One interesting fact about the line is that it appears to be logarithmic, meaning that as the size of the numbers being examined increase, so must the difference between the two in order to successfully distinguish and approximate the numbers. In other words, the larger the difference between the two numbers, the faster the response time for quantity. This is an important aspect when factoring in the use of the number sense and its relation with the quantity code in the triple code model (Dehaen & Cowen, 1997) [14].

Development[edit | edit source]

Numerosity[edit | edit source]

While the development of math does not seem to be as innate as the development of language, some mathematical properties, such as numerosity, tend to appear earlier than language acquisition. In this case, children are more likely to subtize at a younger age than develop a sensitivity for phonemes (Binet, Simon & Kite, 1916) [15]. As numerosity appears to be an early function in the brain, its development at a young age seems to pave the way for the development of more complex functions later on in life.

Xu and Spelke conducted a series of experiments to examine the innateness of numerosity in five to six month-old infants (2000) [18]. Infants were placed in a situation in which they examined an array of dots that varied in set size from eight to sixteen objects. Infants were first habituated to the pattern of dots and then shown the second series of dots (a preferential looking design). In the cases where the two trials differed, infants spent a greater amount of time staring at the changed dot display. Conversely, participants did not spend a great deal of time looking at the second trial if it matched the first, as they were already habituated to the sight. Once all other factors were controlled for, it was found that children did, in fact, discriminate between the dot sets due to an innate approximation rather than independent representations of the objects. These findings were significant as the participants had not yet reached an age of phoneme acquisition and were not able to count. Effectively, Xu and Spelke illustrated that numerosity is one of the first skills developed by infants.

How many stars are there in this diagram? If you can identify the number without counting you are subitizing!

Subsequently, Xu and colleagues expanded on their research to see if discrimination depended on the ratio between items, and if it was dependent on large or small numerosity (Xu, Spelke &Goddard, 2005) [19]. As with their previous study, infant participants were tested with a preferential looking task. Four experiments were conducted to reexamine infants’ abilities to discriminate numbers. The results, once controlled for extenuating factors, illustrated that infants have the ability to subitize with larger sets, but have issues with smaller numerosities. These findings are congruent with the theory that there are separate brain systems that facilitate numerosity depending on size, as postulated by Binet, Simon and Kite (1916) [15].

The Relationship Between Language and Arithmetic[edit | edit source]

Social and Cultural Contexts[edit | edit source]

As with most aspects of life, societal context has an effect on the development of language and arithmetic. In a study on parental and teacher influence on mathematical development, it was found that social context had a large effect on student performance (Levpušček, & Zupančič, 2009) [20]. To study the impact that parents and teachers had on mathematic success, student participants were asked to fill out a questionnaire that documented their feelings on pressure, support, and help given by both adult groups. Indeed, parental pressures were correlated with poorer math results, while supportive teachers tended to have a positive effect on a child's arithmetic understanding. Moreover, it has been shown that other social factors, such as a lower socioeconomic status, residing in a disadvantaged neighbourhood, and race can have negative effects on academic achievement (Kohen, Leventhal, Dahinten, & McIntosh, 2008) [21].

All of these social contexts have been shown to have a great effect on language development as well. One of the best, and most extreme, examples of the negative effects of parental upbringing and abuse is the case of Genie (Rymer, 1993) [22]. Genie, a small girl who had been abused for the first thirteen years of her life, was developmentally delayed both physically and linguistically due to her upbringing. Without any sort of social training or context, Genie did not develop a great understanding for language. Despite intensive therapy and years of effort, Genie still did not develop language in the way that a socially nourished child would. Such a case study illustrates the notion that language isn’t completely innate, but dependent on an array of social factors and context to help it flourish.

Cultural influences also affect the mathematical and linguistic worlds. For instance, just as there are universal governing rules of language, there are many aspects of arithmetic that are similar across cultures. In their review of Peter Gordon’s work on language nativism and the Pirahã tribe, Laurence and Margolis discuss issues that arise with cross-cultural studies (2008) [23]. Through Lawrence and Margolis’ review it is evident that while they believe there are issues in doing cross-cultural studies for language nativism, these studies could help demonstrate that numerosity is universal. While not all cultures and languages have the same number set or system, all humans seem to have a general understanding of size comparison and approximation. Moreover, the factors of the number line, including its logarithmic properties, seem to hold true cross-culturally. Studies such as these illustrate the fact that language and math are not only similar at neural, syntactic and social levels, but at cultural levels as well.

Math and Language Deficits[edit | edit source]

As math and language share similar neural circuits and syntactic properties, it would seem that there is a great chance that both subjects are severely impacted by specific learning disabilities. For instance, there are disorders, such as acalculia and dyscalculia that solely affect math skills and not language skills (Rosca, 2009) [24]. While these disorders seem fairly rudimentary, there are different grades of severities and arithmetic difficulties within each deficit. One such example is a case that Rosca examined in which the patient had acalculia with preserved functions in mathematic facts, but deficits in calculations. When examined, it was found that the patient had a lesion situated in the left parietal lobe, an area known to be involved in mathematic operations and language comprehension. Thus, when further analyzed, it is evident that while some disorders may typically highlight deficits in one field (i.e. math skills), other areas may be involved (i.e language comprehension).

Another disorder includes alexia, a problem in which individuals cannot read or multiply, but can add and subtract (Sakurai., Asami, & Mannen, 2010) [25]. Based on the type of disorder, one could pinpoint some of the communal cognitive areas of language and math that must be affected. For example, in the case of alexia, one could postulate that the left parietal lobe was affected, as individuals affected have difficulties with complex math calculations. Further empirical evidence on the relationship between language and math could aid in developing techniques to create programs to help individuals suffering from math and language difficulties.

Deficit Explanation
Acalculia Deficit in math skills, such as problems with calculations
Dyscalculia Deficit in math skills, such as problems with calculations
Alexia Cannot read or multiply

Arithmetic and Phonology[edit | edit source]

One contemporary study that examines the influence of the angular gyrus was conducted by De medt, Taylor Archibald and Ansari (2010) [26]. This research team examined the relationship between phonology and arithmetic in grade-school children. The study found that poor phonological awareness and processing was related to issues with small-sized math problems. Moreover, the findings suggest that both difficulties in phonological awareness and math skills had to do with the retrieval of information.

Words and Numbers[edit | edit source]

In a study on the relationship between mathematic skills and poor comprehenders, Pimperton and Nation compared the mathematic abilities of typically developing readers and poor comprehenders (2010) [27]. Participants were tested in two areas of arithmetic, one that dealt primarily with simple calculation and another that dealt with complex mathematical reasoning. Poor comprehenders did show lower scores in mathematical reasoning than did their controls, as their verbal abilities and comprehension were lower than that of their peers.

What This Could Mean in Terms of Research[edit | edit source]

These contemporary studies are integral in understanding the ways in which a reading disability could affect math skills. Furthermore, the findings could extend to creating a clinical program that could increase phonological retrieval, comprehension, and thus math problem accuracy. As the relationship between phonology, poor comprehenders and math can be found in the angular gyrus, a plan that would focus on the stimulation of this brain’s functions could aid children falling behind in the school system.

Test Your Knowledge[edit | edit source]

Fill out this crossword to reinforce your understanding of studies and concepts before further applying and evaluating your knowledge in the next section.

Applying and Evaluating Your Knowledge: Thinking About Math and Language[edit | edit source]

These exercises are designed to solidify your understanding of the major concepts in the chapter and apply your knowledge to different situations.

1. If you were conducting an fMRI experiment on individuals with difficulties in phonological awareness, what areas of the brain would you expect to be activated? Moreover, what mathematical deficits would you expect to be present? Integrate this information with your understanding of the cognitive resources used in language and math to design a treatment to aid these individuals.

2. Identify and explain the major brain areas responsible for language and arithmetic. Imagine that the following image is a brain scan from someone performing both math and language tasks at the same time. Highlight the areas on the picture that affect both language and math.

Human brain.png

3. Based on your understanding of the triple code model, design an experiment to illustrate that arithmetic development occurs before language acquisition.

4. Jodi is a 7 year old girl from a low socio-economic status family. Her parents each work two jobs, and as a result Jodi is often left unsupervised. Moreover, Jodi is a student in an inner city school that offers very few extra academic resources. While Jodi has always had some academic issues with reading, recently she's been having issued with simple mathematic calculations, such as addition and subtraction. Identify some possible solutions that will aid Jodi in achieving academic and arithmetic success (use examples and refer to social and cultural aspects of language and math).

5. Are there genetic components involved in mathematic and language abilities? If so, how could you test the biological nature of these two processes?

6. Design an experiment (unlike that of Spelke and Tsivkin) that illustrates the way in which language has an effect on math. Keep in mind some of the properties that are shared by both language and mathematics, such as syntax.

7. Look at the following image and try to rapidly and accurately judge the amount without counting. How many dots can you see? How are you able to do this? Describe this process (Think of one of the principles discussed in the development section). Is there a similar theory relating to language? Or, can you apply this theory to language processes?

Subitising 1-10.svg

8.Watch a portion of the following video. After watching part of the tutorial, would you qualify the type of math equation as simple or complex? Based on your answer, what area of the brain do you think would be active? Now, imagine that you suffer from dyslexia. Would you have normal mathematic abilities, still be able to only answer these kinds of math equation, or would you have issues with another form of mathematics?

9. Now let's do a bit of web surfing- find a website dedicated to math or language (i.e., a support group for individuals with dyscalculia). Learning from a first hand experience is often a great way to have a better understanding of a subject. What are some of the biggest concerns surrounding the disorder? How do individuals overcome these difficulties? Are there any treatment opportunities available that you were unaware of? Try to integrate this information with the knowledge you gained from this chapter.

10. Essay Question: How is language lateralized? Is it the same for everyone? What aspects can change the lateralization of language, and, if it isn't lateralized to one side which areas of the brain correspond to what language functions?

11. Group Discussion: Theories can be changed and disproved over time, and in order to fully understand a theory one must know both sides of the story. Choose one of the theories that are discussed in the chapter and look up a meta analysis on the subject. Has the theory changed, or added any new components to its reasoning? What are the current thoughts on the theory? Does everyone agree with the viewpoint? What are your thoughts on the subject.

12. Imaging that you are a researcher studying the language centers in the brain. While you are very interested in the subject, you do not have the funding for neuroimaging machines. Design a study that will allow you to test the language centers in the brain.

13. Debate: Break up into two groups and choose one of the controversial topics associated with language and arithmetic (such as new treatments, hemisphere activation, etc). One group should argue for the "for" side while the other group argues for the "against" side. Make sure to include empirical evidence in your argument.

14. Now that you've read the chapter and had a chance to solidify your understand on the topic, it's time to integrate this information together. Write a one minute thought paper on everything that you can remember about the chapter (including details). Have a friend time you and make sure that you write for the entire 60 seconds. Whatever you do, DO NOT remove your pen from the page and DO NOT stop writing. Even if you lose your train of thought just write down "I don't know what to write right now." You'll be impressed with how much information you've absorbed. Once you're done your thought paper, read it over and compare it with the information in the chapter. Were you able to remember the key points of the chapter? Were you stronger in some subjects than in others? This is a great way to find out just how much you've learned.

References[edit | edit source]

  1. Spelke, E. S., & Tsivkin, S. (2000). Language and number: A bilingual training study. Cognition, 78(1), 45-88. doi:10.1016/S0010-0277(00)00108-6.
  2. 2.0 2.1 Lancker, D. (2000). Brain structures in verbal communication: A focus on prosody. Topics in Stroke Rehabilitation, 7(1), 1-23. doi:10.1310/B235-90FE-0KAH-9HGN
  3. Broca, P., 1861. Remarques sur le siège de la faculté du langage articulé; suivies d'une observation d'aphemie. Bull. Soc. Anat. Paris 6, 330–357.
  4. Wernicke, 1874 C. Wernicke, In: Der aphasische Symptomenkomplex, Breslau, Cohn, Weigert (1874).
  5. Lamendella, J. T. (1979). Neurolinguistics. Annual Review of Anthropology, 8373-391. doi:10.1146/
  6. Stuss, D. T., & Alexander, M. P. (2000). Affectively burnt in: A proposed role of the right frontal lobe. In E. Tulving, E. Tulving (Eds.) , Memory, consciousness, and the brain: The Tallinn Conference (pp. 215-227). New York, NY US: Psychology Press.
  7. 7.0 7.1 7.2 7.3 Dehaene, S., Molko, N., Cohen, L., & Wilson, A. J. (2004). Arithmetic and the brain. Current Opinion in Neurobiology, 14(2), 218-224. doi:10.1016/j.conb.2004.03.008
  8. Simon O, Mangin JF, Cohen L, Le Bihan D, Dehaene S: Topographical layout of hand, eye, calculation, and language- related areas in the human parietal lobe. Neuron 2002, 33:475-487.
  9. 9.0 9.1 9.2 9.3 Dehaene, S. S., Spelke, E. E., Pinel, P. P., Stanescu, R. R., & Tsivkin, S. S. (1999). Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence. Science, 284(5416), 970.
  10. 10.0 10.1 Lee KM. Cortical areas differentially involved in multiplication and subtraction: a functional magnetic imaging study with a case of selective acalculia. Annals of Neurology 2000;48:657–61.
  11. van Harskamp, N. J., Rudge, P., & Cipolotti, L. (2002). Are multiplication facts implemented by the left supramarginal and angular gyri?. Neuropsychologia, 40(11), 1786-1793. doi:10.1016/S0028-3932(02)00036-2
  12. Rosenberg-Lee, M., Lovett, M. C., & Anderson, J. R. (2009). Neural correlates of arithmetic calculation strategies. Cognitive, Affective & Behavioral Neuroscience, 9(3), 270-285. doi:10.3758/CABN.9.3.270
  13. Carreiras, M., Carr, L., Barber, H. A., & Hernandez, A. (2010). Where syntax meets math: Right intraparietal sulcus activation in response to grammatical number agreement violations. NeuroImage, 49(2), 1741-1749. doi:10.1016/j.neuroimage.2009.09.058
  14. 14.0 14.1 14.2 Dehaene S, Cohen L. Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex 1997;33:219–50.
  15. 15.0 15.1 15.2 15.3 Binet, T. h. Simon, E. Kite (Eds.), 1916, The intelligence of the feeble-minded. Baltimore, MD US: Williams & Wilkins Co. doi:10.1037/11070-009
  16. Jevons, W.S (1871). The power of numerical discrimination. Nature, 3, 281-282.
  17. Kaufman, E.L., Lord, M.W., Reese, T.W., & Volkmann, J. (1949). "The discrimination of visual number". American Journal of Psychology (The American Journal of Psychology) 62 (4): 498–525. doi:10.2307/1418556.
  18. Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1-B11. doi:10.1016/S0010-0277(99)00066-9
  19. Xu, F., Spelke, E. S., & Goddard, S. (2005). Number sense in human infants. Developmental Science, 8(1), 88-101. doi:10.1111/j.1467-7687.2005.00395.x
  20. Levpušček, M., & Zupančič, M. (2009). Math achievement in early adolescence: The role of parental involvement, teachers' behavior, and students' motivational beliefs about math. The Journal of Early Adolescence, 29(4), 541-570. doi:10.1177/0272431608324189.
  21. Kohen, D. E., Leventhal, T., Dahinten, V., & McIntosh, C. N. (2008). Neighborhood disadvantage: Pathways of effects for young children. Child Development, 79(1), 156-169. doi:10.1111/j.1467-8624.2007.01117.x
  22. Rymer, R. (1993). Genie: An abused child's flight from silence. New York, NY US: HarperCollins Publishers. Retrieved from EBSCOhost.
  23. Laurence, S., & Margolis, E. (2008). Linguistic determinism and the innate basis of number. In P. Carruthers, S. Laurence, S. Stich, P. Carruthers, S. Laurence, S. Stich (Eds.) , The innate mind Volume 3: Foundations and the future (pp. 139-169). New York, NY US: Oxford University Press. doi:10.1093/acprof:oso/9780195332834.003.0008
  24. Rosca, E. (2009). A case of acalculia due to impaired procedural knowledge. Neurological Sciences, 30(2), 163-170. doi:10.1007/s10072-009-0029-7
  25. Sakurai, Y., Asami, M., & Mannen, T. (2010). Alexia and agraphia with lesions of the angular and supramarginal gyri: Evidence for the disruption of sequential processing. Journal of the Neurological Sciences, 288(1-2), 25-33. doi:10.1016/j.jns.2009.10.015
  26. De Smedt, B., Taylor, J., Archibald, L., & Ansari, D. (2010). How is phonological processing related to individual differences in children's arithmetic skills?. Developmental Science, 13(3), 508-520. doi:10.1111/j.1467-7687.2009.00897.x
  27. Pimperton, H., & Nation, K. (2010). Understanding words, understanding numbers: An exploration of the mathematical profiles of poor comprehenders. British Journal of Educational Psychology, 80(2), 255-268. doi:10.1348/000709909X477251