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Psycholinguistics/Semantics in the Brain

Introduction to Semantics in the Brain
The study of semantics in the brain is a branch of psycholinguistics that incorporates the understanding of semantics and the neurological structures that are involved. It attempts to answer the unanswered question of “how objects and concepts are represented and processed in the human brain” (Noppeney, Price, Penny, Friston, 2006). This field of study has received an enormous amount of research because in essence, semantics is what allows us to verbalize and express ourselves about the people, places, and things in our lives. It is essential to human communication and exists within all human beings across languages. Although widely studied, understanding the neurological side of semantics is highly controversial. Researchers agree that the inferior frontal (Figure 1a), inferior parietal (Figure 1b), and temporal cortex (Figure 1c) are all involved in processing semantic memory however the exact involvement of the specific areas is not necessarily agreed upon (Visser, Jefferies & Lambon Raplh, 2010)

Distributed Networks
Although it has been widely accepted that the left hemisphere has a higher association with language than the right hemisphere, the specific areas of activation have been a continued source of debate. Recent research has identified distinctive regions of the left lateral prefrontal cortex and the anterior temporal cortex that are associated with object processing (Martin & Chao, 2001). More specifically, the area of activation depends on characteristics such as the object's form, movement, and uniqueness. In regards to object form, studies have shown that images such as faces, animals, and landmarks, produce a greater activation in the lateral fusiform gyrus and the right superior temporal sulcus, both of which are close to areas that mediate object motion. In contrast, when shown images of man-made tools, there was a greater activation in the medial fusiform gyrus, the left middle temporal gyrus, inferior temporal sulcus, and left ventral premotor cortex, coincidentally near the region associated with object manipulation. Although this specific study entailed the participants naming or viewing the object, the same areas show greater activation when the participant imagined the object or answered questions about the objects (Martin & Chao, 2001).

In agreement with object form, object movement pertains to distinctive neurological networks that correspond to the motor-related regions (Martin & Chao, 2001). For example, there is a greater level of activation in the left ventral premotor cortex when participants name tools rather than animals. Like the object form, this is also true when participants view pictures of tools and animals. Due to the limited availability of research upon patients with left premotor lesions, it is unknown if the left ventral premotor cortex is needed for naming and retrieving information that is related to object movement such as tools (Martin & Chao, 2001).

The study of object form and movement both examine generic items. Studies such as Gauthier, Anderson, Tarr, Skudlarski and Gore, 1997, examined the difference in activation when objects are unique rather than generic. For example, is there a difference in neurological activation when shown a picture of your dog Rudy compared to a random Siberian husky? The answer is yes. The anterior regions of the temporal lobes show higher activation when the object is unique and differentiable to the individual, see Figure 2. There are several theories as to why unique items show higher activation in the anterior region of the temporal lobes. One commonly accepted assumption is that objects are organized hierarchically with more generic objects being represented first in the posterior, and specific unique items being represented in the anterior (Martin & Chao, 2001).

It is important to note that although distinct areas have been reported for higher levels of activation, this does not mean that other areas are not also highly activated. There is considerable overlapping among activation within the ventral cortex (Martin & Chao, 2001). The examples provided here demonstrate the distribution of neurological networks in regards to semantic memory and provide the basis for future research.

Neural Network "Hub"
Within recent years the thought of semantic memory deriving from a central “hub” has been widely debated. The term “semantic hub” or “neural network hub” refers to the anterior temporal lobe (ATL), which “acts as a semantic hub, combining information from different sensory and motor areas to form amodal semantic representation” (Visser, Jefferies, & Lambon Ralph, 2010). Researchers such as Lambon Ralph, Pobric, & Jefferies, (2009) agree with the notion of the semantic hub, whereas others such as Martin, 2007, do not place a great deal of significance on the ATLs.

In 2010 a meta-analysis was conducted by Visser et al. to examine the differences in the methodologies between studies that support the significance of the ATLs, in comparison to those that do not. Using predetermined criteria such as imaging modality, field of view (FOV) size, stimulus type, and the magnitude of the semantic task, 164 experimental studies were examined. The results showed significant differences among all criteria.

Studies that were conducted using positron emission tomography (PET) scans had a significantly greater activation in the ATL when compared to functional magnetic resonance imaging (fMRI) scans (Visser et al., 2010). In addition, recent fMRI scans (studies conducted between 2004-2008) had a higher rate of ATL activation that fMRI scans studies which were conducted between 1999-2003. Visser et al. (2010) suggest that this may be due to technological improvements in fMRIs, methodological improvements, or the use of more auditory stimuli (which are known to show higher ATL activation).

Whole-brain coverage was operationally defined as having an FOV of 15 cm or greater (Visser et al., 2010). Although only 22 of the studies within the meta-analysis displayed whole-brain coverage, those that did were more likely to report ATL activation. This is a possible factor that may have inhibited the remaining studies with a smaller FOV to miss ATL activation. Similarly, studies that documented the ATL as a region of interest (ROI), reported higher rates of ATL activation. This means that due to a lack of ROI ATL activation may have been overlooked or unnoticed due to insensitive neuroimaging studies (Visser et al., 2010). The type of stimulation used across the studies varied from pictures, words, to sentences. In agreement with previous research (Rogalsky, 2009), greater ATL activation was recorded when the participants were presented with auditory sentences rather than pictures or words (Visser et al., 2010).

The notion of the semantic hub remains debatable however, meta-analyzes such as Visser et al., 2010 have provided an insight as to the possible methodological differences that may account for the difference within ATL activation during semantic neuroimaging studies.

The Hierarchical Network Model
Collins and Quillian developed the Hierarchical Network Model in 1969. This model of semantic memory is based on the principle known as the “cognitive economy" (Conrad, 1972) . Cognitive economy implies that knowledge about concepts is only stored once so it is not repeated throughout memory (Conrad, 1972) . For example, the concept that cats have tails is stored within the definition of a cat, therefore specific details such as a ragdoll cat has a tail and a maine coon cat also has a tail do not need to be stored (Collins & Quillian, 2004) . Collins and Quillian’s model can be visualized using a hierarchical structure. At the top of structure are general concepts with more specific details on the lower levels of the structure (Collins & Quillian, 2004) . Figure 3 illustrates the cognitive economy principle with the examples of canaries having wings and sharks being able to swim.

The Spreading Activation Model
The spreading activation model was developed by Collins and Loftus in 1975. As with the hierarchical network model, the purpose of this model was to demonstrate how words are accessed within in mental lexicon (Collins & Loftus, 1975). According to the spreading activation model, concepts are connected via nodes and the strength of the connection is represented by the distance between the nodes. For example, the concept of a “fire engine” may have a strong connection to “ambulance” and “fire”; which is represented visually with a short distance between the nodes. Whereas, “fire engine” may have a weaker connection to the words “street” and “vehicle”, which is represented with a larger distance between the nodes, see figure 4 (Collins & Loftus, 1975). Although the spreading activation model is more advanced than the earlier hierarchical network model (1972), it does not include all aspects of lexical access, such as the phonological, syntactic and morphological characteristics of word (Levelt, 1989).

Semantic Priming
Semantic priming is the unintentional increase in speed or accuracy when responding to a stimulus such as a word or a picture. It occurs when a prime word is shown before a target word (McNamara, 2005). For example, the prime word may be nurse and the target word may be doctor. Many studies such as the lexical decision task incorporate semantic priming. In a lexical decision task the participant is shown a prime word with a specific stimulus-onset-asynchrony (SOA) such as 500ms then the prime word is removed and a target word is shown. The participant then must decide if the target word is a real word or a pseudo-word such as “bempal “. Studies have shown that when the participant is shown a prime word that is semantically related to the target word the participant’s reaction time decreases. Other uses of semantic priming include cognitive tasks and in the study of perception and cognition (McNamara, 2005).

Loss of Semantic Memory
The loss of semantic memory can result from a number of different factors however, there are four generally accepted categories: Alzheimer’s disease, herpes simplex encephalitis, severe head injury with localized effects, and semantic dementia. Below we will briefly examine each category.

Alzheimer’s Disease
Alzheimer’s disease is commonly associated with the loss of memory due purely to its high rate of prevalence. Alzheimer’s disease is the most common cause of dementia affected 1 in 8 Americans (Hebert, Scherr, Bienias, Bennett, & Evans, 2003). According to the Alzheimer’s Association of America, it is estimated that an American develops Alzheimer’s disease every 70 seconds (Alzheimer’s Association, 2010). Dementia of Alzheimer’s type (DAT) usually begins with the loss of episodic memory, autobiographically events (Patterson & Hodges, 1995). As the damage spreads into the bilateral posterior association cortices, such as the lateral temporal structures, semantic memory is affected. As the disease progresses the damage within the semantic memory increases becoming more easily detectable (Patterson & Hodges, 1995).

Herpes Simplex Encephalitis
Simply defined, herpes simplex encephalitis (HSE) is inflammation of the brain caused by the herpes simplex virus type 1; the same virus that causes cold sores or blisters around the mouth (National Institute of Neurological Disorders and Stroke, 2004). Although fairly uncommon, affected approximately 2 people per million a year, HSE can be fatal if untreated. During the critical phase, a patient with HSE will experience retrograde and anterograde amnesia of episode memory (Patterson & Hodges, 1995). This loss of memory is caused from damage to the medial temporal lobe and frontal lobe. The loss of semantic memory among patients with HSE is becoming more common. Within these patients damage is usually seen in the temporal neocortical areas (Patterson & Hodges, 1995).

Severe Head Injury
Head injuries are assessed objectively with the use of the Glasgow Coma Scale (GCS). Patients who receive a GCS rating of eight or lower are classified as suffering from a severe head injury (Goldstein & Levin, 1995). Unfortunately, suffering from a severe closed head injury typically results in some form of anterograde amnesia. The duration of the memory loss can vary, however studies have shown that the duration is positively correlated with severity; meaning that those who suffer a more severe injury will experience a greater loss of memory (Vilkki, Poropudas & Servo, 1988). The long-term impacts of head injuries vary amongst individuals, however issues such as impaired delayed recall, susceptibility to interference, contextual and source errors as well as issues with semantic priming are all fairly common. Studies (Crosson, Novack, Trenerry & Craig, 1988) have shown that patients who suffer from severe head injuries are less likely to utilize semantic tactics such as clustering and recategorizing. Although head injuries usually disrupt episodic memory leaving semantic memory in tack, there have been reports of impairment within semantic memory (Patterson & Hodges, 1995). For example, the case of patient M.P., a young woman who was involved in a car accident, is an example of how severe head injuries can cause extensive damage to semantic memory (Patterson & Hodges, 1995).

Exercises
{| cellspacing="0" cellpadding="0" style="margin:0 0 1em; width:100% background:white;" Exercises
 * style="width:50%; vertical-align:top; border:1px solid #afa3bf; background:#D8BFD8;"|

Part I: Knowledge
Compete the following statement by filling in the blanks with the appropriate word. All of the correct words are within the word search below!

{ Semantics in the brain incorporates the understanding of semantics and the { Neurological (i) } structures that are involved.
 * type="{}"}

{ Formal Thought Disorder is highly prevalent among those with { Schizophrenia (i) }.
 * type="{}"}

{ The neural network hub concerns the debate surrounding the function of the { Anterior (i) }. temporal lobe.
 * type="{}"}

{ The neural areas of language activation are clearly defined. True or False? { False (i) }.
 * type="{}"}

{ The area of neural activation for language depends on characteristics such as the object's form, { Movement (i) }, and uniqueness.
 * type="{}"}

{ Object movement pertains to distinctive neurological networks that correspond to the { Motor (i) } -related regions.
 * type="{}"}

{ Positron emission tomography (PET) scans have significantly { Greater (i) } activation in the ATL when compared to functional magnetic resonance imaging (fMRI) scans.
 * type="{}"}

{ Semantic priming is usually examined using a { lexical (i) } decision task.
 * type="{}"}

{ The { Thought (i) }, Language, and Communication Scale is used to clinically diagnose patients with FTD.
 * type="{}"}

{ Researchers have concluded that FTD is the result of impairments in semantic { Priming (i) }.
 * type="{}"}

Answers to Part I
Click here to view the answers for the Part 1 Exercise.

Part II: Comprehension
Based on the information provided within this chapter, expand your understand of Semantics in the Brain by thinking about the following questions:

 * 1) What is the neural network “hub” debate? Based on the information presented in the meta-analysis by Visser, Jefferies, and Lambon Ralph (2010), what is the consensus on the neural network “hub” debate? How do methodological differences affect the support for or against the neural network “hub”?
 * 2) How do you think research will evolve within the next 20 years surrounding the study of semantics? Will we still be using PET and fMRI studies? Will new technologies become available? (Hint: think about how long PET and fMRI studies have been around).
 * 3) Semantic priming studies have been widely used to study semantics. What other types of studies could be used to study semantics?
 * 4) FTD affects about 90% of those with schizophrenia, how could we ensure that those affected by FTD are given the support they need?
 * 5) Other aspects of language such as speech production and speech acquisition have been given more attention than semantics. Why do you think this is? How could we ensure future research involves the study of semantics?

Part III: Application
The following questions are designed to test your knowledge of the material presented in this chapter, while extending to application and comprehension. The following questions should be addressed in essay format while answering all parts of the question.

 * 1) What side of the "hub debate" do you agree with? Why? Provide additional evidence to support your position.
 * 2) Identify two models of semantic memory access. What are the key components to each model? How are the models similar and how do they differ? Provide a diagram of each model that illustrates the key components of the model.
 * 3) Design a study that incorporates the use of semantic priming. Based on empirical evidence, provide a hypothesis of what you would expect to see within your results.
 * 4) Identify three factors that can contribute to the loss of semantic memory.  Briefly explain each while concentration on how they differ in regards to affecting semantic memory. Based on the factors you described, which factor do you think would have the largest impact on the patient’s day-to-day life? Why?
 * 5) As demonstrated within the chapter, semantics in the brain is not easily identified within one specific neural structure. Identify various neural areas that are may negatively affect semantics if damaged. Research various causes of damage, and specific treatments that are available to facilitate recovery. Do you feel that the available treatments sufficient? Why or why not?
 * }


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User:Eml5526.s11.team3/Homework 1

 Homework 1

= Homework 1 = Formulation of Partial Differential Equation (PDE) for a 1- Dimensional Problem.

Refer to lecture notes 6.1

--

Problem Statement
1.1 To find the 1-Dimensional Partial Differential Equation(PDE) of an elastic bar of length L and varying cross-section, subjected to a body force or distributed force f(x,t) along the axis of the bar.

1.2 Now if the bar has a rectangular cross-section where the breadth is constant, perform the balance of forces to formulate the 1-D PDE for this case.

Solution 1.1
Consider an element of length $$d\left(x\right)$$ of the elastic bar, the free body diagram of the bar is shown as below.

In the above figure $$A\left(x\right)$$, $$ \sigma\left (x\right)$$ and  $$n\left(x\right)$$ represent the area, stress and normal to the plane on the left cross-section of the bar and

$$A\left(x+dx\right)$$, $$\sigma\left(x+dx\right)$$ and   $$n\left(x+dx\right)$$ represent the area, stress and normal to the plane on the right cross-section of the bar.

For the left cross-section $$n\left(x\right)=-1$$ and for the right cross-section $$n\left(x+dx\right)=1$$

For the element of the elastic bar shown, we have $$m\left(x\right)=\rho(x) A(x)dx$$

By using Newton's 2nd law of motion, we can have

 * {| style="width:100%" border="0" align="left"

|$$\displaystyle N(x+dx)-N(x)+f(x)dx= \rho(x) A(x)dx\frac{\partial^2u }{\partial t^2} $$ $$
 * $$\displaystyle (Eq. 1)

 * }
 * }

where $$N\left(x+dx\right)$$ is the net force due to the stress $$\sigma\left(x+dx\right)$$ which is given by

 * {| style="width:100%" border="0" align="left"

|$$\displaystyle N\left(x+dx\right)=\sigma(x+dx)A(x+dx)n(x+dx)$$ $$
 * $$\displaystyle (Eq. 2)

 * }
 * }

$$N\left(x\right)$$ is the net force due to the stress $$\sigma\left(x\right)$$ which is given by $$N\left(x\right)=\sigma(x)A(x)n(x)$$

Substituting for $$N\left(x+dx\right)$$,$$N\left(x\right)$$,$$n\left(x+dx\right)$$,$$n\left(x\right)$$, we get

 * {| style="width:100%" border="0" align="left"

|$$\displaystyle \sigma (x+dx)A(x+dx)-\sigma (x)A(x)+f(x,t)dx=\rho (x)A(x)dx\frac{\partial^2u }{\partial t^2} $$ $$
 * $$\displaystyle (Eq. 3)

 * }
 * }

Dividing the above equation by $$d\left(x\right)$$, we get

 * {| style="width:100%" border="0" align="left"

|$$\displaystyle \frac{\sigma (x+dx)A(x+dx)-\sigma (x)A(x)}{dx}+f(x,t)=\rho (x)A(x)\frac{\partial^2u }{\partial t^2}$$ $$
 * $$\displaystyle (Eq. 4)

 * }
 * }

Applying the limit as $${dx}{\to0}$$ in the above equation,by the definition of partial derivative, as we know that

$$ \lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}= \frac{\mathrm{d} f(x)}{\mathrm{d} x} $$

the first term on the LHS reduces to $$ \frac{\partial\left[\sigma (x)A(x)\right]}{\partial x} $$, the second term on the LHS remains the same as there is no $$d\left(x\right)$$ term. the RHS also remains the same.

Equation (4) reduces to

 * {| style="width:100%" border="0" align="left"

\frac{\partial\left[\sigma (x)A(x)\right]}{\partial x} +f(x,t)=\rho (x)A(x)\frac{\partial^2u }{\partial t^2} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)

 * }
 * }

The stress $$ \sigma\left (x\right)$$ can be expressed in terms of elastic modulus $$ E\left (x\right)$$ and strain as $$\sigma (x)=E(x)\frac{\partial u}{\partial x}$$

Hence the PDE for 1 Dimensional elastic bar is

$$ \frac{\partial\left[E(x)A(x)\frac{\partial u}{\partial x}\right]}{\partial x}+f(x,t)= \rho (x)A(x)\frac{\partial^2u }{\partial t^2} $$ $$ $$
 *  $$ \displaystyle
 *  $$ \displaystyle (Eq. 6)

Solution 1.2
The Partial Differential Equation (PDE) of an elastic bar subjected to an axial load derived in the section (1.1) is

Now, a particular case is taken into consideration, where the bar has a rectangular cross-section. The breadth of the bar is a constant and is equal to 'b'. But the height of the bar varies along the length of the bar and hence it is function of x, which is 'h(x).' (For figure, refer page 6-1 of [[media:fe1.s11.mtg6.djvu|meeting 6]]).

Let us consider the inertial force at the center of the thin strip of the elastic bar. Therefore the mass at that location is given by,

Performing the balance of forces similar to part 1.1 and rearranging the terms we get

Applying the limit as $${\frac{dx}{2}\to0}$$ in the above equation,by the definition of partial derivative

$$\displaystyle \lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}= \frac{\mathrm{d} f(x)}{\mathrm{d} x} $$

we get,

By Hooke's Law, the stress $$ \sigma\left (x\right)$$ can be expressed in terms of elastic modulus $$ E\left (x\right)$$ and strain $$\frac{\partial u}{\partial x}$$ as

$$\displaystyle \sigma (x)=E(x)\frac{\partial u}{\partial x} $$

Hence the PDE for 1 Dimensional elastic bar is

$$ \frac{\partial \left[E(x)bh(x)\frac{\partial u}{\partial x}\right]}{\partial x}+f(x,t)= \rho (x)bh(x)\frac{\partial^2u }{\partial t^2}$$ $$
 *  $$ \displaystyle (Eq. 6)

$$\frac{\partial\left[E(x)h(x)\frac{\partial u}{\partial x}\right]}{\partial x}+\frac{1}{b}f(x,t)= \rho (x)h(x)\frac{\partial^2u }{\partial t^2}$$ $$
 *  $$ \displaystyle (Eq. 7)

Reference
1. Newton's 2nd law of motion 2. Partial derivative 3. Hooke's Law

Contributing Authors
EML5526.S11.Team3.risher 21:35, 24 January 2011 (UTC)

Eml5526.s11.team3.akj 16:50, 24 January 2011 (EST)

EML5526.S11.Team3.vnarayanan 16:52, 24 January 2011 (EST)

Eml5526.s11.team3.ushnish12 16:54, 24 January 2011 (EST)

Eml5526.s11.team3.hylon 16:56, 24 January 2011 (EST)

Eml5526.s11.team3.perry 11:57, 26 January 2011 (UTC)

Eml5526.s11.team3.sahin 09:27, 26 January 2011 (EST)

title: User:Eml5526.s11.team3/Homework 1


text:
Sociolinguistics/Ebonics

Introduction & History
Ebonics refers to a unique and interesting English dialect that has captured a lot of attention in film, media, and academics. In 1975, Professor Robert Williams at Washington State University first coined the term Ebonics from ‘ebony’ (meaning black) and ‘phonics’ (the sound system of linguistics). . The dialect known as Ebonics is known by other names: Black speech, Black English, Black English Vernacular, African American English (AAE), and/or African American Vernacular English (AAVE). In academia, the preferred terms are Black English, AAE, or AAVE. Ebonics is used to name the dialect for the context of this page.

Ebonics is spoken largely by African North Americans and often misunderstood to be broken or uneducated Standard English (SE). However, these types of misinterpretation only lead to stereotyping, bias debate, and lack of understanding of the serious system of rules in Ebonics. Unlike popular media and mock Ebonics has portrayed, Ebonics is not spoken by lower class, uneducated people that are too lazy or throw words around in a random fashion to communicate. Rather, Ebonics is filled with the richness that every spoken language has such as phonology, semantics, syntax, and lexical structures. In addition, this language has been through great debate since the 1970’s until present date. It is often debated whether its existence is real or current slang, whether teaching children to be bi-dialectal would be beneficial and sustain positive self-esteem, and should schools reinforce/ punish the use of Ebonics. Although, many have argued against such language, it is very crucial and important to understand that Ebonics like all languages is not primitive or underdeveloped. It’s important to understand that this language spoken is not a disorder or deficiency, it is simply a difference. This difference has foundations of language to communicate the same content that is conveyed by SE speakers.

Components of Ebonics
Unlike common belief, Ebonics has been proven to be a systematic and rule regulated dialect. There is much regularity in Ebonics that confirms its existence as a dialect and not a ‘broken-language’ that just throws words together. It is important to recognize that Ebonics like other languages have a system of phonology, lexical structures, semantic and syntax which is not random, or irrational, but Ebonics carries methods of speech that non-Ebonics speakers would miss or misinterpret.

Phonology
Although the sounds that are in SE are quite similar to Ebonics, there are often pronunciations, endings of words, and sound combinations that solely occur in Ebonics. The most frequent occurrence of sound /consonant combinations at the end of words that are uniquely to Ebonics are st,sk,sp,pt,kt/ct,nd,and ld. These consonant combinations are usually pronounced with the single consonant in words by dropping the second consonant. By dropping the last consonant, Ebonics speakers are not being lazy or improper, they are omitting a constant of a combined pair that have the same voicing value or simply put, using Voicing Generalization. Voicing Generalization does not imply that all of these paired clusters drop their second consonant, only if these clusters are ending the word (the suffix). In Ebonics, when these clusters are in the middle of the word and preceding vowels, they are pronounced. For example, the word acceptable, the t is voiced and present in front of a vowel sound of a. Moreover, the plural form of these words follows the same endings as those seen in SE. The voice generalization of these words show that words ending in st,sk,and sp, add an s to the ending to illustrate plurals. Also, the plural es is added to words such as ‘wrises and dises’, like census' plural is censuses. For examples refer to Table 1.1.

Table 1.1

The consonant combination of th is important in Ebonics, whereby this sound may produce the sounds t,d,f,and v. In SE, the th has two sounds, where th sounds like they or thigh and the difference is underlined in the dictionary. However, where the th would occur at the beginning of a word in SE, a d would occur in Ebonics. When th is in the middle of a word, a f sound is pronounced and in other instances where th is in the middle the a v sound is pronounced. In addition, words that end with the voiced suffix th, v is applied, where voiceless th suffix is a f or a single t in Ebonics. Also, mostly in children, a remarkable skr sound is produced that replaces the str in SE. This rule applies to the beginning of words as well as in the middle of words. See table 1.2 for examples.

Table 1.2 Ebonics Pronounciation/Spelling

Source: Green(2002)

Lexical Items
Ebonics is not a language that is full with slang or mispronounced words. Although most words are similar to SE vocabulary, how a word is pronounced may have a different meaning in Ebonics. Like all languages of the world, Ebonics has a variety of slang words, general phrases, and grammatical/ verbal markers.

Slang
Slang words are subject to constant and continuing changes with time. Most slang words are produced through the teenage generation and are constantly reinventing itself. Slang words are commonly used by cohorts that range from pre-teen to adolescents. Example 1, Sayin Somthin is used usually for describing an enjoyable event, place, person, or thing. Example 2,B, is used usually in greeting and replacing a person’s name with the consonant. Examples of Current Slang in Nova Scotia, Canada, 2010-2011:

Examples:

1.	Sayin’ Somthin’. (Adjective). Beautiful, very good, great. Ex: That dress you’re wearin’ is sayin somthin’.

2.	B. (Noun). Term for a friend, acquaintance. Ex: What up, B?

General Words and Phrases
General words and phrases may seem to be similar to slang, however these words and phrases occur in all age groups and throughout time. For example, the words ashy, and some + adjective are common sayings that have been around across North America and Nova Scotia, respectively, for decades. The adjective ashy has been used exclusively to describe the nature of whitish dry skin. In Nova Scotia, Canada, Ebonics speakers commonly use the phrase some + adjective to describe a noun. The phrase works by placing some in front of the adjective of choice to note intensity. The word some in this sense are similar to adverbs, too or very. In Nova Scotia it is common to hear this use of some + adjective used across all types of English and ethnicities, not only Ebonics speakers.

Examples:

1.	“''Did you cream your feet today? Your feet look really ashy!''”

2.	“''Wow, Ma’! Your cookin’ is some good''”

Verbal Markers
In popular media and for listeners that are not familiar with Ebonics, the verbal markers are the key makers that stand out compared to SE. First, the marker be is the most noticed and common marker misunderstood and stereotyped. However, the use of be is used systematically to describe a habitual or reoccurring state/event. This is sometimes confusing to SE speakers because they do not comprehend that be is not the present tense rather a reoccurring event.) Researchers present a useful and witty dialogue demonstrating this confusion (See Appendix A). Similarly, done is a verbal marker that refers to the past tense of an event that has already passed or is in the past . In Ebonics, done means that the action or event is completely over. Also, done seems to work as a primer for past tense verbs or referring to the past tense. Closely linked to the term done is the marker been (pronounced BIN) which refers to the recent past. This marker is usually accompanied by an -ing or past tense verbs depending on the context . It is important to note that this marker is stressed, so in the example it is capitalized . It’s important to remember that these examples are not deficiencies or inadequacies; rather they are differences.

Table 1.3 Verbal Markers

Appendix A: ''Seymour, H.N., Abdulkarim,L. & Johnson,V. (1999). The Ebonics Controversy: An Educational and Clinical Dilemma. Topics in Language Disorders, Vol 19(4), pp. 66-77.''

Scene: White teacher enters the room with an African American child. It appears the teacher wanted to telephone the child’s parents.

“Bobby, what does your mother do?” the teacher asked.

“she be at home,” Bobby replied.

“You mean she is at home,” the teacher said.

“ No she ain’t,” Bobby offered, ‘Cause she took my grandmother to the hospital this morning.”

The teacher snapped.

“You know what I meant. You aren’t supposed to say, ‘She be at home.’ You say, ‘She is at home.’”

But Bobby, incredulous, could only reply,

“Why you trying to make me lie? She ain’t at home.”

Marking the Past
Ebonics has at least six distinctive past tense markers. These consist of simple past, preterite had, remote past, past perfect, remote past perfect, and resultant state which all describe the past tense. Ebonics and SE share in common the simple past and past perfect, whereby “He slipped on a rock” and “ He had slipped on a rock” resemble SE. Even though, past perfect resembles the preterite had, the preterite had in the past tense has a distinct meaning in narrative contexts; “he had slipped on a rock” means the point of climax before the present. The remote past and remote past perfect is a way of talking about the long ago past events and actions. The remote past perfect refers to an event that has happened in the past that occurred before another past event. The remote past refers to the action/event that has happened a while ago. For example, “Suzy had BEEN fell on the hill” and “Suzy BEEN fell on the hill”, illustrates the remote past perfect and remote past, respectively. In both examples, the event of Suzy falling is happening in the past but the first has happened earlier. Also, it is important to note the stress on BEEN, this is a verbal marker and that is why it is capitalized. Finally, the resultant state in Ebonics refers to an event or action that is over and usually marked by the word done. For example, “Helen done read ‘The Psychology of Language’” which means in SE that “Helen has already read ‘The Psychology of Language’”. Done is unstressed and it is important to recognize the difference of done unstressed and the verb “done” is stressed.

Negation
Negation marking in Ebonics can have two negatives in one sentence that indicate a single negative event/action. It has been found that just like math the fact that two negatives become positive, this is just not the case in Ebonics .This is formed through a negative indefinite noun (NIN) and a negated auxiliary (NA), in most cases it is don’t / can’t and nobody. The placement of the NIN and NA can appear in different order, whereby the meaning of the sentences will refer to the same thing but said in different ways. Ebonics speakers understand word order and these sentences are not put together randomly, but systematic rules like SE are imbedded. See Table 1.4 for examples.

Table 1.4 Negations

Source: Green (2002)

Stereotypes
The use of Ebonics or Black English in American history and culture has been used to stereotype and falsify the lifestyle of many Afro-North Americans. Firstly, mock Ebonics are a commonly used stereotype to discriminate and continue to portray Ebonics as an inferior, less intelligent means of communication. Mock Ebonics by definition are fictional and mimic ways of communicating in Ebonics that have no phonological, semantic, and pragmatic sense or regularity. Researchers list the strategies that mock Ebonics consist of; which are the frequent use of be, semantic and pragmatic irregularities, using vulgar phrases, and non-systematic representations of the phonetic segments. The frequent use of be is inappropriate in most mock Ebonics because of the placement and emphasis it holds in a sentence or discourse. The placement of be is overly used to mock the dialect and as noted in the phonology section, Ebonics has a system of rules that regulates itself.

Also, the use of vulgar language throughout the parodies and comic episodes are very negative towards Ebonics speakers. The vulgar terms are associated with SE words to portray that Ebonics speakers use derogatory language in every sentence spoken. For example, “B, eh yo, done kill a ho’ tomorrow, ya dig?”, demonstrates that these mockeries of Ebonics, not only make no semantic or syntax sense, imply that African North Americans use vulgar language to express simple sentences in everyday language. This example also illustrates the pragmatic problems where a non-Ebonics speaker randomly places greetings and terms in a sentence. Stereotyping occurs from the ignorance and the ethnocentric view that there is a better dialect upon the nation, but this is completely linguistic racism that needs to be critically understood.

Power & Elitism
Power in the media worked against many languages and dialects to prove or provide false evidence that all non-standardized English are not intelligent/proper. Kochman (1981) argued that dialects and ethnicities are differences in style and not intellect. Put simply, what the speaker is saying is not less intelligent if s/he is talking in SE or Ebonics, this is the style of the speaker. However, when students are rating speakers that speak in SE versus Ebonics, who are telling the same exact message, it has been found that students will rate the SE speaker to be “well-spoken” or “proper”. The notion of talking “proper” is socially constructed and a direct result of elitism in the working. Elitism refers to the idea of a language to be proper and more intellectual than another. It is emphasized at a young age that how you say things is just as important as what is being said. However, the how is simply style of the speaker not the inadequacies of speaker. The use of elitism continues to reinforce prejudice and discrimination across non-standardized English, such as Ebonics. Those in power continue to try and maintain these anti-Ebonics ideologies through the discrimination in discourse by describing Ebonics and other sub-English dialects a disorder. Although, when critical thinking and further knowledge is applied, society will be able to see passed the surface of what they are hearing as different and hear the content of the message. Above all, researchers argue that racism and elitism constantly work side by side, thus psycholinguistics’ students should not ignore this dilemma

Code switching/bi-dialectal
An interesting part of being an Ebonics speaker is the idea of code switching. Code switching refers to speakers switching from one dialect to another. Whites are not enforced or pressured to switch from the natural speech they speak at home to the SE in schools. However, when African North Americans Ebonics speakers are in certain situations or conversations, they are encouraged and often forced to switch from their natural dialect into the SE of the situation. Ebonics speakers usually talk about personal events and emotions, while talking to friends in the classroom but, switch to SE while talking to teachers. Code switching is heavily situated and facilitated by the pragmatics of speech and communication. Put simply, it is socially constructed when a child/person should switch to which dialect s/he should use in which context.

In educational debates, it was suggested that being bi-dialectal should be encouraged by giving options for students to communicate in SE, but to use their dialect that they bring from school. This would encourage teachers to understand the components and pragmatics of Ebonics and then, help students to use correct code switching. This teaching strategy is to help lessen the pressure and racist ideology that Ebonics is incorrect or broken SE. This technique is put in place to celebrate diversity and encourage people to respect the use of this dialect for conveying messages in their native tongue.

Teaching Ebonics
Teaching Ebonics in educational institutes has been in constant debate existing since the 1970’s. In the late 1990’s, the Oakland California School Board passed a movement that would integrate Ebonics into the school system to assist with SE. Ebonics integration was met with tons of critical debate. Mostly, people who seen Ebonics as being “laziness”, “broken”, or “slang” that should not be taught in the classroom. However, by teaching Ebonics or letting students use their dialect from home, educators are trying to help students use SE and are trying to eliminate the linguist racism that one dialect or language is less sufficient or accurate than another. There is no primitive language, all languages are equal whereby they convey a message and communicate.

This movement was placed into school systems to celebrate the diversity of dialects that are commonly used outside the classroom. The teaching methods in schools would have the same focus on teaching SE, but SE would be treated as a second dialect used. Most African North American, Spanish decent, Latinos, Whites and other ethnicities often learn SE as a second dialect while in school. This method to maintain the native dialect in the classroom is seen as how bilingualism is taught. Although, the efforts of linguists and the Oakland California School Board have made very positive steps towards attitudes and power structures of Ebonics, it has yet to be formally immerged in schools as learning and teaching tools.

Continuing and the Future of Ebonics in Education
It has been found that Ebonics is spoken over 80 % of the time by black students while in school and outside of school, so one may ask: why is it still inferior to society’s expectation? The answers are still unclear, even with all the research advocating language and cultural diversity. However, an important question to ask is to WHO’s society? Blacks, like other races, have a society to which they have their own cultural norms, expectations, laws and traditions that one must follow.

Moreover, language is a part of one’s identity, and especially Ebonics to black identity. The racism and discrimination against Ebonics and the style of which blacks speak is equivalent to attacking one’s identity. Many black students try and resist the dominant norms (SE), however some efforts are at times harming to them. It was found that many black underachievers resist the dominant cultural norms by avoidance and in turn by not going to class or doing homework; these students fail or drop out. Thus, suggesting that language and communication at school versus at home needs to be culturally sorted, so that students are receiving the proper and relevant education, so that they can academically prosper and succeed. Also, the proper evaluation for students needs to be in place since this would eliminate the bias and wrongly accused deficiency that a standardized test would detail an Ebonics speaker to have. Although, there has not been any major changes to the curriculum of many schools where Ebonics is spoken by the majority, the efforts to place Ebonics in classrooms is still a fiery debate and hope for many black communities.

In conclusion, as optimistic as can be, through linguistic consciousness and education on diversity, Ebonics will be a subject that will continue to be judged and talked about. The combination of linguistics, cultural studies, history, politics, sociology, and psychology will help the research on Ebonics in the classroom and code-switching. This research and studies are continually growing to promote black identity and well-being. The efforts and research of the past, the present and the future on linguistic diversity will help the acceptance and linguistic richness of Ebonics for ‘black folk’ and all sub-dialects.

Learning Exercises
You read all the information...

and understood the linguistics of Ebonics...

Now it's time for you to show what you learned!! Here's a few exercises to strengthen your Ebonics' muscles for battle against linguistic discrimination!

Exercise # 1
Think About It...

Standard English was made so it could be used for a common communication among all English speakers. However, we all do not speak in this standardized form, rather use it as a guideline for our own speech. Put simply, if language was put on a spectrum of Standard English to non- Standard English, where do you stand? To celebrate diversity of language, in this exercise you are free to give and EXPLAIN your own examples of non- Standard English phrases or words that you use or hear in your daily life. Don't forget to state your city or town and country. ALSO, pull evidence from the Ebonics text that support/link to how your example proves the argument that there is no Stone Age language.

Answers can be submitted to the discuss pageTalk:Psycholinguistics/Ebonics with your signature

Exercise # 2
Refresh Your Ebonics!

Answer the following questions on Ebonics as best as you can. Try to fill it out without looking first, and then go back for help later!

Answers can be submitted to the discuss pageTalk:Psycholinguistics/Ebonics with your signature

 * 1) In the academic world, Ebonics is also called?
 * 2) Robert Williams thought of the term Ebonics, he composed this word with the colour related to blackness and what component of linguistics?
 * 3) Often, people will use this to oppress and discriminate against other dialects and languages.
 * 4) Regardless of what popular media has portrayed negatively about Ebonics, linguistics have proven that it is__________?
 * 5) By dropping the last consonant, Ebonics speakers are not being lazy or improper, they are using ____________?
 * 6) “well, well, well…Your prom dress is sayin somethin’ bestie!”
 * 7) I’m not to be confused with general words or phrases; I’m only here to be distinctive.
 * 8) Often, I mark the past.
 * 9) The overly stereotyped and misinterpreted marker of Ebonics. But when do you use this marker, what does it signify? Example: “I be writing papers all year for school.”
 * 10) The idea of a language to be proper and more intellectual than another.
 * 11) “Nobody don't be workin' on Monday's” and “don't nobody be workin’ on Monday's” are examples that are unique to Ebonics negation.
 * 12) I am often used in media incorrectly and slandered in a lazy and stereotypical manner to degrade a dialect.
 * 13) At times this could be a benefit in a capitalist world of communication but the opposition argues the negative effects on cultural identity and confusion.
 * 14) How humans communicate is the ____ of the speaker and this does not change the content.

Exercise # 3
Show Your Understanding!

Rephrase these nursery rhymes in Ebonics. After you finish, try and use another tense! Feel free to use current slang and general phrases of Ebonics, but you must justify why you used a certain term and how it fits (use the information given above).

IMPORTANT: Unlike the pop-media, YOU ARE A PSYCHOLINGUIST and you are aware of linguistic racism, discrimination, and elitism in Mock Ebonics. Therefore, creatively use the components of Ebonics and cultural aspects of true Ebonics to transform these childhood favourites! Have fun

Answers can be submitted to the discuss pageTalk:Psycholinguistics/Ebonics with your signature

1

Hickory dickory dock

The mouse ran up the clock

The clock struck one

The mouse ran down

Hickory dickory dock

2

An apple a day keeps the doctor away

Apple in the morning - Doctor's warning

Roast apple at night - starves the doctor outright

Eat an apple going to bed - knock the doctor on the head

Three each day, seven days a week - ruddy apple, ruddy cheek

3

Little Miss Muffet sat on a tuffet

Eating her curds and whey,

Along came a spider,

Who sat down beside her

And frightened Miss Muffet away

4

Pat a cake, Pat a cake, baker's man

Bake me a cake as fast as you can; Pat it and prick it and mark is with a 'B',

And put it in the oven for baby and me.

5

Wee Willie Winkie runs through the town,

Upstairs and downstairs in his nightgown,

Tapping at the window and crying through the lock,

Are all the children in their beds, it's past eight o'clock?

6

Twinkle twinkle little star, how I wonder what you are?

Up above the world so high, like a diamond in the sky

When the blazing sun is gone, when he nothing shines upon,

Then you show your little light, twinkle, twinkle all the night.

Then the traveller in the dark, thanks you for your tiny spark,

He could not see which way to go, if you did not twinkle so.

In the dark blue sky you keep, and often through my curtains peep,

For you never shut your eye, 'till the sun is in the sky.

As your bright and tiny spark lights the traveller in the dark,

Though I know not what you are - twinkle, twinkle little star.

Exercise # 4
Questions, Questions, Questions...

Answer the following questions in a paragraph or more on Ebonics by integrating numerous aspects of this chapter. Remember, being a critical thinker is a positive skill as a psycholinguist!

Answers can be submitted to the discuss pageTalk:Psycholinguistics/Ebonics with your signature

 * 1) When talking about the past, Ebonics speakers use a variety of ways to describe the past. Most SE speakers would not understand the subtle differences in the past. How could you describe the importance and helpful strengths to the numerous ways to describe the past in Ebonics? Show examples to support your answer.
 * 2) Education and Ebonics has a continuing debate, whereby the discussion to implement Ebonics in the classroom for students and teachers to learn from the native language would be the strategy to increase academic achievement for students. What would be the benefits of this strategy in the classroom? Offer some productive suggestions on how this process could be properly done in the school system. Is there a critical age where Ebonics should be placed in schools?
 * 3) With regard to teaching Ebonics in the classroom, many opposition states that the school is not responsible for teaching children culture, rather the family is. Explain the social and psychological effects of separating the home and school in teaching culture; is culture not supposed to be encouraged in school or is it only certain cultures?
 * 4) Code-switching has its benefits and disadvantages for Ebonics speakers. However, most of the time it goes unnoticed. How does power and elitism in languages devalue or suppress code-switching? Or is there any effect on code-switching? Explain in detail.
 * 5) We all stereotype and mock things, people, and places; however, by learning we escape ignorance. Talk about your past bias, thoughts, and judgements on Ebonics before reading the chapter and explain in which ways the chapter helped or did not help enlighten your knowledge on the subject matter. What do you think about the study of Ebonics now?
 * 6) The history of SE purpose was to overcome barriers of communication among many different speakers, however in time it became a language that became apart of elitism and power structures. Discuss how the change of SE’s purpose has an everlasting effect on other languages, especially Ebonics.

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Psycholinguistics/Connectionist Models

Connectionist models of the mind (a subclass of which is neural networks) can be used to model a number of different behaviors, including language acquisition. They consist of a number of different nodes that interact via weighted connections that can be adjusted through by the system through different ways, the most common being backpropagation of error.

Connectionism as a Cognitive Theory
Connectionism is a cognitive model that grew out of a need to have a model that allowed for and built upon the interaction between the biologically coded aspects of the brain and the learned aspects that humans receive from their environment. Connectionist models focus on the concept of neural networks, where nodes or units function as neurons with many connections to other nodes. A connectionist model is based around the idea that very simple rules applied to a large set of nodes can account for rules and categories that exist in other models, such as associationist models, spreading activation models, or hierarchical models. Instead of starting out with initial categories in the brain, connectionist models allow for the emergence of categories, based on experience. The units of these models are all very simple, consisting of a weight of an activation threshold, and a weight for its interactions with other units, as well a function for determining changes of weight for correction. One of the main benefits of the connectionist model, is that it can be simulated on computers, and has been found to closely mirror effects found in humans, such as the U-shaped performance curve for learning, and similar deficiencies when nodes are removed to those found when a brain is lesioned.

The simple functioning units of the connectionist model are often based on the neuron, despite many connectionists not always operating on a low enough level to actually have a plausible one-to-one relationship between unit and neuron (most high level cognitive models do not work on this level. Many models for language do not function on this level, instead treating stimuli as already processed, such as a whole letter when modeling reading instead of having to also train the network to recognize individual letters as well.

An important aspect to the connectionist model is the ways that it can be used to explain the emergence of particular properties with only simple learning rules instead of requiring bioprogrammed (highly specific and genetically determined) areas of the brain such as Chomsky's Language acquisition device. There has been a push to consider the interaction between the individual and the environment as an important aspect in innateness, and that simple rules likely evolved that, because of the environment, allow very complex networks to form. Connectionist models that have been trained to learn tasks like reading have mirrored the natural progressions of individuals, and also mirror human behaviour in the ways that they process new words. . This also means that when training these networks, they have to be given similar input as what humans get if they are going to actually be used as comparison models.

Layers in a model for word recognition
Connectionist models have their different units organized into layers and the connections between these layers are how the output is calculated. There is a layer of input units, which are activated depending on what characteristics are present in the input stimulus, which is the word that is to be recognized. The connections that the activated units have with the next layer determine the activation of that layer, and if that is the output layer, will determine what the system determines the word to be. Hidden layers allow for better discrimination between similar words.

Input Units
Input units are where the network first accesses the stimuli for the system. This is the first analysis of the information, and can vary widely on how processed the information is to begin with. Input can be in the form of letters that are present, sounds that are present, even whole words. It really depends on what kind of processing the system is expected to do. A model that is attempting to analyze sentences is likely going to have inputs that are words, or at least morphemes, rather than letters, or letter strings, such as were used in Seidenberg and McClelland (1989), discussed later.

Output Units
The output units are the most important for evaluation, because this is how the model can be judged for its efficacy. It is also where correct/incorrect judgments are made, and this feedback is essential for the learning of the program. The output can either be a discrete unit, so that result is only one active, or most active unit, or it can be a pattern of active units. To judge differences between patterns, a non-linear hyperplane is used, and distances are calculated from that to determine how far off the output was from what was desired.

Hidden Units
Hidden units are used to connect the input and output units. The most important reason to have these is to allow for non-linear distinctions between outputs. The added layer between the input and output allows for a different dimension to contrast outputs, and so each additional hidden unit, and levels of hidden units (the number of hidden units that an input goes through before the signal gets to the output) will increase the possible dimensions for differentiating the input. The weights of these hidden units can be adjusted through backpropagation, which uses an algorithm to recalculate weights based on the distance between the output, and the expected output. A good learning resource is available here. Often, the weights of these units are randomly generated to be within +/-.5, and as the system is given thousands of learning trials, it adjusts the weights so as to reach the correct answers. This has lead to the suggestion that connectionism equates the mind with a tabula rasa.

Semantic/Meaning and Context Units
Some connectionist models integrate another module of computation, such as a semantics module, or a meaning module. These modules are used to help the system differentiate between heteronyms, which are words that have the same spelling, but different sounds like lead, which is a metal, and a verb. One method used by Joanisse and Seidenberg had a semantic module that contained 600 verbs, and a past tense node to assist the system in learning the correct past tense of verbs. Despite having all of these preprogrammed verbs, the model was still capable of learning novel words, including non-words, and was capable of giving a past tense similar to a human response, (ex. "wug" and "wugged").

Interactions
Units interact along connections that have particular weights. This means that the effect of any particular unit (a) is defined by its relation with another unit (b), and cannot be reduced to anything within (a), and is not directly affected when any other connections are changed. 'Recurrent networks' have connections that can affect the original unit. This can be direct, in that the unit, when activated, actually modifies its own activation, or it can be that in activating, it activates other units that in turn modify the first unit again. This makes it difficult to determine what units are responsible for any particular output, because of the recursive nature of the processing. Although on the first step, a certain number of units were activated, determining which ones were activated because of the inputs, and which ones were activated because of the recurrent propagation is difficult. This makes coming up with an equation that will properly adjust the weights difficult as well. A 'Feed Forward' network does exactly what it says, it feeds activation forwards. This means that during a particular calculation, a unit cannot affect itself, because each unit always affects another downstream unit, and there is no looping backwards. The inputs feed the next level of units, and proceed to the outputs. This network is very simple to look at and by examining the weights, makes it easy to see which units are driving particular calculations. However, because there is no looping back, it does not have a way of using the connections themselves to adjust the weights, which is how the system can learn from its errors. This can be solved through the use of an algorithm to calculate the backpropagation of error.

Backpropagation of error is a method through which a network can be trained. The desired output is compared with the actual output, and if they are the same, then the connections that were used become strengthened. If it turns out that they were different, and the network was wrong, then the connections are weakened in proportion to how active they were. If they had a large weight, then the weight will be decreased a lot more than if they only had a small weight. In this way the network can learn without requiring connections that loop back within the network, and maintain its feed forward nature.

Another method of modifying weights is Hebbian Learning, named after Donald Hebb who first described it. The basic rule is that units that fire together are strengthened, while units that fire singly would have their weights decreased. Units that usually fire at the same time are likely firing because of the same stimulus, and so giving them more weight allows the system to maintain its structure. This makes the system less susceptible to changes if there are a number of anomalous stimuli that occur in succession.

McClelland and Seidenberg Model
McClelland and Seidenberg made a connectionist model for word recognition. The model was trained on four letter monosyllabic words. Designed in 1989, it was capable of not only reading words from orthographic to phonological representation, but its learning process also mirrored a number of human characteristics, such as: having the same difficulties with difficult to process words, transferring from basic to skilled reading, pronouncing novel items (ones not presented during training), and similar differences in performance on naming and lexical decision tasks. Originally, their model for reading included a number of interconnected levels: the orthographic input, the phonological output, a 'meaning' level, three hidden levels between each of these, as well as a 'content' level that was connected to the 'meaning' level. All of these levels are theoretically capable of being altered by, and altering any on that they are directly connected to. In practice, only the input and output layers, as well as one hidden layer was used. This was in large part due to the length of time that it takes to simulate parallel processes on a serial processing computer.

The input level for this model consisted of a series of triple characters, and so if that series was present in the orthography, that unit became activated. There were 400 input units, and they each contained a list of ten possible letters for the first, second and last positions of the word being analyzed. This means that there are 1000 possible three-letter strings that could activate a particular unit. Although this might seem like it would be hard to differentiate words this way, each three letter string only activates about 20 units, and had effectively no chance of two different strings activating the exact same units, allowing the model to effectively distinguish between these strings.

The output units were triples of types of phonemes, so that the string "vowel fricative word boundary" would be activated by both 'laugh' and 'scoff'. There were 460 of these units, and each phoneme string activated 16 of them on average. The importance of reporting this distribution is to show how the distributed activation is at work in this model. There is not a particular unit that corresponds to a particular letter or sound, but a particular activation pattern.

This simulation was capable of learning, over 150,000 training trials, to correctly distinguish 2,820 out of the 2,897 words that it was trained on. This model also correctly read novel words, and traits that were similar to human readers when irregular pronunciations were presented, such as PINT, and pronounced it as rhyming with LINT. Non-words were also pronounced in the same fashion that Humans generally pronounce them, so that the word MAVE rhymed with GAVE, which is more common than HAVE.

Pathological Evidence
Another piece of strong evidence for the use of connectionist models is their ability to replicate deficits found in humans with particular brain injuries. Joanisse and Seidenberg found that they could recreate the double dissociation of the correct conjugation of irregular past tense verbs and the conjugation of non-words by 'lesioning' the model, which consisted of severing a number of connections after training. The results were consistent with individuals who had anterior lesion aphasia, which showed phonological deficits, and posterior lesions aphasics, who had semantic deficits. Patterson et al. had also shown that there are similarities in effect between lesioned brains and lesioning their models accounted for similar deficits as those of individuals who had surface dyslexia.

Nativism vs. Tabula Rasa
Connectionist models are often viewed as inherently accepting the theory that our minds are tabula rasa, blank slates. This view makes sense due to the fact that initial weights are generally randomized. If they were not, it would both take a long time to program, and then also not show the emergent learning properties that make connectionist models so interesting. However, this is not necessarily the case. Some connectionists suggest that it is possible that initial weights may favour particular pathways such that, barring incredible intervention, a particular network will form. This suggests that while the particular networks are not innate, because of what normal interactions between the organism and the environment, the genetically determined neurons will almost always for that network. An example of this is how Roe et al. redirected a pathway from the eyes to the auditory cortex, and a retinotopic map formed. In this experiment, the retinal inputs in a number of ferrets were redirected from the occipital lobe to the temporal lobe. When this happened, the new input changed the structure of the auditory cortex so that it resembled a visual cortex. This example shows that despite the apparent natural organization of the brain, it can adapt to rerouting, suggesting that it is the processing and connections of neurons that gives shape to the networks. The input that the neurons get, and the feedback that they receive, is instrumental in the formation of the brain.

Localization vs. Distributed Activation
The debate about the function of single units in connectionist models has its roots in the search for and engram, a place where a specific memory is stored. Karl Lashely, in an attempt to locate this by lesioning rat’s brains in the hopes of their forgetting a task they had learned, concluded that the engram did not exist, and that memories consist of neurons throughout the brain. The colloquial name "Grandmother Cell" (GMaC) is seen to be similar to this, where a particular concept is encoded, or located in a single neuron. The name comes from the idea that there could be a cell that fired at the memory of one's grandmother, and that this firing is responsible for one's remembering. The opposing view is that variations in activation over distributed neurons cause patterns, and the neurons in these patterns have particular weights associated with their firings, and this weighted pattern of activation is what would cause a memory. The localist view must be distinguished here, because it is not merely the case that if the GMaC had not fired in this pattern, then perhaps a memory of fruitcake or one's grandfather would have been remembered. Instead, it is that the GMaC is a dedicated cell for that particular idea, and will not fire as part of a pattern for another idea, or require other neurons to fire to differentiate the idea of grandmother from a different idea. This argument stems in large part from single cell recordings where particular stimuli elicited highly selective responses for particular cells. This theory is also suggested because of the possibility of the problem that is sometimes encountered in connectionist models that is called "superposition". This is when two different stimuli result in the same output pattern, although this can be achieved through different activation of preceding units. A few prominent connectionists reject this notion that there are dedicated cells for particular ideas, words or concepts. The possibility that one neuron could be dedicated to an idea runs the risk of what constitutes an idea. If there were a Grandmother cell, what about a 'Grandmother's ear' cell? And if there were one, why isn't the idea of 'grandmother' made up of all these parts of her. These arguments, while certainly not definitive, show the value of distributed, weighted activation. The fact that the activation is weighted is especially important, because one heavily weighted cell could affect the whole pattern, while a highly active one could have little effect of it had no weight.

Conclusion
Connectionist models, utilizing very simple rules are capable of simulating very complex behaviours and learning progressions that have been observed in humans. The fact that these models have been so successful without requiring a lot of specific rules applied to them suggests that the method of connectionist models could be good approximations of what occurs in the human brain. This does not mean that the connections are simulating exactly what neurons do, particularly in language processing models, but it could be similar to the activity of groups of neurons. The really important part of connectionist models is not what is learned from how well a model can mimic the function of a brain, but how simple the rules are that are required to produce this simulation, and that the emergence of complex connections can occur from randomly generated connections.

Learning Exercises
These two activities are designed to facilitate the understanding of two important concepts for connectionist models. The first one will involve the importance of the hidden units for differentiating between inputs. The second involves emergent properties in the relationship between a structure and its environment.

Hidden Units
Step one is to take a sheet of graphing paper, and three different coloured markers. It is preferable if you can at least partially see through the paper. For the first example, draw two groups of dots in different colours, and try to separate them with a straight line.

Next, draw a series of dots in the centre in colour A, and a group of colour B dots in the upper right and lower left hand corners. If you only have one colour, different shapes can be used. Now, like before, attempt to draw a single straight line to separate the two. Now try folding the paper, bottom left up to top right. This is an example of how adding a hidden layer can add another dimension to the calculation of weights, and allow a system to differentiate between two stimuli that might have the same variable. Now, unfold the paper and draw two groups of dots on the remaining two corners.
 * If you cannot do this, why not?
 * How does this relate to the problem of having only a set of connected inputs and outputs?
 * Try drawing a line to separate the two colours again, is it possible?
 * Is it possible to draw just two straight lines to separate the three colours?
 * Does one set of hidden units help? Might adding a a second layer of units help, allowing you to add an even further dimension?

Constraints and Innateness
This exercise will get you to think about the ways that the environment can put constraints on inputs that do not necessarily have to be programmed genetically. This can probably be done just with thinking about how things might change, but will probably be more amusing if you actually try it. Take 30 roughly spherical objects. Marbles work well, as do jellybeans. Now, try and form them into as high a pile as you can on a regular tabletop. Try and list five properties for both the marbles and the environment. What about trying to make the stack on a sandy or dirty surface, like a beach or a garden bed, or even a snow bank. What about if the stack was made in a different fluid besides air, might the pile's maximum height change when under water? You don't have to try this one, but what if you made the pile in molasses, do you think it would be a lot higher? In this exercise, the properties of the marbles, or whatever you were stacking, can be considered to be what is programmed genetically. This does not change over the course of the experiments. On the other hand, there are different things that can be done to increase the size of the pile. Changes in the environment evidently contribute to the way that the marbles can interact with each other. This is quite simplistic, but it hopefully gets the point across that the height of a pile of marbles is not something that is determined merely by the properties of the marbles, but also by the properties of the environment that they are piled in. Write down a list for each of these.
 * How high can you get the pile?
 * What attributes of the marbles (or whatever) that you are using might be limiting or helping the stack grow larger?
 * What aspects of the environment are contributing to the size of the stack?
 * How has the change in the environment changed the height of the stack?
 * How does this relate to connectionist models?
 * What parts of the model are analogous to the marble or neuron, and what parts to the environment?
 * Where does the weighting, input, output, and changing of weights lie in this dichotomy?
 * If you had a language learning program that was able to interpret words and give the correct orthography, what constraints that are a part of language would help the program to learn?
 * What aspects might cause difficulty?

title: Psycholinguistics/Connectionist Models


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PreK Skills

 * Created: 2006 09 29 | Percent completed: [[Image:00%.png]]

What is PreKindergarten Skills? [[Image:00%.png]]
PreKindergarten Skills is the teacher's guide for teaching science at the kindergarten level.

Each unit is numbered and can be taught in one day. Each unit also contains scripted instructions for the teacher. Many units contain workbook pages and activities.
 * 1) This is item 1
 * 2) This is item 2

 * This is bullet 1
 * This is bullet 2

Unit 1 The Human Body - Overview [[Image:75%.png]]
September science lessons deal with taking care of the body through cleanliness, rest, and healthy eating. The teacher can develop healthy habits at the beginning of the year and assist children in following procedures all year long that are developed in September. It is strongly suggested that the teacher request children wash their hands prior to any food consumption.

Review proper ways to cover the mouth and nose when coughing and sneezing and to wash after sneezing and coughing. Provide opportunity to eat healthy foods at snack time. Emphasize the importance of rest and how the body requires time to 'recharge' if nap time is a part of your day.

Authors and contributors
This Wikicurriculum has been written by (please add your name here if you are a contributor in order to comply with licensing):

 * Harriska2 (Contributions)

title: PreK Skills


text:
Template:PreK Skills

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title: Template:PreK Skills


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Preschool Language and Skills

 * Created: 2011 01 25 | Percent completed: [[Image:100%.png]]

What is Preschool Language and Skills?
Preschool Language and Skills is a teacher/parent guide for teaching children with autism and developmental disabilities language and school readiness skills at the preschool level.

Each guide section is numbered and can be taught or reviewd in one day. Some sections also contains scripted instructions for the parent, teacher, or therapist. Many sections contain workbook pages and activities.

What gets taught?
This program will focus on Pairing, Manding, Tacting, and Intraverbals for language skills up to the age of 4-5.

Early Learner - Pairing
In the very beginning, it will look like much time is spent playing. This is not playing but Pairing. The focus is on early play and social skill development by teaching the child that it is fun to interact with parents, teachers, therapists, and peers along with fun items and activities.

Natural Environment Teaching (NET) and Manding
The focus will slowly morph to teaching functional communication skills. Here, you will teach the student that communication makes their world better. We do this by developing manding skills by either shaping verbal or sign language in order for the child to communicate requests of items or activities. In this way, the student learns that talking, signing, or exchanging pictures (PECS) is fun and provides immediate benefit. These skills are often present in 1-2 year old children but must be discretely (directly) taught in most children with autism.

Natural Environment Teaching (NET) takes advantage of a more loose teaching environment (generally not a table with chairs). In NET, you can find more items and activities to motivate and reinforce the student. You are able to take advantage of a larger array of learning environments: kitchen, backyard, playground, store. Although it may look like teaching is taking place in a haphazard way, there is always a plan with reinforcers at hand.

Intensive Teaching/Discrete Trial Training (DTT) and Echoics, Imitation, and Tacting
The focus then morphs to gaining and maintaining instructional control and compliance. Often at a table with cards, objects, puzzles, and toys, students learn to respond quickly and accurately to teacher directed demands for increasingly longer periods of time.

By pairing reinforcers with low preferred activities/toys and using errorless teaching (don't let the student guess at answers) a therapist is able to build the number of demands (finishing a puzzle, identifying flashcards), the effort of the response, and the complexity of the task(s). It is here where tacting (labeling and identifying) is mostly taught. The student tacts (labels) items because he or she can see, hear, small, taste, or feel something.

Echoic - repeating what someone else says - you say it because someone else says it

Imitation - repeating someone else's motor movements - you move because someone else moved the same way

Listener Responding/Receptive - following directions - you do what someone else asks you to do

Advanced Natural Environment Teaching (NET)
The focus moves to teaching conversation skills and generalizing skills learned in intensive teaching. The student learns to ask and answer "Wh" questions, expand sentence structure and length with adjectives, adverbs, and discuss items and activities without them being present. These skills all fall under the Intraverbal skills. At some point, the student can answer "How was school today."

Social Skills with Peers
Students will need to learn to give and take in discussions, sharing or objects, and cooperative groups with peers.

Academic and Group Skills
A focus on teaching basic and advanced academic skills for home schooling or schooling in the classroom. Often Direct Instruction materials by SRA are used to teach reading, math, writing, and spelling.

Who is the teacher?
A teacher is anyone who interacts with the student with disabilities. One parent doesn't have to be the only teacher - you can also recruit the other parent, grandparents, aunts/ucles, church members, neighbors, high schoolers, and even college students. Sometimes you can find volunteers. You may also be able to hire individuals ($8-$15 per hour) from your local college, university, even high school. You’ll need to train everyone, regularly supervise each therapist via video taping or sitting in on sessions, and ensure quality control through regular team meetings.

Do I need a consultant?
It is highly advisable to hire a consultant to review the student's progress and ensure appropriate teaching procedures. A team (therapists and parents) can lose valuable teaching time by getting off track.

Finding qualified professionals can be difficult. It is best to find a Certified Behavior Analyst (BCBA) or, in the minimum, a Certified Assistant Behavior Analyst (BCaBA). However, because of the lack of certified professionals and the high demand for ABA consultants, some families may be forced to go out of state for ABA expertise. Another way to bring in experts is by banding together to bring an outside consultant for periodic visits and share the associated travel expenses. Email listserves are one way families can get in touch with each other to hire a consultant.

According to Association of Behavior Analysts, it is recommended that at minimum consultants:

 * 1) Hold a master’s degree in psychology, special education or related field;
 * 2) Have two years of experience using ABA to treat autism in young children;
 * 3) Provide customized program design and monitoring;
 * 4) Provide training of line therapists. Expert consultation is essential to successful ABA programs.

Authors and contributors
This Wikicurriculum has been written by (please add your name here if you are a contributor in order to comply with licensing):

 * Harriska2 (Contributions)
 * Margoschaefer

title: Preschool Language and Skills


text:
User:Fem.tm7

HW 1, version Jin

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title: User:Fem.tm7


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Talk:Preschool Language and Skills

=To Do= Still working on this... Harriska2 03:52, 21 January 2011 (UTC)

 * 1) Add specific items/examples to work on for QA
 * 2) Add lesson on shaping
 * 3) Add info about Quick Assessment (QA) - why do one, who should administrate it
 * 4) Add info after QA (lesson 5) - talk about each area and where to start teaching - not to teach in order of QA
 * 5) Add template
 * 6) Move reinforcer items to lesson 1
 * 7) Move tact and mand items to appropriate lessons
 * 8) See if wikiversity has a plug in for youtube or like - no but can use youtube which is limited to 320 x 240 - wikis generally do not support plug ins - this is a plus for our own server using drupal.  See http://www.videoonwikipedia.org/howto.html
 * 9) Add notes, pictures/graphics, and links for video
 * 10) Eventually move project to a different name such as Autism Early Learner Language and Skills, maybe Autism Early Learner Assessment and Curriculum (ELAC)

=3/21/11 Meeting= Harriska2 14:27, 22 March 2011 (UTC)

Notes from Kathy:
 * Add info about Quick Assessment (QA) - why do one, who should administrate it
 * Add info after QA (lesson 5) - talk about each area and where to start teaching - not to teach in order of QA
 * Rename lesson to guide?
 * Adapt longer assessment to matrix?
 * Lesson 4, can make spreadsheet for people to print and take data on
 * Wondering about book format for rest of "lessons" so that people can jump to the area the student needs
 * Illustrations would be used to break it up (just a bit) but mostly for the parent handbook

Reactions from Margo:
 * "Don't do what don't like first"
 * Balance of information vs. amount of information - don't let parents get bogged down in too much information
 * Could get Oregonian involved in Oregon standards
 * Lesson 2, #5 - duplication? example?
 * Lesson 4 - define tact/intraverbal
 * Wondered if videos/pictures more useful than illustrations

=2/28/11 Meeting= Harriska2 18:46, 1 March 2011 (UTC)

Reactions from Margo:
 * What is 10 reinforcer - description?
 * Find - how to - reinforcers
 * Maybe a reinforcer program
 * Add encouragement for parents - how to get started, can be overwhelmed, sadness, you child need you to pull yourself up/together - maybe a "Success Module"
 * Ease parent into teaching
 * One parent doesn't have to be the only teacher: other parent, grandparents, aunts/uncles, church, volunteers, neighbors, high school volunteers (service learning credit), college (community service thru frat/sorority or even college credits)
 * How do you ID the appropriate teacher (sorry Margo, I can't remember what this is in reference to)

Margo To Do:
 * See where illustrations/charts/images might be helpful

Kathy To Do:
 * Put in spot for images (ie reinforcer giving up would be a picture of the student handing back a toy to the teacher with a hand out)
 * Try to limit track (which at the beginning might be the whole lesson) to 3-7 minutes. Think of breaking up info/lesson for parents, too.
 * Switch inventory and assessment
 * Change from Unit to something else
 * Do not teach tacting (switch)
 * Add "Giving up reinforcer" lesson plan/track - lesson 3?
 * Add "Do this" program
 * Make note on front page of Instructional Control
 * Explain NET
 * Explain teaching opportunities
 * Think about learning climate/environment - this is anytime when not a melt down (and even then you can arrange that!)
 * Things have to be in place - ie reinforcers
 * Then work on comm/compliant things

Tacting (Identifying)
Do not start a tacting program until at least 10 or more mands are consistently made by student.

 * Definition

Tacting Assessment
Identify if child can identify any items via verbal language, sign, or PECS.
 * Start children with very limited language (5 or fewer words) by introducing items that they WANT.
 * Pair identification with giving the item.
 * You are now doing Tact to Mand transfer, the identification of an item leads to requesting behavior. Manding (requesting) gives motivation and tacting (identification of item/activity) gives words.
 * You are now teaching the fact that communication pays -> their world gets better when they communicate their needs and wants.

Manding (Requesting)
Verbal behavior (spoken, sign, PECs) of a request for an item or information that benefits the requestor. A baby cries and is reinforced by getting changed or fed. A Mand is a Terrible Thing to Waste
 * Definition
 * Example
 * Slogan
 * Description
 * 1) Pair teacher with getting things that make their world better.  It teaches that it pays to communicate your wants and needs.
 * 2) Manding is needed for social interaction: conversation, getting information, making friends
 * 3) Manding is the fastest way for the student to get reinforcement therefore strengthening communication.
 * 4) Student must have a lot of mands under his/her belt in order to move onto tacting (labeling/identifying) and intraverbals (fill in the blanks, "wh" questions and answers)
 * 5) Manding is the first thing you should teach.  Do not teach labeling first.
 * 6) By teaching manding first, you can get rid of problem behaviors due to lack of communicating wants and needs.
 * 7) Teach manding by doing a reinforcer assessment and then controlling strongly reinforcing items.  However, don't make it too difficult to ask for the items and ensure that when you take the item back, it doesn't make you a "this person makes me work too hard and I don't like the item anymore" aversive.
 * 8) Mand items can change moment by moment to month to month.  You have to be flexible and always on the lookout for new reinforcers. For instance teach asking for a tub toy during bath time.  Teach asking for a favorite snack during snake time.

Manding Assessment
Identify current and potential mand items

 * Can look at an item to request it – you’ll know because the child will play with the item
 * Stand by requested item or pull to item
 * Request the item by pointing to item
 * Request item through verbal language, sign language, or PECS

How to teach:

What do you want? [Give the student a word, sign, or PECS for the item. Do not expect perfection]

title: Talk:Preschool Language and Skills


text:
Template:Elementary Science K2

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title: Template:Elementary Science K2


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User:Fem.tm7/HW

Problem Description

 * Part 1: Derive the one-dimensional partial differential equation (PDE) for an elastic bar under dynamic loading conditions. (Refer to equation 3 of lecture page 2-3.)

 * Part 2: Discuss the case in which the bar has a rectangular cross-section.

Given

From lecture page 2-3 equation 3 ,the PDE for an elastic bar under dynamic loading conditions is:

 * $$\frac{\partial{}}{\partial{x}}[E(x)A(x)]\frac{\partial{u}}{\partial{x}}+f(x,t)=m\frac{\partial^2{u}}{\partial{t^2}}$$

Since the elastic bar is loaded dynamically, the applied force, f, is a function of distance, x , as well as time, t. This is shown in the free body diagram to the right. Details of the remaining forces will be discussed in the solution below.

Solution
Assumptions :
 * The rod is in stress equilibrium.
 * The rod satisfies Hooke's Law, $$\sigma (x) = E(x)\epsilon (x) $$.
 * Compatible displacement field.
 * The rod obeys the strain displacement equation.

Following Newton's Second Law:

 * $$\displaystyle F=m*a $$

Where:

 * F=Force

 * m=Mass

 * a=Acceleration

Additionally,

 * $$\displaystyle m=\rho * V$$

Where $$\displaystyle\rho = $$ density and V=Volume.

and,

 * $$a=\frac{d^2u}{dt^2}$$

Where u is displacement and t is time.

Performing a force balance on the rod using the free body diagram above, the net force is equal to the internal forces plus the external, or applied, forces.

 * {| style="width:100%" border="0"

$$ \displaystyle \Sigma \text{ Internal Force } + \text{ External Force } = \text{ Net Force}$$ (1.1)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle \Sigma \text{ Internal Force }=-N(x) + N(x+\Delta x)$$ (1.2)
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 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle \text{ Applied Force }=f(x+\frac{\Delta x}{2},t)\Delta x$$ (1.3)
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 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle \text{ Net Force}= m*a =\rho (x)A(x)\frac{\partial^2{u}}{\partial{t^2}}\Delta x$$ (1.4) Note: Acceleration is a partial derivative since the displacement, u, is also dependent on the axial position, x.
 * style="width:95%" |
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 *
 * }

Substituting equations 1.2 through 1.4 back into the force balance equation, 1.1:

 * {| style="width:100%" border="0"

$$ \displaystyle -N(x) + N(x+\Delta x) + f(x+\frac{\Delta x}{2},t)\Delta x =\rho (x)A(x)\frac{\partial^2{u}}{\partial{t^2}}\Delta x$$ (1.5)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

The balance of forces at position x can be expressed as:

 * {| style="width:100%" border="0"

$$ \displaystyle N(x)=\sigma (x)n(x)A(x)=-\sigma (x)A(x)$$ (1.6)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

The corresponding force at x + Δx:

 * {| style="width:100%" border="0"

$$ \displaystyle N(x+\Delta x)=\sigma (x+\Delta x)n(x+\Delta x)A(x+\Delta x)=\sigma (x+\Delta x)A(x\Delta x)$$ (1.7)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Substituting equations 1.6 and 1.7 into 1.5 yields:

 * {| style="width:100%" border="0"

$$ \displaystyle -\sigma (x)A(x) + \sigma (x+\Delta x)A(x+\Delta x) + f(x+\frac{\Delta x}{2},t)\Delta (x) = \rho (x)A(x)\Delta x\frac{\partial^2{u}}{\partial{t^2}}$$ (1.8)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Dividing this equation and the internal forces (1.2) through by Δx and recognizing that as Δx approaches zero:

 * {| style="width:100%" border="0"

$$ \displaystyle \lim_{\Delta x \to 0} \frac{N(x+\Delta x)-N(x)}{\Delta x}=\frac{dN}{dx}.$$ (1.9)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Equation 1.8 can be rewritten as:
 * {| style="width:100%" border="0"

$$ \displaystyle \frac{\partial{\sigma (x)A(x)}}{\partial{x}}+f(x,t)=\rho (x) A(x)\frac{\partial^2{u}}{\partial{t^2}}$$ (1.10)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

The strain-displacement equation is
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\varepsilon \left( x \right) = \frac

$$     (1.11) By Hooke ‘ s Law
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\sigma \left( x \right) = E\left( x \right)\varepsilon \left( x \right) = E\left( x \right)\frac

$$     (1.12)
 *
 * }

and substituting into equation 1.10 yields:

 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

\frac{\partial }\left[ {E\left( x \right)A\left( x \right)\frac} \right] + f\left( {x,t} \right) = m\left( x \right)\frac

$$     (1.13)
 *
 * }

Given
Drawing inserted here to show rectangular cross-section and define h(x).\

Solution
The cross section area
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

A\left( x \right) = b \cdot h\left( x \right)

$$     (1.14) The mass
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

m\left( {x + \frac{2}} \right) = \rho \left( {x + \frac{2}} \right)\frac{1}{2}\left[ {h\left( x \right) + h\left( {x + \Delta x} \right)} \right]b

$$     (1.15) Take the limit $$\Delta x \to 0$$
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

m\left( x \right) = \rho \left( x \right)h\left( x \right)b

$$     (1.16) Substituting Equation 1.14 and 1.16 into Equation 1.13
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

\frac{\partial }\left[ {E\left( x \right)h\left( x \right)b\frac} \right] + f\left( {x,t} \right) = \rho \left( x \right)h\left( x \right)b\frac

$$     (1.17)
 * <p style="text-align:right">
 * }

Authors
Johnathan Whittaker Bullard

Philip Flater

Brandon Hua

Jiang Jin

Braden Snook

Srilalithkumar Swaminathan

Zongyi Yang

title: User:Fem.tm7/HW