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Logic for Computer Scientists/Propositional Logic/Equivalence and Normal Forms

Equivalence and Normal Forms
Until now, we only discussed single formulae and their semantical properties. In this section we start investigating whether formulae can be transformed into another form, without changing their semantics. For this we introduce the concept of ''logical equivalence, which can be used to investigate the transformation of a given formula into its normal form''.

Definition 6
The formulae $$F$$ and $$G$$ are called (semantically) equivalent, iff for all assignments $$\mathcal{A}$$ for $$F$$ and $$G$$, $$\mathcal{A}(F)= \mathcal{A}(G)$$. We write $$F \equiv G$$.

Formulae containing different subformulae can be equivalent, e.g. all tautologies are equivalent. More interesting is the following theorem:

Theorem 3 (Substituitivity)
Let $$F \equiv G$$ and $$H$$ a formula which contains at least one occurrence of the subformula $$F$$. Then it holds $$H \equiv H'$$, where $$H'$$ is obtained from $$H$$ by substituting any occurrence of $$F$$ by $$G$$.

Proof: The proof is by induction over the structure of the subformula $$H$$:

Assume for the induction start, that $$H$$ is atomic, hence $$H=F$$ holds, and the result of substituting the only occurrence of $$F$$ by $$G$$ results in $$H' =G$$ and because $$F \equiv G$$, we have $$H \equiv H'$$. Assume the theorem holds for all proper subformulae of $$H$$: If $$F = H$$ we have the same argumentation as above in the start of the induction. If $$F \neq H$$ we have three cases:
 * $$H = \lnot h_1$$: Because $$h_1$$ is a subformula of $$H$$ we can  conclude that $$h_1 \equiv  h_1'$$, where $$h_1'$$ is constructed by substituting any occurrence  of $$F$$ by $$G$$. From the definition   of the semantics of $$\lnot$$ we conclude that $$\lnot h_1 \equiv  \lnot h_1'$$ and hence $$H \equiv  H_1'$$. Where H_1 is equivalent formula constructed by replacing F in H by G, resulting in H_1
 * $$H = (h_1 \lor h_2)$$: Assume without loss of generality, that $$F$$ occurs in $$h_1$$. Then, again we can conclude from the  induction assumption, that $$h_1 \equiv h_1'$$ and from the definition of the semantics of $$\lor$$ we conclude, that $$(h_1 \lor h_2) = (h_1' \lor   h_2)$$.
 * $$H = (h_1 \land h_2)$$ is similar.

Theorem 4
The following equivalences hold:

$$(F \land F)$$ $$\equiv$$  $$F$$

$$(F \lor F)$$   $$\equiv$$  $$F$$   (Idempotence) $$(F \land G)$$ $$\equiv$$  $$(G \land F)$$

$$(F \lor G)$$   $$\equiv$$  $$(G \lor F)$$  (Commutativity) $$((F \land G) \land H)$$ $$\equiv$$ $$(F \land (G \land H))$$ $$((F \lor G) \lor H)$$ $$\equiv$$ $$(F \lor (G \lor H))$$ (Associativity) $$(F \land (F \lor G))$$ $$\equiv$$  $$F$$

$$(F \lor (F \land G))$$ $$\equiv$$  $$F$$   (Absorption) $$(F \land (G \lor H))$$ $$\equiv$$ $$((F \land G) \lor (F \land H))$$

$$(F \lor (G \land H))$$ $$\equiv$$ $$((F \lor G) \land (F \lor H))$$ (Distributivity) $$\lnot ~\lnot F$$ $$\equiv$$  $$F$$   (Double negation) $$\lnot(F \land G)$$ $$\equiv$$ $$(\lnot F \lor \lnot G)$$

$$\lnot(F \lor G)$$ $$\equiv$$  $$(\lnot F \land \lnot G)$$  (deMorgan's rule) $$(F \lor G)$$ $$\equiv$$  $$F$$, if  $$F$$ is a tautology

$$(F \land G)$$ $$\equiv$$  $$G$$, if  $$F$$ is a tautology (Rule of Tautology) $$(F \lor G)$$ $$\equiv$$  $$G$$, if  $$F$$  is unsatisfiable

$$(F \land G)$$ $$\equiv$$  $$F$$, if  $$F$$  is unsatisfiable  (Rule of Unsatisfiability)

Proof: All equivalences can be proved by truth tables. This is done here for the second rule of absorption:

Let us now use these equivalences together with the theorem of substituitivity (TS) to prove the following equivalence: $$ ((A\lor (B\lor C)) \land (C \lor \lnot A))\equiv ((B \land \lnot A) \lor C) $$

 * $$((A \lor (B \lor C)) \land (C \lor \lnot A))$$

$$\equiv$$ $$((A \lor B) \lor C) \land (C \lor \lnot A))$$ Associativity and  TS $$\equiv$$ $$((C \lor (A \lor B)) \land ((C \lor \lnot A))$$ Commutativity and TS  $$\equiv$$  $$(C \lor ((A \lor B) \land \lnot A))$$      Distributivity $$\equiv$$ $$(C \lor (\lnot A \land (A \lor B)))$$     Commutativity and TS

$$\equiv$$ $$(C \lor ((\lnot A \land A) \lor (\lnot A \land B))$$     Distributivity and TS

$$\equiv$$ $$(C \lor (\lnot A \land B))$$      Unsatisfiability and TS

$$\equiv$$ $$(C \lor (B \land \lnot A))$$      Commutativity and TS $$\equiv$$  $$((B \land \lnot A) \lor C)$$      Commutativity and TS

Problem 9 (Propositional)
The binary junctor $$\stackrel{\cdot} \vee$$ for exclusive disjunction is defined by $$ F \stackrel{\cdot} \vee G = F \leftrightarrow (\lnot G)$$. Show that the following holds for propositional logic formulae $$F$$, $$G$$ and $$H$$: $$\Box$$
 * 1) $$F \stackrel{\cdot} \vee G \equiv G \stackrel{\cdot} \vee F$$ (Commutativity).
 * 2) $$(F \stackrel{\cdot} \vee G) \stackrel{\cdot} \vee H \equiv F \stackrel{\cdot} \vee (G \stackrel{\cdot} \vee H)$$ (Associativity)

Problem 10 (Propositional)
Show the following by equivalence transformation. Give to all remodelling steps the used rules! $$\Box$$
 * 1) $$(A \to (B \lor C)) \equiv (A \to B) \lor (A \to C)$$
 * 2) $$(A \to B) \to A \equiv (B \to A) \land (B \lor A)$$
 * 3) $$A \leftrightarrow \lnot B \equiv \lnot (A \leftrightarrow B)$$
 * 4) $$(A \to (B \land C)) \equiv (A \to B) \land (A \to C)$$
 * 5) $$(A \to B) \to (B \to A) \equiv B \to A$$
 * 6) $$(A \lor \lnot B) \lor ((\lnot A \land \lnot C) \land B)\equiv (A \lor \lnot C) \lor \lnot B$$

Problem 11 (Propositional)
Prove with equivalences:
 * 1) $$p \lor \lnot(p \land q)$$ is a tautology.
 * 2) $$(p \lor \lnot q) \land (\lnot p \land q)$$ is unsatisfiable.

$$\Box$$

Problem 12 (Propositional)
The binary junctor $$\downarrow$$ with the meaning neither-nor is defined by $$F \downarrow G = \lnot(F \lor G)$$. Let $$F$$ be a propositional logic formula, which contains only the operators $$\land$$ $$\lor$$ and $$\lnot$$. Prove that for every formula $$F$$ a formula $$ \mathcal{T} (F)$$ exists, such that $$F \equiv \mathcal{T} (F)$$ and $$\mathcal{T} (F)$$ is built by using only the junctor $$\downarrow$$.

$$\Box$$

Problem 13 (Propositional)
Let $$F$$ be a propositional logic formula, which contains only the operators $$\land$$, $$\lor$$ and $$\lnot$$. Prove that for every formula $$F$$ a formula $$\mathcal{T}(F)$$ exists, such that $$F \equiv \mathcal{T}(F)$$ and $$\mathcal{T}(F)$$ is built by using only the junctors $$\to$$ and $$\stackrel{\cdot} \vee$$. $$\Box$$

Problem 14 (Propositional)
The following formulae are given: $$\Box$$
 * $$A \equiv D \lor B$$
 * $$B \equiv \lnot B \lor \lnot D \lor \lnot S$$
 * $$C \equiv (\lnot S\land D) \lor \lnot B$$
 * 1) Simplify $$A \land B \land C$$ by using $$B \land C \equiv C$$.
 * 2) Simplify $$A \land B \land C$$ by using equivalences but not $$B \land C \equiv C$$.

Definition 7 (Normal Forms)
A literal is an atomic formula or the negation of an atomic formula (positive or negative, resp.) A formula $$F$$ is in conjunctive normalform (CNF) iff

$$F = (\bigwedge^n_{i=1} (\bigvee^{m_i}_{j=1} L_{i,j}))$$ where $$L_{ij}$$ is a literal.

A formula $$F$$ is in disjunctive  normalform (DNF) iff $$F = (\bigvee^n_{i=1} (\bigwedge^{m_i}_{j=1} L_{i,j}))$$ where $$L_{ij}$$ is a literal

Theorem 5
For every formula $$F$$ there is an equivalent formula which is in DNF and an equivalent formula which is in CNF.

Let us formulate an algorithm to transform a given formula F into an equivalent normalform:

Given: A formula $$F $$

1. Substitute in $$F$$ every subformula of the form
 * $$\lnot ~\lnot ~G \text{ by } G$$
 * $$\lnot ~(G ~\land ~H)  \text{ by }   (\lnot ~G ~\lor ~\lnot ~H)$$
 * $$\lnot ~(G ~\lor ~H) \text{ by } (\lnot ~G ~\land ~\lnot ~H) $$

until there is no subformula of this kind. 2. Substitute in the result from the above step  every subformula of the form
 * $$(F ~\lor ~(G~\land~H))  \text{ by }   ((F~\lor~G) ~\land ~(F~\lor~H))$$

 * $$((F~\land~G) ~\lor ~H) \text{ by } ((F~\lor~H) ~\land ~(G~\lor~H))$$

until there is no subformula of this kind. Result: An equivalent formula in CNF

Until now, we investigate the transformation of a propositional formula into an equivalent normal form. Another problem in the context of normal forms is, to construct a normal form formula from a given truth table; i.e. the formula itself is not known, but its behaviour is given by a truth table. Let's read a  normalform formula from a truth table: Assume a formula $$F$$, which is given by the following truthtable.

In order to construct a formula in DNF, which is equivalent to $$F$$, we have to take into account, that every line of the table which yields the truthvalue $$true$$ gives one conjunction: if the assignment of the literal $$A_i$$ is $$true$$ it is included as $$ \ldots \land A_i \land \ldots$$, if  is $$false$$ we include $$\ldots \land \lnot A_i \ldots \land \ldots$$. For the above example we get as a DNF:

$$(\lnot A \land  \lnot  B\land  \lnot C )  \lor $$

$$( A \land  \lnot  B\land  \lnot C )   \lor    $$

$$( A \land  \lnot  B\land      C ) $$

If we change in the above procedure the roles of $$true$$ and $$false$$ and $$\lor$$ and $$\land$$ we arrive at a CNF:

$$( A \lor B \lor\lnot C) \land $$

$$(A\lor\lnot B\lor C) \land $$

$$(A \lor \lnot B \lor \lnot C) \land $$

$$( \lnot A \lor \lnot B \lor C) \land $$

$$(\lnot A \lor \lnot B \lor \lnot C) $$

We introduce a special representation for formulae in  normalform. In our circuit-example (circuit) from the introduction we already used a very special form of normalforms, namely the implication form for formulae in CNF: every subformula $$F_i$$ of a conjunction $$F_1, \cdots, F_l$$ is written as: $$ A_1 \land \cdots \land A_n \to B_1 \lor \cdots \lor B_m $$ It is easy to see, that this implication is logically equivalent to a disjunction $$\lnot A_1 \lor \cdots \lor \lnot A_n \lor B_1 \lor \cdots \lor B_m$$. Sometimes the implication is written as

$$ A_1, \cdots, A_n \to B_1; \cdots ; B_m $$ Even the following ambiguous notation is used in some cases, where the comma in the premiss stands for a conjunction and the comma in the conclusion for a disjunction: $$ A_1, \cdots, A_n \to B_1, \cdots , B_m $$ For some important procedures for logical reasoning it is mandatory to represent the formulae not only in one of the above notations for a normalform, moreover, it is necessary to use the so-called clause-form from the following definition.

Definition 8
If $$F$$ is a formula in CNF, i.e.  $$ F = (L_{1,1} \lor \ldots \lor L_{1,{n_1}}) \land \ldots \land (L_{k,1} \lor \ldots \lor L_{k,{n_k}}) $$ then its corresponding representation in clause form is given as $$ F = \{ \{L_{1,1}, \ldots, L_{1,{n_1}}\}, \ldots, \{L_{k,1}, \ldots, L_{k,{n_k}}\} \} $$

The sets $$\{L_{i,1}, \ldots, L_{i,{n_i}}\}$$ are called clauses.

This representation as sets of literals has the advantage that literals occur in no special order and that multiple occurrences of a literal in a disjunction are "merged" in its clause form. Note that as a consequence we have built in associativity, commutativity and Idempotence into the representation. $$ ( (A_1 \lor \lnot A_2) \land (A_3 \land A_3) )$$            $$\{\{A_1, \lnot A_2\}, \{A_3\}\} $$ $$(A_3 \land (\lnot A_2 \lor A_1)) $$

Problem 15 (Propositional)
Create formulae in (a) conjunctive or (b) disjunctive normal form which are equivalent to: $$ A \stackrel{\cdot} \vee B \stackrel{\cdot} \vee C $$ where $$\stackrel{\cdot} \vee$$ denotes exclusive or. $$\Box$$

Problem 16 (Propositional)
The theorem prover OTTER uses the following optimization rules for clause sets:

Subsumption: If the literals of a clause $$K$$ are a subset of an another clause $$K'$$ remove $$K'$$ from the set of clauses. Deleting by unit clause: If the set of clauses contains a unit clause $$L$$ -here $$L$$ is single literal unit-clause- every occurrence of a complementary literal $$\overline{L}$$ in a clause is deleted.

Which laws of the logic justify these procedures?

$$\Box$$

Problem 17 (Propositional)
Generate truth tables for the following formulae. Give the conjunctive and disjunctive normal forms of the formulae. Which one of the relations $$F \models G$$, $$G \models F$$, $$F \equiv G$$ and $$F = G$$ holds?
 * 1) $$F = (A \land \lnot (B \lor C)) \stackrel{\cdot} \vee ((A \to B) \lor (A \to C),)$$ whereas $$F \stackrel{\cdot} \vee G = F \leftrightarrow (\lnot G)$$ applies.
 * 2) $$G = ((A \lor (B \land A)) \to \lnot A) \lor C$$

$$\Box$$

Problem 18 (Propositional)
Generate a CNF and a DNF form the following truth table for the formula $$F$$.

$$\Box$$

Problem 19 (Propositional)
Generate a CNF from the formula $$(P \land(Q \to R)\to S)$$

$$\Box$$


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Belarusian/Lesson 5

Consonant Assimilation
There are occasions when voiced consonants are pronounced like their surd mates (e.g. п instead of б) and vice versa (б instead of п).

Voiced consonants being at the end of a word, before a surd or assimilated voiced consonant are pronounced as surd sounds.

го́рад "horat" a town (final д pronounced as т)

мар́оз "maros" frost (final з pronounced as с)

Пары́ж "Parysh" Paris (final ж pronounced as ш)

адка́з "atkas" an answer (final з pronounced as с; д before к pronounced as т)

дождж "doshch" rain (final дж pronounced as ч; ж before assimilated дж pronounced as ш)

This process causes a number of homophones, i.e. words having different spelling but the same pronunciation.

кот a cat - код a code

ка́зка a tale - ка́ска a helmet

The opposite phenomenon takes place when a surd consonant occurs after a vowel before a voiced (and not assimilated) consonant. Then it is pronounced as a voiced consonant itself.

футбо́л "fudbol" football (т after у and before б pronounced as д)

вакза́л "vagzal" a railway station (к after а and before з pronounced as English g)

This process does not take place before в, л, м, н and р sounds though they are voiced.

патрэ́бны "patrebny" necessary (т not assimilated to д)

Noun Gender
Each noun belongs to one of the three genders. The "theoretical" rule for noun gender is the following.

 * nouns ending by а or я are feminine: вада́ water, Ірла́ндыя Ireland
 * nouns ending by о, ё; э or е are neuter: вакно́ a window, ра́дыё radio
 * nouns ending by a consonant are masculine кот a cat, го́рад a town

In practice, however, the things are far not so simple. First of all, with neuter nouns, final unstressed о or э due to akannie transform into а to get an ending typical for feminine nouns.

мо́ра a sea (а is used instead of unstressed э)

ко́ла a wheel (а is used instead of unstressed о)

There is a large number of nouns ending by a consonant but belonging to the feminine gender.

мыш a mouse

ноч a night

Finally, there exist a number of nouns ending by а or я but being masculine.

мужчы́на a man

саба́ка a dog

валацýга a tramp

Plural Nouns
Plural nouns usually end by ы.

With feminine and neuter nouns, the plural ending is never stressed. If a feminine noun has a stressed ending in singular, it becomes unstressed in plural.

вада́ - во́ды water (waters)

акно́ - во́кны a window (windows) (в here appears only before o, because native Belarusian words can't begin with o-)

гара́ - го́ры a mountain (mountains)

каза́ - ко́зы a goat (goats)

цана́ - цэ́ны a price (prices)

царква́ - цэ́рквы a church (churches)

With feminine and neuter nouns having unstressed endings, the stress remains unshifted.

варо́на - варо́ны a crow (crows)

дрэ́ва (neuter) - дрэ́вы a tree (trees)

жа́ба - жа́бы a toad (toads)

жанчы́на - жанчы́ны a woman (women)

каро́ва - каро́вы a cow (cows)

ко́ла (neuter) - ко́лы a wheel (wheels)

краiна - краiны a country (countries)

кра́ма - кра́мы a shop (shops)

мо́ра (neuter) - мо́ры a sea (seas)

мыш - мы́шы a mouse (mice)

ноч - но́чы a night (nights)

фа́ра - фа́ры a headlight (headlights)

ха́та - ха́ты a house (houses)

шко́ла - шко́лы a school (schools)

Masculine nouns may have either stressed or unstressed ending in plural. The first case applies to most monosyllabic nouns. But there is no strict rule to determine the stress position in plural for masculine nouns.

Notice the last word, тэатр. The э letter here violates the rule of akannie - a situation which is common with borrowed words.

Gender
Make three sentences with the given pair of words: "This is a "; "It/he/she is "; This is a ". Take note that nouns and adjectives must be coordinated by gender, i.e. masculine adjectives are used with masculine nouns etc.

Sample church, new: Гэта царква. Яна новая. Гэта новая царква.

(1) frost, strong

(2) tree, old

(3) woman, ill

(4) voice, beautiful

(5) goat, young

(6) palace, old

Clue: (1) Гэта мароз. Ён моцны. Гэта моцны мароз. (2) Гэта дрэва. Яно старое. Гэта старое дрэва. (3) Гэта жанчына. Яна хворая. Гэта хворая жанчына. (4) Гэта голас. Ён прыгожы. Гэта прыгожы голас. (5) Гэта каза. Яна маладая. Гэта маладая каза. (6) Гэта палац. Ён стары. Гэта стары палац.

Number
Transform all sentences from the previous exercise into plural.

Sample church, new: Гэта цэрквы. Яны новыя. Гэта новыя цэрквы.

Clue: (1) Гэта маразы. Яны моцныя. Гэта моцныя маразы. (2) Гэта дрэвы. Яны старыя. Гэта старыя дрэвы. (3) Гэта жанчыны. Яны хворыя. Гэта хворыя жанчыны. (4) Гэта галасы. Яны прыгожыя. Гэта прыгожыя галасы. (5) Гэта козы. Яны маладыя. Гэта маладыя козы. (6) Гэта палацы. Яны старыя. Гэта старыя палацы.

Translation
Translate into Belarusian.

(1) I am a strong man.

(2) These are strong men.

(3) What is this? This is an old town.

(4) Who are these? These are women. They are beautiful.

(5) They are young students.

(6) This is a wheel. It is new. This is a new wheel.

Clue: (1) Я дужы мужчына. (2) Гэта дужыя мужчыны. (3) Што гэта? Гэта стары горад. (4) Хто гэта? Гэта жанчыны. Яны прыгожыя. (5) Яны маладыя студэнты. (6) Гэта кола. Яно новае. Гэта новае кола.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Belarusian/Lesson 5


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User talk:Cwalck

My IB Biology students are the best internationally!

title: User talk:Cwalck


text:
Statistical Analysis: an Introduction using R/R

Topics in R
All the R topics in the wikibook Statistical Analysis: an Introduction using R are listed here.

 * /R fundamentals/ (Reading these topics should give you a good grounding in R)
 * /R as a calculator/
 * /Storing objects/
 * /Functions/
 * /Getting help/
 * /Quitting/
 * /Vectors/
 * /Factors/
 * /Missing data/
 * /Matrices and arrays/
 * /Lists/
 * /Simple plotting/
 * /Univariate plots/
 * /Data frames/
 * /Getting data into R/
 * /Bivariate plots/
 * /Programming R/
 * Blocks of code
 * Your own functions
 * if/ifelse

The full list of topics is below
/All


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Statistical Analysis: an Introduction using R/R/R as a calculator

R topic:  ../../R basics Text marked like this is used to discuss an R-specific point. The basics of R can be learned by reading these sections in the order they appear in the book. There will also be commands that can be entered directly into R; you should be able to copy-and-paste them directly into your R session. Try the following to see how to use R as a simple calculator R commands can sometimes be rather difficult to follow, so occasionally it can be useful to annotate them with comments. This can be done by typing a hash (#) character: any further text on the same line is ignored by R. This will be used extensively in the R examples in this wikibook, e.g.


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Talk:PHP Programming/Cookies

hi i am vaibhav

title: Talk:PHP Programming/Cookies


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Autistic Survival Guide/Developing Social Learning Skills

As an autistic person struggling with social situations, it often feels like there is a list of rules whose memorisation would solve all one's problems. Sadly, trying to memorise rules and remember them later is probably not the most helpful. A better technique is to try to understand the reasoning behind common neurotypical social behaviors and to teach yourself new skills in that context. To do so, it is helpful to adopt tools that best enable you personally to learn, understand, and apply these behaviors. Marc Segar explains that he has found success by always looking for new things to think about; however, for some people, simply thinking about things never actually solves a given problem, so returning to them with new approaches may be necessary. As such, this page provides a basic toolkit that autistic people can draw on when developing new social skills.

The Development Cycle
While hyperfocusing on solving social interaction problems is attractive, it is unfortunately destructive to the process of actually learning about social interactions. Simply spending time thinking things through in one's own head is often quite useful, but it's easy to waste that time trapped in logic loops and other circuitous, unproductive pathways. To escape such loops, questioning and testing the logic can help—you can learn a lot from doing this. If it is not possible to develop and test the logic, then finding something else to think about for a while is probably better than wasting your time. The basic method behind developing anything, from an invention to an idea, is to go through the phases of a development cycle. The phases of this cycle are 1) design, 2) build, 3) test, and 4) post-mortem assessment. For example, to develop a recipe for a great cup of coffee, you would start by thinking about the qualities of a good cup (designing), followed by making a cup yourself (building), tasting it (testing), and finding room for improvement (post-mortem). Restart the cycle by changing the recipe to make it better (designing), then repeat the process until you have a really great cup of coffee.

These steps may seem rather obvious, but it is very easy to lose track of where you are in this cycle, especially when developing more complex things like guidelines for social interaction. Skipping steps in the cycle or not doing them properly often results in wasting lots of time, and it is better speed up the process by shortening the time taken for each step while still performing them all properly.

The development cycle can also be supplemented by the "reduction algorithm". This posits that having figured something out, it is often possible to learn even more by eliminating the irrelevant from what you have learned and dissecting the rest to its simplest possible components. Those smaller components are often more applicable to radically different aspects of life, and they are also much easier to explain and to prove to other people.

Attitude
When trying to understand people and social situations, your attitude will go a long way. One of the most helpful things you can do is to adopt the perspective that nobody ever does anything without a reason, even if that reason may not be obvious to you. Try cultivating an easy manner and showing interest in a calm way. Never pretend to be all-knowing—if you want to talk to someone but don't understand what they are talking about, your ignorance is valuable. Admit that you don't know what the conversation is about but that you are curious. When confronted in good faith with an error you've made, accept criticism with sincere thanks. Moreover, gratitude is also your best weapon against sarcastic comments and rudeness. Take pains to find and fix your mistakes. Apologise for errors with a straightforward, unashamed manner without apologising for yourself—you are not your mistakes. Running away from social threats can mean surviving, but possibly running forever. On the other hand, running into threats can mean suffering serious loss. Instead, keeping a safe distance from threats can mean gaining the opportunity to study them. Remember: if life isn't like falling off a log, you're probably doing something wrong.

 * When you criticize someone, incorporate a compliment into your criticism with optional humor. This is better than lying or being evasive. Example, "You have a wonderful sense of musical phrasing. I didn't even notice that you kicked me twice. Can we dance again?"


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Examples
One argument (a simple box) Lorem Lorem

Two arguments (a box with a title) Box 1.2 Box 1.2

A collapsible box Box 1.3 Box 1.3

A collapsible box, hidden by default, with a left-aligned title of heading level 4 Box 1.4 Box 1.4

The most complicated (and most realistic) Box 1.5 A collapsible box at the side, in a different colour Box 1.5

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User:Padraic

See my main page at Wikipedia.

title: User:Padraic


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Statistical Analysis: an Introduction using R/R/Functions

Apart from numbers, perhaps the most useful named objects in R are functions. Nearly everything useful that you will do in R is carried out using a function, and many are available in R by default. You can use (or "call") a function by typing its name followed by a pair of round brackets. For instance, the start up text mentions the following function, which you might find useful if you want to reference R in published work: Many R functions can produce results which differ depending on arguments that you provide to them. Arguments are placed inside the round brackets, separated by commas. Many functions have one or more optional arguments: that is, you can choose whether or not to provide them. An example of this is the function. It can take an optional argument giving the name of an R add-on package. If you do not provide an optional argument, there is usually an assumed default value (in the case of, this default value is , i.e. provide the citation reference for the base package: the package which provides most of the foundations of the R language). Most arguments to a function are named. For example, the first argument of the citation function is named package. To provide extra clarity, when using a function you can provide arguments in the longer form name=value. Thus citation("base") does the same as citation(package="base") If a function can take more than one argument, using the long form also allows you to change the order of arguments, as shown in the example code below.


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Template talk:Code:Transclude

PrettyTextBox
Should this use the PrettyTextBox css class?

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User talk:Ssugioka

"Speak the speech I pray you . . . trippingly on the tongue" (Shakespeare's Hamlet).

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Statistical Analysis: an Introduction using R/R/Quitting R

To quit R, you can use either the function or its identical shortcut,, which do not require any arguments. Alternatively, if your version of R has a menu bar, you can select "quit" or "exit" with the mouse.


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User talk:62.171.194.43

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Herby talk thyme 14:02, 24 May 2007 (UTC)

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User talk:204.138.9.1

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Herby talk thyme 14:56, 24 May 2007 (UTC)

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Logic for Computer Scientists/Predicate Logic

5.1 Predicate Logic 5.2 Syntax 5.3 Semantics 5.4 Equivalence and Normal Forms 5.5 Herbrand Theories 5.6 Semantic Trees 5.7 Resolution 5.8 Strategies for Resolution 5.9 SATCHMO


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Social and Cultural Foundations of American Education/Multiculturalism/Differences

One may realize very early in life that everyone is different. For most of us as little children, our very first day of school was scary. We saw children of different gender, color, and ethnicity just to name a few. This was just the beginning. Once we sat in class we realized that other children talked and even laughed differently than us. At a very young age we realize how different we all are. As we grow older we realize that these same differences that once scared us are the very same differences that help us coexist.

Respecting Students who are "Different"
In the world of education, the biggest difference arises through cultural or racial differences. “Problems can occur when teachers—most of whom are white and English speakers—expect minority and limited English-language students to act and learn in a way that does not mesh with their cultures”(www.edweek.org). A Caucasian teacher may have more difficulty relating to an Asian student then she would to another Caucasian student. For example, Susan Black writes in her article "Respecting Differences," "A third-grade teacher informed Mexican immigrants that their daughter was 'insecure and overly dependent.' The teacher didn't realize the parents taught their little girl to be quiet and obedient and to seek approval while working on her assignments." This is only one example of how ethnicity can be a barrier in the classroom if the teacher does not know how to keep an open-mind and respect his or her students' differences. These are serious issues because they block learning on most levels. If a teacher cannot relate to his or her students the children are less likely to take anything away from the lecture, especially if they feel the teacher is looking down on them or scolding them for things they cannot help or do not understand. When teachers step into the classroom they need to be prepared to deal with students who do not think, look, act, or speak like they do. They need to keep an open mind, and be willing to learn about their students and their differences. All students deserve to be treated with an equal amount of respect. If teachers understand their students and their differences, many barriers will be crossed and learning will be a lot more enjoyable and effective for everyone involved.

Respecting Teachers who are "Different"
The difference between a foreign teacher and students is, in my opinion, much greater. While, according to the article "Respecting Differences" teachers are "overwhelmingly white and English speaking," there are some classes that are taught by a foreign teacher. The same rules apply to teachers that are ethnically different that apply to students: Respect, understanding, and open-mindedness is key. When you have a teacher who does not understand a culture and a language you have a much greater degree of difficulty. For example, a foreign teacher trying to teach Civics would more than likely be a daunting task. It is almost impossible for a teacher who has never been in the country to teach a topic such as civics. As a society, we put ourselves in a difficult position by doing such things. I will say, however, that it is not the teachers fault, since they weren’t necessarily the ones who put themselves in that particular position. However, this is a difference that, though easy to respect, is not necessarily easy to overcome.

How Self Esteem Helps Children Respect Differences
Differences among the students can also take away from a student’s education. Most importantly, this process of helping students deal with each other starts from within. “When children respect themselves and others, they feel good about who they are. When children learn to value people who are different from themselves, they are better prepared to live peacefully in a diverse world.” Kids who have no self worth or esteem want no part in respecting differences. It is logical to think that children who are uncomfortable with themselves are very uncomfortable with being different. However, it is also noted that those obvious differences could be an escape or a way of connecting with others. When a child sees another who is a different race or a different ethnicity, he or she could see this as an opportunity, or a chance of making a connection with another student who, in his or her eyes, is different too. However, is it good for those children see this as a way out of their discomfort? In most peoples eyes' the answer is no.

Special Education
Another major difference in education is special education. IEP’s and 504’s give some students special attention, which creates a distinct difference between students. These kids get special treatment and most of the time it is rightfully so. However, these kids are less likely to want to succeed when they have a teacher reading them a test in the classroom, or if they are wearing headphones to listen to a recorded test. What makes the situation even more frustrating is that sometimes it does not help the student. This could be because of the negativity the student may feel from the other kids, or it could be from lack of support at home. “When I ask parents how their child’s progress toward goals and objectives is being monitored and reported to them, most often the response is ‘I’m not sure’ or ‘I don’t know’”. This is sad because we are creating many differences among children, yet we are not utilizing the choice we are making.I believe that the 504 plans are very helpful, “The concept and practice of "inclusion" has resulted in new challenges for teachers, administration, and parents to provide effective educational experiences for students”. In doing my observation I found that one of the students was on the 504 plan and she was very embarrassed by it. I feel that we as teachers need to stress the fact that the students are as smart as everyone, they just need a little more assistance. The young girl felt as if she was not as smart as the rest of the class, but she was very intelligent. Not only is it useful to the students, but to the teacher as well because it makes it easier to teach that student knowing that particular child need more attention. If they did not have this plan, then the teacher would be holding up the rest of the class trying to make sure that that one student gets the material, but with the 504 plan it is already established when they go into that classroom.

Conclusion
Differences throughout the world are prevalent, but none more so than in education. Whether it be race, gender, or education levels and capabilities, these differences remain. Differences are two fold, mainly because they could help children in so many ways yet hurt or cause them stress in others. If we can be successful in helping today’s children understand that being different is acceptable, then we could help them see that once they appreciate themselves then we, as teachers, would be helping countless kids everyday.

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Logic for Computer Scientists/Predicate Logic/Predicate Logic

Predicate Logic
When we introduced propositional logic we argued in treating the electrical circuit example (circuit), that a string like $$high(inv1, o)$$ can be read intuitively as "the output of inverter 1 is high", but then we abstracted from this internal structure of the proposition and further on we represented the proposition by a simple string like  "the_output_of_inverter_1_ is_high". In the formal parts of the calculus we even assumed more abstract just a given countable set of propositions $$\{ P_1, \cdots, P_i, \cdots \}$$. Until now, we investigated a logical language that was able to deal with connecting these propositions, In this section we will have a closer look into the structure of propositions. We will be able to explicitly refer to objects like the inverter $$inv1$$ or its output $$o$$ in the sentences of our language,. Besides this domain, we will introduce the new concept of a variable, which stands for an arbitrary object of our domain. Hence, we will be able to write e.g. $$high(x, o)$$ to denote the fact that the output of a $$x$$ is high. As a consequence of this concept we will have quantifiers, which allow for sentences like $$\exists x \; high(x,o)$$, with an intended meaning that there exists an object $$x$$, whose output $$o$$ is high, or $$\forall x \; high(x,o)$$ with an intended meaning that for all objects $$x$$ output $$o$$ is high,

Together with these new concepts it will be possible to express rather abstract properties of elements from the domain. Assume, e.g. that we want to introduce the concept of elements in a circuit to be connected. We could say $$ connected(inv1, inv2)$$ and $$connected(inv2, inv3)$$. Now, with the intended meaning of being connected, we can assume, that $$inv1$$ and $$inv3$$ are connected as well. Of course, we could introduce the new sentence $$connected(inv1, inv3)$$ to express this; but the disadvantage is obvious: this have to be done for all objects from our domain. In predicate logic this property of transitivity of connectedness can be expressed without referring to explicit objects by the following sentence: $$ \forall x \forall y \forall z ((connected(x,y) \land connected(y,z) ) \to connected(x,z)) $$


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Logic for Computer Scientists/Predicate Logic/Syntax

Definition 1 (Syntax of predicate logic - Terms)
Assume a The set of terms is defined by the following induction:
 * countable set of function symbols $$\{ f{_i}^k \mid i,k =1,2,3,\cdots \}$$
 * a countable set of $$\{ x_i \mid i = 1,2,3, \cdots\}$$
 * Variables $$x_i$$ are terms.
 * If $$t_1, \cdots, t_k$$ are terms and $$f_i^k$$ is a function symbol, then $$f_i^k(t_1, \cdots, t_k)$$ is a term.

Terms of type $$f_i^0$$ are special ones, they are called constants. In this case we omit the braces and denote them as $$f_i^0$$.

Terms are the syntactic counterpart of the above mentioned objects. Constants will denote the elements of the domain and function symbols will denote a way to refer to such objects.

The following definition introduces the formulae.

Definition 2 (Syntax of predicate logic - Formulae)
Assume a countable set of predicate symbols$$ \{ p_i^k \mid i = 1,2,3, \cdots\} $$. The set of (well-formed) formulae is defined by the following induction: Formulae of type $$P_i^k(t_1, \cdots, t_k)$$ are called atoms or atomic formulae.
 * If $$t_1,\cdots, t_k$$ are terms and $$P_i^k$$ is a predicate   symbol, then  $$P_i^k(t_1, \cdots, t_k)$$ is a formula.
 * If $$F$$ and $$G$$ are formulae, then $$(F\land G)$$ and $$(F\lor G)$$  are formulae.
 * If $$F$$ is a formula, then $$\lnot F$$ is a formula.
 * If $$x$$ is a variable and $$F$$ a formula, then $$\forall x F$$ and   $$\exists x F$$ are formulae.

Note that the concept of subformulae applies exactly like in the propositional case (Syntax (Propositional Logic)).

We introduce the following abbreviations, which will be used with indices as well:

Note the arity of function and predicate symbols is ommited in these abbreviations; we assume that it will be obvious from the context.

Example: Assume we want to represent the following equation, which holds for arbitrary elements in a field: $$ x * (y + z) = x*y +x*z $$ The two operators $$*$$ and $$+$$ are represented in a predicate logic formula as binary function symbols $$f_1^2$$ and $$f_2^2$$, the three variables are $$x_1, x_2$$ and $$x_3$$, and the equality relation $$=$$ is the binary predicate symbol $$P_1^2$$. Altogether we have the following formula in predicate logic: $$ \forall x_1 \forall x_2 \forall x_3 (P_1^2(f_2^2(x_1, f_1^2(x_2,x_3)), f_1^2(f_2^2(x_1,x_2), f_2^2(x_1,x_3)))) $$ In the following we will use the obvious and more liberal notation as in the propositional case.

Definition 3: Bound and Free Occurrences of Variables
An occurrence of a variable $$x$$ in a formula $$F$$ is called bound, if it occurs in a subformula of $$F$$ which is of the form $$\exists x \;G$$ or $$ \forall x\;G$$. Otherwise we call the occurrence free.

A formula, which does not contain a free occurrence of a variable is called closed.

Example: The following formula contains both free and bound occurrences of $$x$$ and $$y$$.

$$\forall z(Q(z) \land \forall x(P(x,y)) \lor \exists y(P(x, y)))$$


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Social and Cultural Foundations of American Education/Assessment/Testing

Are these statements reasonable or unreasonable? That question was asked of each student in Mrs. Platt’s fifth grade classroom, followed by a moderated group discussion about the topics. Having an open discussion required the students to see things from different points of view. Getting her troublesome student, Gertrude, to stand up front and moderate the conversation, helped get Gertrude engaged in the classroom and made her gain a new respect for the teacher (Myers, R.E., Torrance, E.P., 1970). Is it important for students to learn creativity, and working with abstract ideas, like Mrs. Platt tried to do? What are the benefits of this type of learning, and if any, how do we test it?

Importance of Creativity and Teamwork
It is of course hoped that experiences in creative thinking at their own level will help students develop a characteristic way of looking at things that will, for some individuals, ultimately result in the creation of ideas and products that are truly original and useful for the culture at large (Osborn, 1963, Cited: Ford, B.G, Renzulli, M.J., 1976). In life, being creative and looking at abstracts from numerous perspectives is imperative. Not only does working creatively with teams help children maneuver through life, but it will be a tool needed in many of the areas they choose to work after high school and in college.

Preparation
In countless professions, such as marketing, entertainment, science, engineering, and entrepreneurship, thinking “Outside of the box” is necessary for success. There is a good deal of research evidence which shows that persons who have engaged in systematic creativity training exercises can be expected to increase their capacity for creative thinking in a variety of fields (Osborn, 1963, Cited: Ford, B.G, Renzulli, M.J.,1976). There have been special teachers throughout the years, that we (Generation X, Y, and M) can remember, that used creative projects and student-centered work to teach our classes. But at one time student-centered work was rare, and teacher lead direct style teaching was dominant in the schools. You will still find direct style teaching within the classroom today. This can be a classroom mangagement or control issue. If a lesson is well thought out, the teacher can incorporate creative group work and still be able to maintain classroom control. The extra time spent to prepare equipment and special lessons are worth the effort. Studies have shown students retained more of what they learned when they were actively involved in assignments. This suggest Student-centered and hands-on lessons can be more productive than teacher-centered lessons. Rob Sternberg, a Yale Psychology professor, came up with the “Rainbow Project” which provided a supplementary test of analytical, practical, and creative skills to augment the SAT in predicting validity for college GPA relative to high school GPA and the SAT and decreased ethnic-group disparities in test scores (Cavanagn, S., 2003). His premise is the belief that the SAT, measures primarily memory and analytical ability, rather than creative or practical skills. He has also come up with the “Aurora Project” that uses creativity and problem solving to identify gifted students. WIC, acronym for Wisdom, Intelligence, and Creativity- which was also designed for college admissions offices to use to assess a student in a way that is beyond just academics. WIC suggest attributes like good citizenship and positive leadership should be considered along side grades and test scores (Sternberg, R.J., 2007). Since college admissions incorporate creative, practical, and elaborative thinking, shouldn’t we as teachers at primary levels, teach and prepare our students with those skills?

Equalization
One study called “Effects of Cooperative Learning and Perceived status of Male and Female Pupils” found varying results, depending on if the students were put in groups with or without having designated roles for each student. When put into groups to work on a problem solving project with no assigned duties for each team member, the male students tended to dominant the team overtime and the female students played less of a role in the decisions of the team. But once teams were divided and each person had a part to contribute to the team, an equalizing effect occurred. When the students had to cooperate to accomplish the task, overtime there were no significant differences between female and male group members in achievement, verbal participation, perceived leadership, and status(Peterson, R.P., p.717-735). This study proves, that teamwork with individual accountability is crucial for developing critical thinking and problem solving skills among boys and girls. Another study, called PACER, done among a 6th grade Physical Education classroom, found when working in teams low skilled students performed as well as their average and highly skilled counterparts. The study also found that cooperative efforts resulted in higher achievement on higher level learning task, than individualistic efforts did. As well, engaging in and receiving verbal comments are significantly related with achievement (Barrett, T., 2005).

Peer Acceptance
Besides bridging gender boundaries and higher achievement, teamwork and cooperative learning can also facilitate peer acceptance of students with disabilities. Studies have shown problem solving and group work create confidence among diverse types of students. That is why a set of alternative instructional methods, called CL, were developed. CL stands for Cooperative Learning. With CL, students, in general and special education settings, work in small groups to help one another learn academic material (Johnson, 1981; Kagan, 1992; Slavin, 1995). CL has demonstrated academic gains. For instance, it increased student achievement and social gains, such as improved inter group relations, acceptance of academically handicapped classmates, and increased self esteem (Johnson et al, 1981). So as we can see, creative and collaborative learning not only help students learn, but can also help with social acceptance issues, and confidence.

Lesson Plans
Incorporating creative and abstract thinking into lesson plans engages the students. It can change:

 * Troublesome behavior into outstanding performance
 * Apathy to enthusiasm about learning
 * Lack of self expression to creative self expression
 * Non-reader to average or superior reader (Myers, R.E., Torrance, E.P., 1970)

Since we now know creativity, working with abstracts, and teams are essential to a well rounded student; how should we create lesson plans and test to incorporate it into the classroom? In order for students to reap the benefits of thinking abstract, teachers must be creative themselves while making up their lesson plans, quizzes, and test. Lesson plans should work up to the test. For teachers the creative process should start when brainstorming and writing lesson plans and exams. Some suggest writing your final exam first, then writing the lesson plans to incorporate everything that you want your students to retain from the semester(Bartel, Marvin, 2006). Think about what your objectives for the class are, then add the imaginative formats for teaching them. Adding versatility in your lesson plans by using technology, skits, and team projects, keeps the students attentive. There are no longer any excuses to have pupils just sitting listening to a boring lectures everyday!

Testing
One format for testing abstract thinking can be open ended questions, which are questions with no real definite answer. You can give higher grades for less ordinary answers. These questions can allow the student to elaborate. In the “Rainbow Project” students were asked to write 2 stories, taking 15 minutes each, based on a list of fictitious titles. A team of judges rated those stories on originality, evocativeness, complexity, and descriptiveness. In another section, the students were given cartoons from the New Yorker and were asked to write captions for them (Cavanagn, S.,2003). Various categories you can test are logical reasoning, ideational fluency, spontaneous flexibility, originality, elaboration, redefinition, sensitivity to problems, and judgment (Myers, R.E., Torrance, E.P., 2007). Torrance's Tests of Creative Thinking, is highly recommended in the educational field. You can use these pre-formatted test as a guide, or you can be clever and invent your own. See goshen.edu for ideas.

One benefit of making up your own test, is you can formulate team test. On example of assessing cohesiveness, & problem solving skills within teams could be: a class projects which requires each team member to find a piece of a puzzle in order to find the final answer. This could be used in a history class with historical facts on each piece or in English with different quotes on each section from one author. Then when the teams fit together the completed puzzles, they would reveal a shape or picture of something representing a major event in time or the outline of something representing the author. Then the students can guess, which event the picture, along with all the cumulatively individual facts on each piece represents. The teams that put the puzzles together and come up with the answer in the fastest time bracket get As, and second fastest bracket Bs, and so on down. You can make the clues more difficult for higher grade levels. Determining standards for each grade will be decided beforehand, providing the teacher with a guide to grade upon. The students should also know in advance the requirements and how the project will be graded. In addition, teachers can calculate in participation points for each individual’s contribution within the group.

Conclusion
With so many resources available to teachers today, the sky is not the limit for creativity in the classroom. I challenge you to expand your student’s minds. Press them to see perspectives bigger than themselves. Being creative might take more time to plan, but the end product will be well worth it. Who knows you might grow and learn something new yourself.

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Social and Cultural Foundations of American Education/Assessment/Standards

National standards play a critical role in today’s educational community. These standards shape what is taught to students, and how teachers teach. When developing national standards it is necessary to examine every view of education. This includes advanced programs including Advanced Placement and International Baccalaureate.

National Standards Ideals
In today’s education world there is a constant battle to decide what should be taught to students. Each different branch of education has a customized way of forming ideals upon which to focus teaching. Among the top programs in the American school system are the Advanced Placement program, and the International Baccalaureate program. When creating national standards these programs must be held in high regard. Not only do the highest high school students participate in this program, the teachers, with considerable extra training, show a true display of care for the students. Once these programs are looked upon intently national standards can be formed which challenge both teacher and student, and ultimately advanced education on a national basis.

International Baccalaureate Standpoint
The IBO states that educational standards are a key part in the development of sound students and teachers(IBO.org). In this regard IB provides an excellent base for national standards. The standard IB student completes a rigorous program, and is constantly challenged on every level. While the students travel through the program they are forced to meet objectives and solve problems accordingly. Within the IB program students reach above and beyond what is typically expected. Students learn on various levels, and are challenged accordingly in each area. As students advance in the program areas begin to overlap, and students are given the opportunity to understand how each subject area affects the others. With this overlap students understand how the learning system s not separate parts, but one giant unit. National standards which require students to be challenged, and reach above in beyond are constantly being discussed in America’s various educational communities.

Chris Fischer states “the International Baccalaureate Theory of Knowledge program provides students a complete view of education.” In this class students have the chance to examine each area of education. Students begin to realize that while learning is different for each area, the basic methods used are the same. While students learn of each area they begin to realize that the standards affecting each intertwine. They are also given the chance to examine the various ways of knowing. In examining the various ways of knowing students maintain an understanding of education as a whole. In this way students are pushed to understand how each content area interacts. Within each way of knowing is an intermittent part of the others. Each area in and of itself directly affects the others. Once students understand the standards of how these ways of knowing work they achieve a better understanding of their learning experience. These standards which can be transferred directly to the national system enhance the learning experience. In this portion of the program students begin to breakdown the walls that divide learning, and view the educational instrument as a whole. Through this portion of the International Baccalaureate program students understand that each standard directly affects the others.

Advanced Placement Standpoint
Looking upon the other side of the spectrum the Advanced Placement program also provides students an excellent chance to develop themselves on a higher level. In this aspect students once again are given the chance to develop standards which lead to their betterment. In the Advanced Placement program students are given college level work, and upon completion given the chance to receive college credit for their efforts. Since these advanced students are learning on a college level they are exposed to college-level material. Being that national standards must push the envelope in the world of education, it is pivotal for advanced placement to play a role in the creation of these standards. As students study in the AP program they are constantly forced to look outside the box, and approach problems full circle. This skill proves to be vital in the forming of any type of national standard. When education is concerned students must be pushed from all sides to form a complete package. If they are allowed to only approach problems from the most obvious viewpoint they are denied the experience of seeing problems through. Also they do not gain the chance to observe these from different angles, and achieve solutions which shy away from the norm. More often than not educational problems can’t be solved with the most obvious solution. By taking the Advanced Placement outlook students see problems in different way, and thus find more complete solutions.

National standards have always been a key part of education. In every educational community standards vary, and students learn slightly differently. However this strategy proves to be incomplete, and leads to varying educational levels across the country. In viewing this problem it is easily understood that some national standards need to be established in order to give a more complete educational plan. With both the International Baccalaureate and Advanced Placement programs students are challenged to be better than they currently are. Within this view we can see that national standards should be shaped around these programs. Not in that each student will be of the caliber of advanced students, but that students should be challenged in order to form complete views and ideals. When standards are made that require students to reach inside themselves, the ultimate result is students that are better than before. These standards create students who not only solve problems, but understand the issues surrounding the problems.

Conclusion
In the creation of national standards teachers and students must understand that these advanced programs provide a brilliant base upon which to build any educational program. Students who are taught with these standards develop as complete packages in the world’s eye. Teachers who teach with these standards not only better students, but better themselves. In teaching on a higher level, in reference to advanced programs, teachers inspire students, and provide a much needed advancement in the national educational standard. Students pushed to be better themselves ultimately become people who push for a better tomorrow. In establishing a national standard largely influenced by advanced programs there can only be an improvement in education as a whole. With teachers and students constantly pushed to better themselves the entire nation will improve.

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Social and Cultural Foundations of American Education/Acknowledgment/Intelligence

You are intelligent. Doesn’t that make you feel good about yourself? Intelligence is a highly valued quality that most people desire. But what exactly is it? Where does it come from? How is it measured? And most importantly for us, how does intelligence apply in the world of education? This article will address all of these questions.

Definition
Encarta Encyclopedia defines intelligence as a “term usually referring to a general mental capability to reason, solve problems, think abstractly, learn and understand new material, and profit from past experience(1).” Britannica Encyclopedia defines it as the “ability to adapt effectively to the environment, either by making a change in oneself or by changing the environment or finding a new one(2).” Even these definitions, found in widely accepted encyclopedias, are different from each other. Even if we suggest that the general American public would agree on the definition of intelligence, many scholars and researchers have differing opinions. As of yet, experts have not fully agreed on one particular definition of intelligence. Many questions remain as to the qualities, origins, and measurements of intelligence so a consensus is difficult and unlikely.

Cultural Considerations
At a fundamental level, many disagree as to whether intelligence is a general mental ability or if it varies between cultures. We know that different cultures value different things as representing intelligence (1). North American society values mathematical and verbal skills as intelligence markers while some seafaring cultures in the South Pacific value spatial and navigational skills(1). This article will not focus on cultural issues in intelligence, but it is worth noting that differences between cultures do exist and should be considered when assessing intelligence and intelligence tests.

Galton
Interest and research on intelligence began in the late 1800’s with the work of Sir Francis Galton who began to test people on qualities such as fame, awards, reaction time, and body proportions (1). James Cattell soon followed Galton’s ideas and developed over 50 tests that were intended to measure basic mental ability (1).

Stanford-Binet Test
The first intelligence measurement to predict school success was spurred on by the French government, looking for an objective way to determine which students were capable of formal schooling and which classified as mentally retarded and would need remedial help. As a result, French psychologists, Binet and Simon, developed a test measuring practical knowledge, memory, reasoning, vocabulary, and problem solving (1). Binet’s test was revised by Stanford psychologist, Lewis Terman and published in the United States as the Stanford-Binet test(1).

IQ
Terman first coined the term Intelligence Quotient (IQ) as one score which was a function of mental age divided by chronological age. While the term IQ is still used, it does not refer to the same equation. Modern IQ tests determine a person’s score based on the standard deviation (difference from the average person of that age) (1,2,6).

Currently used IQ tests include the Stanford-Binet, Wechsler Adult Intelligence Scale (WAIS), Wechsler Intelligence Scale for Children (WISC), Kaufman Assessment Battery for Children (Kaufman-ABC) (1). Individual IQ tests are expensive and time-consuming, and they can only be administered by specially trained psychologists. Group IQ tests, such as the Otis-Lennon School Ability Test and the Cognitive Abilities Test are more cost-effective and consequently are often used in schools (6).

The “g factor”
Charles Spearman developed an early intelligence theory that involved a general factor, often referred to as the “g factor”. The g factor was considered to be the underlying mental ability that would influence performance on any intelligence test (1).

Primary Mental Abilities
As the g factor was questioned, the next idea was a theory of primary mental abilities, specific to different areas of ability, including verbal comprehension, verbal fluency, number, spatial visualization, inductive reasoning, memory, and perceptual speed (2).

Fluid Intelligence and Crystallized Intelligence
Fluid intelligence is the biological basis of intelligence, involving reasoning ability, memory capacity, and speed of information processing (1,6). Crystallized intelligence involves the knowledge and skills gained through learning (1,6). Since crystallized and fluid intelligence are both measured on most intelligence tests, they are correlated with each other. Consequently, some see this theory as supportive of Spearman’s g factor.

Multiple Intelligences
Recently, there has been an increased focus on the idea of different areas of intelligence(6). In 1983, Howard Gardner proposed a concept of intelligence that includes eight separate intelligences: logical-mathematical, linguistic, musical, spatial, bodily-kinesthetic, interpersonal, intrapersonal, and naturalist. This theory of Multiple Intelligences was not focused on IQ tests but rather on explaining the wide range of abilities that people display(6).

The many critics of Gardner’s theory point to his lack of formal measurements for the separate intelligences, little supporting scientific data, and a negligence of the previous, well-established research suggesting the presence of a g factor(1,5,6,7). Despite all the criticism, the theory of Multiple Intelligences was widely accepted among educators because it gave a new perspective on the way school should be done(5). Perhaps the “slow”, low-achieving students simply weren’t given the right opportunities to display their intelligence in areas other than the subjects traditionally emphasized in school. This expanded the concept of intelligence and suggested the possibility that the traditional curriculum was much too limited. Some schools have attempted to design and implement curricula that assess and develop all eight of Gardner’s multiple intelligences (1).

Some teachers have assessed the effectiveness of their own classroom activities by analyzing how many of the multiple intelligences are addressed in each activity (3). If an activity involves linguistic, spatial, and bodily-kinesthetic intelligences, then it would be considered more effective than an activity that only measures logical-mathematical intelligence.

Triarchic Theory
The triarchic theory is a demonstration of the cognitive perspective of intelligence, focusing on how people use their intelligence (6). Developed by Robert Sternberg, the triarchic theory of human intelligence suggests three types of intelligence: analytical, practical, and creative (6).

Emotional Intelligence
Emotional intelligence “consists of the ability to perceive and express emotion, assimilate emotion in thought, understand and reason with emotion, and regulate emotion (6).” The concept of emotional intelligence was developed in 1990 by Salovey and Mayer, and popularized in 1995 when Daniel Goleman wrote a best-seller entitled Emotional Intelligence. Goreman credited EQ for being 80% responsible for the amount of success an individual will have in their lifetime, with the remaining 20% being dependent upon IQ. Professor Chan Chieh has defined the five major qualities of EQ as being “self awareness, mood management, self-motivation, impulse control and interpersonal skills”. The Multifactor Emotional Intelligence Scale (MEIS) is the most empirically supported of many emotional intelligence tests (6).

EQ and IQ are not dependent upon one another. They also differ in the fact that an individual’s EQ can be “nurtured and significantly strengthen and it is never too late for students to improve.” (Chieh) The challenge as a teacher is to foster that development in a traditional classroom. One option is to have allotted periods or courses that concentrate on self-study or personal awareness. Another option, that is far easier to implement within existing curriculum, is group work. Group is especially effective in EQ development when it is designed to promote creative thinking and intuition.

Origins of Intelligence: Nature vs. Nurture
The debate as to whether genetic factors or environmental factors are responsible for intelligence has gone on for many years(1,2). Many studies, particularly twin studies, point to the influence of genetics while many other studies, including adoption studies, environmental deprivation and enrichment studies, and home environment studies point to the influence of environment on intelligence (6). Both nature and nurture have been scientifically established to have a significant impact on intelligence. The exact percentages of influence can be estimated, but it is clearly a complex formula involving both genetic influences and environmental influences.

Why is Intelligence Important to the Field of Education?
As educators, we must be aware of the different theories on intelligence because they influence the way we see, teach, and test our students. That’s not to mention the fact that intelligence theories can influence the entire school system, as we have seen with Gardner’s theory of multiple intelligences. As teachers and administrators, we must acknowledge the influence of both nature and nurture so that we can be sensitive to children’s biological capability and also determined to teach them well and push them to their highest potential. It is not necessary that we all agree on which theory of intelligence to adopt, but we must remain open to different concepts of how children’s minds work. We all desire to teach well and to positively influence our students. Being aware of intelligence theories can help us do that effectively so that we know how we might reach our students and tap into their potential.

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Social and Cultural Foundations of American Education/Choice/For Home Schooling

Given the enormous benefits and relatively inconsequential sacrifices of home schooling, one could reasonably argue that there more needs to be a case for public schooling than home schooling. However, America has had an historical propensity to question and resist anything that involves change, or that does not comply with the societal norm. Compulsory attendance laws, while initially passed with the intent of making sure every child was afforded an education, have had the effect of qualifying the institution of public school as the norm, thereby making anything other than public schooling “abnormal.” This is reinforced by the fact that any student not attending public school, including those enrolled in private schools, must by law notify their public school district of their choice to pursue an “alternative education program.” (Code of VA § 22.1-254) The monolith that is the National Education Association also propagates the sophomoric notion that public school is the only proper way to educate, and espouses these ideals in the tenets of its Code of Ethics, which states in Principle II: Commitment to the Profession that “In the belief that the quality of the services of the education profession directly influences the nation and its citizens, the educator shall exert every effort …to assist in preventing the practice of the profession by unqualified persons. In fulfillment of the obligation to the profession, the educator--5. Shall not assist a noneducator in the unauthorized practice of teaching.”  (NEA, 1975) Thanks to NCLB, the only “noneducators” that are currently engaged in the practice of teaching are uncertified parents who have chosen to homeschool.

Parental Rights: Father Knows Best vs. Big Brother is Watching
Central to the homeschooling debate is the inherent question, “Who knows what is best for each American student; their parent(s) or the public school system?” Any good parent will tell you that they know what is best for their own child, which is actually counter to the thinking among many public school officials who think they know what is best for the child: after all, public school officials have been “trained.” Author Jennifer Kaufield (2002, p.10) encourages parents, “No matter what your reasons are for wishing to homeschool your children, if they center on what’s best for your family right now, then your reasons are valid and worth pursuing. Home education is all about meeting your child’s needs.”

Education in America: A Historical Perspective
The debate surrounding homeschooling is filled with irony, literally from beginning to end. In early America, almost every child was educated due to the colonists’ desire for their children to be able to read the Scriptures, and parents saw it as their responsibility, not that of the government, to provide education. In fact, The Boston Latin School, the first school established outside the home in 1636, was formed for disadvantaged children and orphans who had no parents to educate them. (Beliles & McDowell, 1989, p.102-103) This line of thinking has now been reversed, with the majority of parents placing the burden of educating their children squarely on the government.

Homeschooling was the prevalent means of formal education until the advent of compulsory public school attendance in the 1870s. (Beliles & McDowell, 1989, p.102-103 and Scheuer, 2004). In 1925, the U.S. Supreme Court’s landmark decision in Pierce v Society of Sisters essentially struck down compulsory attendance laws which required ALL children to attend ONLY public schools, and guaranteed a parents right to choose their children’s form of education; be it public or private, or by modern extension, home schooled. (Pierce v Society of Sisters, 1925).

Homeschooling was still illegal in about half the states during much of the 20th century, but today it is legal nationwide. The current, modern wave of homeschooling is primarily associated with and driven by conservative evangelical Christians, who since the 1980s are increasingly choosing to home school their children in response to what they see as an increasingly secular, anti-Christian public school system. However, the modern home school movement was actually rejuvenated in the 1960s by Liberal parents and the anti-establishment counterculture movement, which rejected the rigid structure and conservative nature of schools at the time. One of its early proponents and pioneers was John Holt, an influential critic of mainstream education, who also happened to be...an atheist. (Scheuer, 2004) Recently, the home school pendulum has begun to swing back to the left. Although most homeschool parents define themselves as Christian, that is not necessarily the primary reason for homeschooling. According to the US Census Bureau and other sources, in 2000 only 33 percent of homeschooling parents cited religion as a reason for homeschooling (and many of these parents cited other reasons as well). On the other hand, 50.8 percent cited a belief that their children could get a better education through homeschooling, while 29.8 percent stated that school offers a poor learning environment, and 11.5 percent said that their children were not being challenged in school. (Gathercole, 2005)

No Child Left Behind vs. My Child as a Guinea Pig
By all accounts, the public school system is in need of major repair and overhaul. The most lauded and concurrently disdained issue in public schools today is the No Child Left Behind act. While noble in its aspirations of accountability, NCLB has now created an entirely new dynamic of problems. Passing the SOL test and meeting AYP has now become the focus of public schools because a school’s future and the livelihood of its employees literally depend on it. Many public school teachers are now out of necessity and self preservation “teaching the test.”

“The Principle Approach” established by Noah Webster as the standard method of education using the principles of research, reasoning, relating, and recording (the “4-Rs”) has been all but forgotten in modern public schools. Until the 1920s, essays were the primary method of testing; now, most modern tests are fill-in-the-blank, multiple choice, matching, or True-False. Students can take and pass these tests for years without truly learning how to reason and think and be prepared for life after completion of school. (Beliles & McDowell, p.98-100) The truest test of knowledge is application, and all we can ascertain from NCLB is that public school students can pass a standardized test, and life, as we all know, is not standardized. With a child’s future in the balance, is it not reasonable for a parent to remove their child from this admittedly flawed system rather than sending them in and “hoping for the best?” In the past two decades the growth rate of homeschooling has increased enormously. Disillusioned parents have been pulling their children out of the secular and in many instances substandard public education system, taking education into their own hands. (Scheuer, 2004)

Religious Freedom
In 1987, a seminal case in the arena of parental rights, Mozert v. Hawkins County Public Schools, was argued before the Sixth Circuit Court of Appeals. The case was brought by the parents of several students who were expelled because they refused to read a series of reading books that violated their religious beliefs. The suit sought to have the students re-instated and allowed an alternative reading assignment. The central issue in the case was: Do parents and children who have sincere religious objections to public school curriculum have the right to alternative instruction for the objectionable material? The answer given by the Sixth Circuit Court of Appeals was clear: Parents have no such right. Once a child has been submitted to the public schools for his education, parents lose all ability to control the course of instruction. (Farris, 2006)

America began as a country whose oldest and greatest institutions of higher learning, among them Harvard 1638, William & Mary 1693, Yale 1701, Princeton 1746, and Rutgers 1766, were founded with the sole purpose of keeping God as the centerpiece of education. (Beliles & McDowell, 1989, p.111) America has since become a country who has removed public school prayer and through the ever broadening interpretation of the “Separation of Church and State,” become increasingly secular and anti-religion. If, as a parent, you support this direction, that is your right. However, the ONLY recourse for a conscientious parent who objects to curriculum for religious reasons is to withdraw their child from public school, and if a private school alternative does not exist, the only reasonable option is homeschooling.

Socialization
Among veteran homeschooling circles, the issue of socialization is referred to simply as “The Question;” as in, have you been asked “The Question” yet: What about your child’s socialization? Before answering this question, the term “socialization” must first be defined: is this an inquiry into a student’s social outlets, or into the more abstract concept of assimilation into society? (Kaufield, 2002, p 29)

Social Outlets: Homeschoolers Aren’t Locked in the Dungeon!
Contrary to popular belief, homeschoolers are often engaged in MORE social outlets than their peers. In part because they receive individualized instruction, they often complete and master their school work in a fraction of the time required in public school. In addition, their often flexible schedule allows for more extracurricular activities. The number of clubs, activities, and social outlets is so numerous, that parents of homeschoolers often need to be reminded not to over-schedule their children to fulfill some nay-sayers delusion that their children aren’t being “socialized.” (Kaufield 2002, p 30, and Gathercole, 2005)

Assimilation into Society
According to Manfred Zysk (1999), “One of the silliest and most annoying comments made to homeschooling parents is, "Aren't you concerned about how your child will be able to socialize with others?" What is being implied here is that the homeschooled child is some kind of introverted misfit who cannot relate to other people, children, and the outside world. In reality, most of the homeschooled children that I have known and met are not only outgoing, but polite and respectful, too. This is a sharp contrast to the public school children that I have known, who can't relate to adults and whose behavior is rude and inconsiderate…in public school, children are segregated by age, and have very little interaction with other adults, except their teacher(s). This environment only promotes alienation from different age groups, especially adults. This is beginning to look like the real socialization problem…homeschooling parents choose to homeschool for a variety of reasons, but I have never heard any homeschooling parent say that the reason they want to homeschool is to isolate their child from all of society. But, it probably wouldn't be a bad idea for homeschooled children to stay away from public school administrators, the NEA members, sociologists, and others who cannot properly "socialize" with children. From all of the research gathered for this article, this rather terse response sums up the general feeling of frustration felt by the majority of home school parents on the subject of socialization. Quite simply, home schooled students are generally better adapted and socialized to society than their public school peers. (Kaufeld, 2002, p33)

Athletic Clubs vs. Interscholastic Athletics
Although many professional educators disapprove of homeschooling and claim that it limits children's exposure to peers or people of different cultures, current laws in 16 states allow homeschoolers "equal access" to activities like music and interscholastic sports at local public high schools. (Scheuer, 2004) In Virginia, the Virginia High School League which governs public school athletics & the Virginia Independent School Athletic Association which governs private school athletics effectively deny homeschoolers the right to participate in interscholastic sports. (personal correspondence, Zimorski, 2007, and VISAA.org, 2007) In Virginia, this is a legitimate sacrifice that homeschoolers must weigh in their decision to homeschool. If one is pursuing an athletic career, this door of opportunity is essentially closed in Virginia. However, these are not the only opportunities for an athletic career in Virginia. Certain athletic clubs more than welcome homeschoolers and in some instances their devotion to the sport requires it. Figure skating is a case in point. A home school co-op group meets at the Hampton Roads Iceplex in Yorktown, VA for figure skaters whose dreams of pursuing Olympic glory require inordinate amounts of time at the rink, which also includes classical ballet training. (personal interview, Caldwell, 2004) Other “club” sports such as ice hockey and soccer have college and professional scouts in attendance at many elite level tournaments. (personal interview, Tucker, 2007, and personal interview, Scrofani, 2004) Home schooling associations are growing to the point where athletic opportunities inside the national home school community are growing as well. The National Christian Homeschool Basketball Championship drew over 200 teams from 25 states to its 2003 tournament. (Scheuer, 2004) As these nationwide tournaments grow, it is only logical that more college and pro scouts will be in attendance.

Co-ops vs. The Classroom
Cooperatives allow groups of homeschoolers to unite and pool their resources. Often, classes will be taught by experts, professors, and/or licensed teachers for subjects ranging from science to woodshop. The classes are usually dynamic, with a focus on learning the subject matter rather than just passing the test. (homeschoolrichmond.com, 2007) The Commonwealth of Virginia is arguably the most historically rich state in the Union. All but forgotten in today’s NCLB/SOL crush are field trips to enhance the learning experience of public school students. Many classes are allowed only one or two field trips per year, and many more never leave the confines of the school building. Conversely, the experiential learning opportunities for homeschoolers through field trips and Co-ops are boundless. (VA is for Field Trips, 2007)

Technology: The Great Equalizer
One of the greatest perceived hindrances to the homeschool movement is its lack of “qualified” parents fulfilling the teacher’s role. With modern technology, homeschoolers are often enrolling in college classes at a very early age. After all, where does a child go after they have completed calculus at 12 years old? (Kaufield, 2002, p14) Cyberspace offers online courses, chat-rooms, and websites devoted to homeschooling. The teacher standing in front of the blackboard is no longer the only acceptable method of teaching. With the tremendous and varied amount of online resources available, parents are finding it easier than ever to make the decision to home school.(Scheuer, 2004) This very wikibook is a testament to the advances in technology and changes to the standard methodology used in education. Logically, one can deduce that it is much easier for a home schooled student receiving individualized and personalized instruction to access these opportunities than those limited by the classroom methodology of public schools.

University Admissions
Another misconception applied to homeschooling is that most colleges and universities will not accept home schooled students. That is just flat wrong. Homeschoolers are now just as likely as other students to attend college. Increasing numbers of homeschooled students are entering even the most elite universities. Harvard's acceptance rate for homeschooled applicants is about 10 percent; the same rate of acceptance as for traditionally schooled applicants. (Scheuer, 2004) In the fall of 1999, Stanford accepted 27% of homeschool applicants, nearly double the rate of traditional students. (Goldman, 2000) Higher admission rates are due in part to new, more accepting attitudes toward homeschooled applicants in college admissions offices. A January 2003 report by the National Association of College Admission Counselors in Alexandria, Virginia showed that 74 percent of colleges surveyed had formal policies to judge homeschool credentials, up from 52 percent in 2000 (Scheuer, 2004).

In 2006, the overwhelming majority of colleges and universities now have either a dedicated homeschool admissions officer or a homeschool admissions policy. In fact, since many colleges have experienced the quality of homeschoolers they often actively pursue homeschooled students (Homeschool Enrollment in Colleges and Universities, 2006). The Home School Legal Defense Association has instituted a tier ratings scale for colleges and universities based on their admission policies regarding home schooled students, with Tier 1 being basically equal to the requirements impressed upon public/private school students, Tier 2 requiring a GED in lieu of a high school diploma, and Tier 3 requiring higher standards (i.e. extra testing (SAT II) or higher scores on standardized tests) for home schooled students than traditional students. The vast majority of colleges and universities in the US are rated at Tier 1, while only Five Towns College in Dix Hills, New York, flatly denies admission to home school students (Rating Colleges & Universities by their Home School Admission Policies, 2007)

Conclusion
In conclusion, making the case for homeschooling is fairly easy. It is, without a doubt, a legitimate, and more often than not, superior alternative to public school. Home schooled students consistently outperform their public schooled peers in EVERY academic area. (Home Schooling Statistics, 2007) Public schooling follows the fatal flaw of credentialism; that only “certified” teachers have the ability to teach. Homeschooling’s individualized, personalized, and usually multi-faceted strategy of education simply outperforms the current methodology used by the public school system.

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Social and Cultural Foundations of American Education/Choice/Against Home Schooling

Tommy and Sara are both second graders and they both live in the same neighborhood. They get up every morning and attend school; however, they do not go to the same school. Tommy attends an elementary school nearby. He sits in a class with his fellow classmates, eats lunch with his friends, and plays on the swings during recess. When he comes home, he tells his parents about what he learned and how his day went. Sara, on the other hand, is learning at home. She gets up, attends a teaching session with her mom, eats lunch with her family, and plays on the computer in her room. Why did her parents choose to teach Sara at home?

According to the Webster's dictionary, home schooling is the act of learning in your own home, not at an institute. Instead of riding a bus to school and being taught by a teacher, you are taught at home with the rules and regulations created by your parent. Usually the parent is the teacher, and they choose the curriculum, pace, and instruction of the child’s learning.

Home schooling is a rapidly growing process in the U.S. The population of K-12 students has grown from 850,000 in 1993 to 1.1 million in 2003 (Green 2007). The movement was developed in the 1960s and 70s in order to ?despise? public school systems, and encourage parents to seek control over their children’s education and future (Collom 2005). Though the idea began in the 60s and 70s, home schooling was not legal in all 50 states until 1993 (Green 2007). The movement has pushed more and more parents to teach their children at home, instead of at school.

Ideologues
Parents in this category believe that the school system is not teaching what needs to be taught, for lack of better words. Today, the school system is trying to steer away from religious acts because of the growing diversity in the classroom. While most parents support this, others are outraged because they are not teaching religious values and character development. There becomes further difficulty with the teaching of such things as evolution or even dinosaurs because it goes against the beliefs of certain religions. A family has certain beliefs, values, and skills that they believe their children should learn. The religious belief seems to be the most influential problem when taking a child out of school and into their own home. "75% of homeschool educators are conservative Christians who stress Biblical values and doctrine for their children and tend to believe that public schools are not able to teach the values they want their children to learn." (Green 2007) The school system does not practice or emphasize their certain beliefs; therefore, home schooling allows the parent to teach their religious views to their children. Also, the children may practice religion throughout the day without getting in trouble or being criticized.

Pedagogues
Contrary to the ideologues, parents are not concerned with what is taught, but how the information is presented. In other words, “they believe that whatever public schools teach, they teach ineptly”(Romanowski, 2005). Pedagogues also include parents of special needs children that were not receiving what they consider to be appropriate care and education for their children. Pedagogy and special needs are the growing motivator for parents to begin home schooling (Green 2007). Teachers in the school systems are not able to look at their particular child and teach them the unique teaching style that would suit them the best. In result, home schooling will allow the parent to focus on what the child’s best learning strategy is and focus on their needs. This also gives the parent the freedom to create the curriculum and pace throughout the day.

The Dangers
Parents would rather teach their children at home in order to keep them safe. Who wouldn’t? These dangers include: bomb threats, drugs, large classrooms, cliques, peer pressure, bullies, etc. Keeping your child at home will not only reduce the risk, but keep them away from the exposure as well. This is home schooling motivated by fear.

Qualification
An argument against homeschooling is the qualification of the teacher. First of all, is the parent even qualified to teach children? According to Romanowski (2001), ten states require that a parent who teaches their child have a high school diploma or equivalent. Does that mean that some parents do not have to have a high school diploma? This creates many question regarding professionalism, organization, and the teaching skills to teach their children properly and efficiently(Romanowsk, 2005). Teaching is a big responsibility and children deserve the best.

Socialization
Social concerns are one of the main disadvantages. A child needs to develop social skills and interact with their peers in order to succeed in life and be prepared for the real world. By being at home all the time, the child is “sheltered” from the real world and are not presented with opportunities for question, debate, or a maybe even an alternate viewpoint. The classroom offers a chance to hear different ideas, questions, interaction, and discussions(Romanowski, 2005). When being home schooled, the child can only hear the views of their parents. There are online courses the child can take, but they will never be able to experience a heated discussion. These are qualities that cannot be obtained at home.

Diversity
Along with the socialization, the child does not encounter diversity. A child does not interact with diversity in their home, unless they are on a computer. Classrooms have diversity from gender, race, sex, to ethnicity. A student needs to be exposed to different beliefs and backgrounds in order to create his own original self. Even though home school children do

Cost
Naturally, if one of the parents is teaching a child all day, they are not employed. Instead of making an income and paying for school, the parent is only spending money to provide the adequate education and resources. These are only a few of the items: software, videos, books, record books, paper, supplies, field trips, etc. Another cost other than money, is the cost of time. Many families lack the time to put in full responsibility and effort into the child’s education. Distractions will occur throughout the day such as: phones ringing, baby’s crying, siblings fighting, unexpected visitors, emergencies, etc. Dodging distraction becomes a task in itself.

Knowledge
This follows up on the qualification of the parent. Does the parent know enough about a subject in order to teach it? What if both the child and the parent do not understand the subject? Or what if the child had the opportunity to excel, but the parent could not teach any further? These questions are very important to consider, because if the problem arises, then the parent will be its own bias. The parent will then be limiting their own child from furthering their education.

Conclusion
In today’s world, every parent has to decide what is best for their child. Once the child is born, a common question is, where should my child go to school? An alternate option is to home school their child. With growing technology and resources, it is now becoming easier and more convenient to pick this option. It is important for parents to know the biases and especially the limitations in order to make a correct judgment on whether or not it will be best.

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