The Relation of Grammar to Critical Thinking: An argumentfor the reintroduction of traditional grammar at all levels

of education and for foreign language teaching as well

Recognizing that a stance in favor of the teaching of traditional grammar is just recently becoming popular again after forty years of disdain, this paper makes two argumentsin favor of the reintroduction of traditional grammar at alllevels of education. The first argument is from the value of a classical education and from the erroneous view that grammar is somehow too difficult or too marginally valuable to play a role in any standard curriculum, and the second isfrom recent studies in Critical Studies which can be used todemonstrate the crucial role grammar plays in Critical Thinking. Traditional grammar skills such as identifying parts of speech, parts of the sentence, and types of sentences and clauses as well as parsing sentences and manipulating different grammatical forms are core aspects ofcritical thinking and as such can serve a valuable function if taught in elementary, secondary, and post-secondary schools.

The first argument, the reappraisal of grammar as an aspect of any classical education: 1) because it is a part of classical education that goes back to the greeks romans and Egyptians, 2) it is just not that difficult,. It should strike any reasonably intelligent college graduate as patently absurd and shocking that any would entertain arguments such as grammar doesn’t help writing, or it is nota real life skill. It is as if we had let the rough kids smoking cigarettes on the far side of the playground set thecurriculum as well as the rationale for it. These are arguments you expect to hear from frustrated students not from responsible educators. The 1993 statement by the NCTE should reconned as a blackeye on American education and it should be recinded utterly, an apology offered, and heads rolled (post mortem if necessary).

The second argument, based on research into Critical Thinking, demonstrates that the core skills of traditional grammar are identical with the core skills of Critical Thinking.

PART 1: The Vilification of GrammarAlthough the teaching of grammar has been ignored, vilified,and argued against for the last forty years, there are stillmany who believe either intuitively or with their own justifications that the study of grammar provides students with important thi nking and analysis skills as well as a heightened awareness of their own language for writing and thinking and for foreign languages they may study. There isalso a recent and growing movement that supports the study of grammar and recognizes its value both as an important intellectual skill in itself and as a building block for or contributor to other intellectual disciplines. The new SAT,to debut in 2005, will include a 25 minute written essay anda 25 minute multiple choice test on grammar.

Studies which contradict these beliefs (Harris Hillock McQade Braddock, Lloyd-Jones, and Schoer 1963) have done little to dissuade many though they have silenced some. Theunderlying belief for all those who still support the teaching of grammar is that it is a core aspect of critical thinking and as such cannot and should not be ignored in elementary, secondary, or post-secondary schools. Many alsobelieve that grammar should also play a significant role in the teaching of foreign languages. The underlying conviction that the value of learning traditional grammar has to do with its connection to critical thinking is perhaps the most important factor. If it could be shown that teaching traditional grammar (parts of speech, parts ofthe sentence, sentence types, and so on) perhaps even sentence diagramming were closely linked to the development of critical thinking other explanations would have to be found for the research results that led to the removal of traditional grammar from many classrooms. Or at least new research would have to be undertaken to determine what and

how to teach traditional grammar in order to capitalize on its role in framing and developing critical thinking.

The arguments against some forms of grammar teaching such asthe memorization of abstract rules divorced from a practicalconnection to the rules usefulness as well as repetitive drills divorced from communicative relevance were well-founded. This argument however popular it has been over thepast forty years fails in significant ways to be convincing,not the least of which is the unstated belief that the teaching of grammar is such a burden on students that it is reasonable to drop it from the curriculum. The vilificationof grammar, whatever its excuses, simply does not hold up inthe face of the reality that grammar simply is not that difficult nor is it such a vast subject. Unlike algebra, trigonometry, history, political science and so forth, grammar represents a small body of knowledge with comparatively simple elements and rules. Eight parts of speech, the parts of a sentence, sentence types, and sentence manipulations can be taught to students in 3 hours a day in 10 days. This cannot be said of math, history, political science or any other disciline in the entirety of the elementary, junior high school, and high school curriculum. We can add dozens of rules like those below andstill have a mole hill in the face of the mountainous numbers of facts that are required for all other subjects. Those five basic rules of grammar are followed by the five postulates of Euclidian geometry.

1. A sentence must have a subject and a verb. 2. A verb must agree with its subject in person and number. 3. A pronoun must have a specific, stated antecedent. 4. An introductory prepositional phrase must refer to the subject of

the clause which contains the phrase. 5. A modifer must be placed as close as possible to the word it

describes.

Euclid based his geometry on five fundamental assumptions which obviously represent a larger intellectual challenge tostudent and teacher both:

Postulate I

For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.

Postulate II For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE.

Postulate III For every point O and every point A not equal to O there exists a circle with center O and radius OA.

Postulate IV All right angles are congruent to each other.

Postulate V If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles areless than two right angles.

  Now someone might want to assert that the grammar rules described above were somehow equivalent of more difficult than the Euclidian postulates that follow, but I doubt that there would be much support for such a position. In the face of the difficulty required to master math subjects, grammar must lower its head and slink off in embarrassment if it had tried to pretend to complexity, obtuseness, or excessive difficulty. It hardly places any demands on students for homework in comparison to the other classes anddoes not require as much class time. In fact, it seems thatgrammar has to play the role of scapegoat to those from mathand writing classes who cannot do grammar do not like it or merely want a target for their negativity other than their own discipline. We could perhaps ask anyone who favored thevilification of grammar, what was it threw you for a loop?. Was it the eight parts of speech? the parts of a sentence? or the sentence types? I can teach those rather thoroughly in 25 - 30 hours of class with some lecture, exercises, and group work and all with minimal homework. Perhaps changing active to passive was difficult or changing sentences to

questions. If you personally did not find it difficult, do you have trouble teaching it? Perhaps if you are teaching sign language in a gorilla cage the concepts of subject and object and noun and verb may be difficult, but I am not sureif it has been tried. Perhaps we can find a vocabulary listfor koko and check what is there. Perhaps we can find an avid young behaviorist who wants to investigate non-human primate ability to learn abstract concepts. Will they be able to handle noun and subject, but not isosceles triangle?Any student who can learn high school algebra can learn grammar. The reverse is not true. The failures that we experience in grammar can only be explained by poor teachingor through the indulgence of whininess and complaining of immature students and perhaps of immature educators. The magical difficulty or wall between simply does not exist, some find it boring many do not. Many find math boring but they are not induldged the way they are in grammar.

Naturally, students will balk, but this s

Here is a copy of what geometry students are expected to know from just one chapter of their book

After completing Chapter 11 the student should be able to:

1) describe the historical evolution and origins of the axiomatic approach to geometry and the power as well as the limitations of the axioms of Euclid’s Elements.

2) identify and use the axioms for incidence and betweenness in Euclidean geometry, and show examples of other geometries that satisfy these axioms.

3) apply the congruence axioms and explain why these axioms, togetherwith the axioms for incidence and betweenness, are not sufficient to completely describe Euclidean geometry.

4) explain the roles of the continuity and parallel axioms in Euclidean geometry, including the identification of properties dependent on, or equivalent to these axioms.

5) explain why the Cartesian coordinate plane provides a model for the axiom system for Euclidean geometry constructed in the text, and use this Cartesian model to deduce theorems of Euclidean geometry.

6) explain what is meant by a categorical set of axioms for a mathematical system, and what that term means specifically for Euclidean geometry.

Just looking at this list I am rather convinced that grammarhas been used as a scapegoat to distract curriculum directors from looking at the demands of mathematics while getting them rabid about the imagined problems, difficulty, and uselessness of grammar. I even half suspect that out ofwork math teachers, quickly boned up on their grammar, became grammar teachers and now spend the rest of their lives doing what they could to bolster math. Rather than asking for school directors and curriculum developers to be cutting grammar, perhaps we should be insisting on the inclusion of teaching the history grammar as well on a parallel with geometry. This again is far more complex thanfinding subject verb agreement and the antecedents of pronouns. There is no sane conclusion concerning the vilification of grammar other than an indulgent attitude of teacher’s in the face of students complaints and whines.

It seems the key to this has been focusing the discussion onthe usefulness of grammar for writing. The second section of this paper demonstrates more clearly the value of grammarnot only for writing but for all aspects of critical thinking but for now, we have to look at this parallel. Thevalue of algebra for calculus and trigonometry are clear butwhat about

Others, such as the knowledge of parts of speech, parts of the sentence and the ability to manipulate related forms (changing actives to passives, questions to statements and so forth) are perhaps more valuable.

Compare these with the definitions required for one semester of geometry. There are only eight presented here as there are only eight parts of speech to be learned. Comparatively, the parts ofspeech are far easier. There is nothing in grammar that matches the complexity of “the line passing through the vertex of the

parabola on which it can be folded to have the two sides coincide.”

Abscissa The First element, or x-coordinate, of an ordered pair.

Acute angle  An angle that has a measure between 0o and 90o

Acute triangle  A triangle that has three acute angles.

Adjacent anglesTwo angles that share a common side but do not overlap each other.

Angle  The union of two non-collinear rays with a common endpoint.

Angle of depressionThe angle formed by the horizontaland the line of sight to an objectbelow the horizontal

Angle of elevationThe angle formed by the horizontaland the line of sight to an objectabove the horizontal

Axis of symmetry

The line passing through the vertex of the parabola on which itcan be folded to have the two halves coincide

http://www.infomath.com/html/geometryglossary1.asp

On the home page of the National Council of Teacher’s of Math offers instructions that let you “Explore incoming calculus students’ understanding of the Cartesian connectionand the graphical representation of functions.” Are these the same high school students who would be hard pressed to understand noun, verb, adjective or subject, object, and prepositional phrase? Granted calculus is not for every high school student but the same web site gives the following for expectations for all high school math students.

In secondary school, all students should learn an ambitious common foundation of mathematical ideas and applications. This shared mathematical understanding isas important for students who will enter the workplace as it is for those who will pursue further study in mathematics and science. All students should study mathematics in each of the four years that they are enrolled in high school.

Because students' interests and aspirations may change during and after high school, their mathematics education should guarantee access to a broad spectrum of career and educational options. They should experience the interplay of algebra, geometry, statistics, probability, and discrete mathematics. Theyneed to understand the fundamental mathematical concepts of function and relation, invariance, and transformation. They should be adept at visualizing, describing, and analyzing situations in mathematical terms. And they need to be able to justify and prove mathematically based ideas. P. 2???

TRANSITIONHowever, rather than throwing out the baby with the bath water, new methods of teaching grammar should have been found instead. In fact, in 1963 the National Council of English Teachers even withdrew its support for the teaching of grammar in the face of studies which seemed to indicate that teaching grammar had either no effect or a negative effect on students writing. This naturally informsus that there are some forms of sharpening the mind that canhave no effect or a negative effect on ones writing. Given this insight we should naturally design studies to investigate which other forms of sharpening the mind might have similar effects: geometry, and . This proposal sounds preposterous as should the first upon which it is based. Hillocks (1986) states the general feeling on this matter,

If schools insist upon teaching the identification of parts of speech, the parsing or diagramming of sentences, or other concepts of traditional grammar

(as many still do), they cannot defend it as a means of improving the quality of writing (Hillocks, 1986).

As recently as 1995, the NCTE in a fact sheet for teachers stated,

The most common reason for teaching grammar as a system for analyzing and labeling sentences has been to accomplish some practical aim or aims, typically the improvement of writing. For decades, however, research has shown that the teaching of grammar rarelyaccomplishes such practical aims. Relatively few students learn grammar well, fewer retain it, and still fewer transfer the grammar they have learned to improving or editing their writing. NCTE (1995).

So, assuming for a moment that the teaching of grammar could actually have no positive effect on a student’s writing, could wesay there was no other practical aim or aims for the teaching of the parts of speech and the parts of the sentence.

Rather than the 1993 statement by NCTE admonishing teachers to not teach grammar, the following would be a more appropriate response. This is also taken from the National Council of

The Standards for high school students are ambitious. The demands made on high school teachers in achieving the Standards will require extended and sustained professional development and a large degree of administrative support.

The justification for this heavy load of math is the students often change their career goals and life directionsafter high school and they should be prepared for such a need. Similar arguments should be advanced for the teachingof traditional grammar.

The number and operations standard alone for high schoolers far exceed anything that would be required for the teaching

of grammar even if it were decided we would teach it all in high school alone.

Instructional programs from prekindergarten through grade 12 should enable all students to—

In grades 9–12 all students should—

Understand numbers, waysof representing numbers,relationships among numbers, and number systems

•develop a deeper understanding of very largeand very small numbers and of various representations of them;

•compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions;

•understand vectors and matrices as systems that have some of the properties of the real-number system;

•use number-theory arguments to justify relationships involving whole numbers.

Understand meanings of operations and how they relate to one another

•judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities;

•develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices;

•develop an understanding of permutations andcombinations as counting techniques.

Compute fluently and make reasonable estimates

•develop fluency in operations with real numbers, vectors, and matrices, using mentalcomputation or paper-and-pencil calculationsfor simple cases and technology for more-complicated cases.

•judge the reasonableness of numerical computations and their results.

http://standards.nctm.org/document/chapter7/numb.htmFortunately, the last forty years has also seen an increase in the research into critical thinking, the results of whichcan now be used to demonstrate beyond a doubt that grammar is indeed a fundamental aspect of human thinking and can and

should be brought back to the table. In fact, it is possible to demonstrate (as is done in the second part of this paper) that grammar is in fact the mold and the model for all critical thinking and as such is crucial to the preparation of students, business men, and professionals forschool and work life. It is to critical thinking what molecual chemistry is to chemistry.; Thus, we can now adduce to arguments for the reemphasis of grammar at all levels of education: 1) because it is a part of classical education that goes back to the greeks romans and Egyptians,2) it is just not that difficult, and 3) it is closely connected to critical thinking skills.

As demonstrated below, the skills of grammar comprehension and analysis such as identification of parts of speech and parts of a sentence, manipulation of grammatical patterns, and identification of types of sentence, and so on are models of core skills of critical thinking.

This paper argues that grammar is not only a good tool to train critical thinking but that grammar is the initial moldand the model for all critical thinking. It further demonstrates the value of teaching grammar for the improvement of critical thinking. The core skills which comprise Critical Thinking (CT) are those of good grammar. In fact, grammar serves as a model of just those skills.

he real question for grammar teachers is not how do we make it acceptable or how do we make it palatable or how do we make it easier or what do we remove from the curriculum. The one and only real question in this debate is how do we stop the whining? You could fit the entirety of grammaticalstudies into one chapter of a typical geometry or math book if you made a comparison of relative difficulty. The only real problem is that we allow people to whine and complain and grouse and we cater to a mentality that constantly lowers the bar in grammar but this is simply not allowed in those areas where the complaining and whining is more to be expected (though again not indulged). If we allowed this in

geometry or algebra we would Questions of real life relevance for algebra and geometry come up all the time and often we simply do not entertain them. These are skills that are expected of high school grads. If it were shown the geometry had no value for learning algebra or algebra had no value for learning geometry we would not decide to throw one of them out. If we found that only 10% of the students ever used geometry or math skills in the real world, again we would not through either of them out. We would simply assert that the students were benefited throughthe added intellectual flexibility and depth and through their membership with others who had studied the same things. In a similar manner we would not want to pass on the rights, benefits and obligations of a high school diploma if these skills were not attained. Exactly the samearguments apply to grammar. Whether real world value is found or not whether a connection to writing is found or not, the fact remains that teaching grammar is a valuable and important skill and needs to be maintained in the curriculum at all levels of education. Later we may ask whether or not it plays a role in the second language classroom but once the students know it and know it well they will gross as loudly at its absence in the second language classroom. This will undoubtedly result in fewer foreign language teachers who do not know their grammar, butas this is not a difficult study (as compared to math, history, or political science) it is not serious to ask language teachers to learn it.

We may reasonably argue that there is more than enough goingon algebra and geometry classes and we could save a little for college by putting the grammar back in the classroom But allowing the vilification of grammar merely as a means to bring in more math is simply not professional and in no way does it constitute an argument for math or against grammar. It is mere political expeditiousness on the parts of those who want more math and fewer grammar classes. The benefit for the student or for the culture has been thrown out the window. And rather than a burden this would be more

like a respite for the students. Subjects must agree with their verbs. A pronoun must have an antecedent. There is no rocket science here. As a matter of fact there is no algebra, no geometry, and when you think about it, no basic math either. None of the arguments that we have tolerated for forty years for lowering the bar in grammar classes would have ever have been tolerated in math. History, geography, political science – all of these pose greater problems than learning grammar does. But still there are those who would argue that grammar (and in some cases math, algebra, ancient history) is irrelevant or that it has no demonstrable value for the improvement of writing skills, orthat they have no clear value in the real world but these arguments are not tolerated in any subjects other than math.An effort to make mathematicians prove that algebra must have a demonstrated value for geometry of vice versa would be justifiably met with derision. So should tests that pretend to investigate whether grammar skills detract or do nothing for writing skills. Send them back to the drawing board without the courtesy of a review: they have landed a ufo in our laps. We can no more tolerate that than we can tolerate the removal of math from our schools because it hasnot demonstrable value.

Who dreamed up this entire anti-grammar attitude in the first place? Was it perhaps inspired by jealous math teachers who wanted extra hours of classroom time? Was it math teachers who wanted to deflect students attention from the realities and the miseries of learning math? Was it just bitterness due to unresolved whining around grammar? My unresolved bitterness toward math has not been pandered to in this way, by the way. If we bring the entirety of the curriculum of grammar that existed before the war against grammar began we would lose nothing. There is no need to respond at all to the whining, beligerant, self-indulgent carping and childishness that constitutes this movement. We should no more induldge it or publish papers about it than we should do so to a similar movement about geometry or algebra. What deep rooted low self esteem must

exist among the cadre of the last forty years of English teachers that they could not stave off this loss of valuableground (and not valuable for them but valuable for the students). We merely tell the students that grammar is goodfor their speech, their reading, their writing, as well as their foreign language study and we don’t feed their attempts to wheedle cajole or even coerce arguemtns that allow them back into the math class, the study hall or the playground.

Part II. Grammar and Critical Thinking.

The importance of critical thinking cannot be underestimatedfor adults in any society and even less so in military, professional, and business environments. It is only the skills of critical thinking that will guarantee successal evaluation and implementation of situations and plans. Without these skills the individual is adrift like an animal, dependent on instinct and environment with no choiceexcept to try and align himself with the herd and the environment in order to stave off fear, minimize stress and distress and participate as best he can in an average way with all other members of his community.

Grammar, the organizational principles of human language, isitself the result of critical thinking and a presentation ofcritical thinking in a rather native form. In perceiving the world and after discovering the possibility of vocalization the world is first ANALYZED into its component parts and the early human responds and tries to identify objects (nouns), actions (verbs), qualities and quantities (adjectives and adverbs); these are complemented by relational principles such as before after from to between and so on (prepositions or case markers); and make phrases possible. Once phrases are recognized they can be strung together (see Bickerton ) made they can be further INTERPRETED as phrases, SUBJECTS, OBJECTS, and so forth. More abstract notions such as and or if and because past

present and future fill out the set of and so on then make more complex sentences possible.

In the last x years research into critical thinking has beenadvanced to a degree. It has taken prominence. This paper draws heavily on the work of Peter A. Facione (200 200 200 )in his role as the chief researcher for the American Philosophical Association’s research report on CT.

What is Clinical Thinking?The goal of this paper is not so much to contribute to the field of Critical Thinking or to make any advances in that area. Rather the goal is to use the state of the art of Critical Thinking to determine whether or not traditional grammar is in indeed an aspect of that important life skill.For this reason, this paper draws heavily on work by the American Philosophical Asspociation, which, using a panel of46 recognized experts in the field of Critical Thinking, provide a consenus opinion of what constitutes critical thinking. In addition, this paper also uses work by the samegroup to determine the disposition ot use critical thinking.

A panel of 46 experts in the area of CT in the Delphi reportusing consensus to make decisions to determine the core skills of CT.

They identified the following six sub-classifications. Eachof which were further sub-classified to

What is a Disposition to Critical ThinkingFacione, 2004 has divided the core Critical Thinking (CT) skills into the following: interpretation, analysis, evaluation, inference, explanation, and self-regulation. Each of these skills are an aspect of grammatical awareness and analysis. More importantly, as these skills are represented in our linguistic knowledge, the linguistic knowledge, the basis of human separation from animals and the basis of all our critical thinking development is actually

Facione 200 4 also points out that those with a disposition to critical thinking are: inquisitive, systematic, judicious, analytical, truth-seeking, open-minded, and confident in reasoning. These are qualities that are valuedin all apects of a mature adult’s life.

Language learning skills in the acquisition stage but more so in the learning stage are precisely those of critical thinking, and therefore any language learning study, whetherit be on one’s first language or on a second will enhance ones critical thinking.

All aspects of the CT can be found in the parsing of a sentence. This same procedure exists in automechanics and chemistry as much as it does in grammar, but in grammar it is harder to see. A good writer doesn’t have to be a good grammarian any more than a good race car driver has to be a good mechanic. However, a good critical thinker needs theseskills and to apply them to languageIT SEEMS THAT CRITICAL THINKING CAN BE DIVIDED INTO TWO CRUCIAL STEPS ONE EARLIER AND MORE BASIC AND THE OTHER “HIGHER LEVEL”

The first or lower level form of critical thinking is more fundamental and perhaps requires a different sort of thinking. It is seminal or deep level. It has to do with the identification of core concepts in a sphere of investigation. The higher level thinking has to do with thepresentation of arguments and evidence in favor or against particular implementations of the field that has been discovered through this. Let’s use automechanics and chemistry as examples. Before we can write papers about excessive pollution or noise or about the values of the use of particular cars; before we can talk about

Critical thinking should be divided into two subfields: definitions of a sphere of investigation. Applications of that sphere of investigation to the larger world. The firsthas far fewer ethical and social implications. It is merelya matter of describing the field. The latter brings in the higher order sciences and ethics and morals. It is in that first division that we find grammar.

In the lower lever we have early steps of a scientific investigation: find the component parts of what we are investigating; analyze the component parts into meaningful groupings, describe the relation of the groupings one to theother. Draw both deductive and inductive conclusions about the material at hand and

Most abstract is naturally the least corporeal. There are no physical models at this point, no reference whatsoever toany five senses measurable reality. It is the basis of all mental functioning. Pure philosophy as well as the tools ofphilosophy are all built on this model. Once this work is done we can apply it to other areas

analysis into objects actions/states qualifiers and quantifiersevaluation into meaningful groupings subjects objects locationsinterpretation: something to do, a query about somethingexplanation: can’t be known, can be done, is being done

And it is precisely at this lower level of cognition that grammar operates as the model and mold of all such thinking.

The language is the model of critical thinking and as such understanding it will enhance all understanding it is the prime from which all other critical thinking is derived. Weorganize into the smallest component parts ANALYZE, then evaluate into meaningful parts that comprise a recognizable entitity or proposed entity EVALUATE. We decide what sort

of response is required: an answer, carry out a command etcINTER; deduce induce

Grammar is the mold, the prime, and the example, out of which ALL critical thinking is formed. The better we understand and relate to our grammar the better will be our ability to do all critical thinking: 1Analysis into Core Categories2Evaluation of the relations between those core categories in particular cases3Interpretation of particular arrangements of those core categories in different ways4Explanation of the understanding of each of those MAKE DEDUCTIONS5 MAKE INDUCTIONS6SELF REGULATION SYLLOGISTIC COMBINATIONS OF INDUCTION AND DEDUCTION Checking ones logic. The core skills of grammar analysis are presented in the table 1:Table 1: Universal Grammar

Parts ofSpeech

Parts of theSentence

Sentence Types Variations on orManipulation of

SentencesNouns Subject Declarative Passive/ActiveVerbs Object of Verb

(Direct and Indirect)

Interrogative (yes/no & question word)

Direct Quotes

Adjectives Complement (Subjective and Objective)

Imperative Indirect Quotes

Adverbs Object of Preposition

Exclamatory Subjunctives/conditionals

Prepositions Adjective Clause

Simple Making relative clauses

Conjunctions Adverb Clause Compound Making yes/no questions

Noun Clause Complex Making question word questions Possessives

Compound/Complex Requests/suggestions/invitations

The Delphi report agreed upon 6 core skills and several sub-skills each for. The Delphi report skills and Sub skills are presented in table 2. An abbreviated form of these skills and subskills is presented in table 3 which compares them with the core grammar skills just described. Table 2: CT Skills and Sub-Skills

CT SKILLS CT SUB-SKILLSInterpretation Categorization

Decoding SignificanceClarifying Meaning

Analysis Examining IdeasIdentifying ArgumentsAnalyzing Arguments

Evaluation Assessing ClaimsAssessing Arguments

Inference Querying EvidenceConjecturing AlternativesDrawing Conclusions

Explanation Stating ResultsJustifying ProceduresPresenting Arguments

Self-Regulation Self-ExaminationSelf-Correction

Table 3: Comparison of CT and Grammar Skills

CT SKILLS CT SUB-SKILLS GRAMMAR EQUIVALENTSinterpretation categorization analysis into parts of

speechanalysis identifying arguments Analysis into parts of

sentence (sometimes called arguments) / clauses

Evaluation assessing arguments analysis of sentence types; e.g. simple/compound; declarative/interrogative

inference querying evidence/conjecturing

making statements fromquestions, questions

alternatives from statements; active from passive paraphrasing, etc.

explanation stating results/presenting arguments

answering questions/responding to statements commands, etc

self-regulation self-examination, selfcorrection

rephrase, edit, error check, etc.

I address each of these in turn:

1. Interpretation – Identifying Arguments – Analysis into Parts of Speech.

2. Analysis – Identifying Arguments – Analysis into part of speech / clauses

3. Evaluation – Assessing Arguments – Analysis of Sentence Types

4. Inference – Querying Evidence, Conjecturing Alternatives – Paraphrasing etc.

5. Explanation – Stating Results / Presenting Arguments– Answering, Responding

6. Self-regulation – Self-examination, Self-correction – Rephrase, Edit, etc.

This near exact correspondence is no coincidence and confirms to what many believe intuitively, that grammar is aform of CT. More importantly, as a reflection of the core organizational principles of human cognition, this correspondence of the APA Delphi report skills lends evidence to their validity. Had there been no correspondence or little correspondence it would be better to reject the CT schema as flawed rather than try to indict grammar for not being a human CT skill.

INTERPRETATION INTO PARTS OF SPEECHThey are CATEGORIZED into the parts of speech which naturally further clarifies their role and their significance and the meaning is further clarified through a more direct conscious look at the word (rather than just responding intuitively and spontaneously as in a casual conversation).

ANALASYS INTO PARTS OF SENTENCEThe phrases are identified in the examining of ideas and then in recognizing them as arguments (phrases conjoined to form complex phrases) and then analyzing those arguments in the sentence as subjects objects and so forth. Clauses are identified at this point as well. Incidentally, the parts of the sentence such are subject and object are referred to as “arguments” of the verb in formal linguistics following XYZ.

EVALUATION OF SENTENCE TYPESIn the phase of assessing arguments the sentences are identified as declarative, imperative, interrogative, or exclamatory , and also as simple, compound, complex, and compound complex. These relations between these sentences can be further determined through the use of punctuation at this point.

INFERENTIAL ANALYSISIn querrying the evidence tests such as changing active, passive, statements to questions and questions to statementscan be applied to see if there are any confusions or problems If all these tests are possible the sentence has been properly analyzed. Making paraphrases and inferences are also possible at this point.

EXPLANATION (DEDUCTIVE ANALYSIS)Here EXPLANATION entails answering questions, responding to statements as appropriate and deciding whether or not to acton commands. Views as to whether or not questions are

answerable, or commands can be carried out would also occur at this point. For example, a command to visit the present king of France could not be followed and would be noted at this point.

SELF-REGULATIONSELF-REGULATION finally allows the speaker/listener to respond, rephrase, defer, pass his turn, take his turn, request another to speak, begin anew and so forth.

In a discourse such evaluation would take place for each individual sentence but also such evaluation could be begun again on elements of the paragraph to evaluate the sentencesas items in logic. It is reasonable that the same CT methods would then be applied in a cycle manner through eachof the following operations except the final one, taking action/modifying belief which is the end result.

GrammarLogicRhetoricCritical Thinking Problem SolvingTaking Action or Modifying/confirming belief

As a mold it is the first iteration of the cyclic activity and forms the basis of the structured device (language) the we will use for all successive iterations. There could be no accurate description of CT that did not include grammar as grammar is our first and most important application of this skill.

As model: whether intuitively accessed or accessed through a conscious awareness of the principles of grammar, grammar is the model for all critical thinking. Language was there before Socrates, before the Greeks and the Romans before theSumerians and the Egyptians the criticl thinking that went into the creation of grammar was the first the article

states that people may not be able to describe their critical thinking skills but they are able to access them intuitively. This is not unlike saying that people do and can respond to grammar intuitively but often do not do so overtly.

Knowing grammar is knowing this moldz: we organize our automechanics and our chemistry according to the core elements (the nouns) their states and actions (verbs) their qualities and quantives (verbs and adverbs) and then we lookat the way they combine and relate via prepositions (peptides and amino acids lubricants and screws)

Our speech reflects this in our grammar and our grammar is reflected in everything else. Without the mold of speech no science or categorization is possible and without grammar – if we use grammar as a model in our approach to anything we can say we need to look at categories of building blocks activitites/states and qualities and quantities and then howthey are related. We can use our prepositions to remind us and to check if we are thorouogh enough. We can use types of adjectives to determine if we have been thorough enough: did we look for demonstrative, qualitative, numerative, and distributative relationships? Did we check locations to from between and around? Did we check time before after andduring? The knowledge of grammar is the best possible tool to approach science, critical thinking and so on. The more succinct and pithy our knowledge of grammar, the more rapid and efficient will be our application of critical thinking to knew areas of our lifes.

Without grammar we must intuit allWith a knowledge of grammar but without a particular knowledge of its usefulness as a critical thinking menmonic our thinking is improved but it has not gone as far as it good.

With a sound awareness of grammar and an awareness of the use of grammar as a criticl thinking mnemonic we have

arrived at the pinnacle of CT and of Grammar and we notice how they are identical.

1. Analysis: break down of a sentence into its SMALLEST component parts as parts of speech

2. Evaluation: noting the connections of the parts of speech in the sentence and their relations within a larger whole (the sentence)

3. Interpretation: has to do with the meaning of the sentence and an assessment of its truth value. All thewords are understood, underlying premises are discerned: does it ask a question make a statement, expect a response or express an attitude.,

4. explanation: to make deductive conclusions in paraphrases from the meaning discovered

5. Inference: is draw inductive conclusions from the meaning discovered

6. Self-regulation: has to do with the “checking back” with oneself on successive interpretations and evaluations until the sentence is understood sufficiently for a given task or for the satisfaction of the learner.

The application of these skills to grammar is a difficult task as the concepts are purely abstract: noun-ness, verb-ness, subject-ness, and so on do not and could never have any measurable, five senses reality that a student can hold on to or look at. It is a purely mental exercise. However,in spite of their purely abstract nature, it is possible to do analysis and evaluation of them and of the hierarchical structures for which the serve as building blocks. But unlike mental exercises like poetry, literture, or philosophy, this study has a far more discorable and predictable regularity and logic. It is indisputably an excellent way to develop ones critical thinking abilities.

With auto mechanics we can see a similar sort of applicationof critical thinking, but in the case of auto mechanics, drawing, photos, and actual engines are more concrete and

limit the “pure” development of critical thinking that is seen in grammar study.

1. Analysis: break down of an engine into its component parts as parts

4. Evaluation: noting the connections of the parts of the engine as carburator, transimission, and so forth

5. Interpretation: note their function and their interconnections

4. explanation: decide how to recreate the engine and how to use it

6. Inference: note the value of using an automation to other modes of transportation and perhaps discover possible innovations for the future.

2. Self-regulation: learn from ones mistakes in the process

Nouns: the core elements or building blocks of our sphereof investigation (drive shaft, gears, steering wheel)

Verbs: the activities and states within that sphere (running, off, idle)Adjectives: numerative:

qualitative:

Adverbs: manner:location:Frequency:

Prepositons: How are other of the core elements (nouns) related to the current actions or states (verbs)locations:time:

Nominal Morphology

casegenderperson

numberVerbal Morphology tense

progressive aspectperfective aspect

Comparison: comparitveSuperlative

Conjunctions: coordinateAndButSo

Subordinate

Phrasal analysis: done in conjunction with PARTS OF SENTENCE ANALYSIS

By familiarizing themselves with the core structure of critical thinking, students will naturally improvein that ability. They may not be aware of the improvement and initial forays into critical thinking may cause some confusion as students overapply or get lost in this organizational structure, but in the long run, the training willbe of value. They will be more aware of the core categories, their subcategories and the way thingsfit together. This can only be an aid to all their reading, writing, discourse and study skills.

Pronouns express our ability to create variables and classes1

1 You could also include the syntactic notion of empty category here as well as perhaps the most abstract of all concepts in our grammar. It isnot even represented as a pause, they merely exist where there is

Given the above discription, this should be falsifiable via an experiment. We merely test the critical thinking abilities of students before taking a grammar course and then measure them again after the course to see if their critical thinking skills have improved. If there is no visible improvement in the short period of time they can be tested again 2 or more months later to see if there has beenany improvement.

Compare against those who do well on an English grammar testand those who do not do well on the English grammar test.

Test 1 first time: before the grammar courseTest 1 second time: after the grammar course

Test 2 first time: AFTER TWO MONTHS IN THE LANGUAGE CLASSTest 2 second time: AFTER FOUR MONTHS IN THE LANGUAGE CLASS

take students who have successfully completed a grammar course and then

The ability to abstract is the main ability in all thinking processes. Animals respond to their world directly and to some degree to their imagination of the world (the dog at the door expecting his owner to return), but they do not create hypothetical situations or less corporeal (= more abstract) situations from which to learn and make decisions.To create a model and that model can be successively less corporeal and in that more or less representative of essential features of what is being studied. In corporeality we respond with our senses to the sensed world in abstraction we respond with our cognitive functions to

missing material for which an interpretation can be found. However, as I am providing an account of traditional grammar, this will not be discussed any further.

imaginal or linguistic representations of items in our world. The abstraction can begin with the images of the phenomenal world dreams and fantasies without speech words and images of the phenomenal world dreams and fantasies with speech, poetry, song just words of the phenomenal world speech about speech discussion of the words of the phenomenal world

The discussion of the words of the world has two parts: 1) the discussion of grammar, 2) the discussion of propositionsand arguments. Rhetoric (pervasiveness) Then what follows is critical thinking then problem solving and so on.

So then how is it that we understand studies that led to theNCET decision? The research on dispositon to use critical thinking provides one possible explanation, one that is better than trying to claim that grammar study could in someway damage critical thinking or writing ability. If this were the case we would have to begin evaluating, math, engineering, and science to see which of those had a negative affect on writing ability.

AbstractionGoing from the corporeal to the incorporeal

Analytical Global/IntuitiveLanguage and image as distinct

GrammarLogicRhetoricCritical ThinkingProblem SolvingEstablishment of belief or taking action

Language and Image asinseperable

Visual art / cave painting,sculptureChant/ritual gestureMusic/dancepoetrysongOpera/ballet/theater

There is a hierarchical relation between all the items in the analytical column. Though they are all independent enough to be done without the previous. That is problem solving can occur without an awareness of critical thinking but it is not as good. Critical Thinking can occur without rhetoric, but it is not as pervasive. Rhetoric can occur without a study of logic but it will be less effective and more prone to error. Logic can be studied without grammar but it will not be as precise. Many people have studied critical thinking without studying logic, many have studied problem solving without a formal understanding of any of the previous skill but a formal understanding of any one of them increases ones informal ability in all of them. To ignore anyone of them as has been done with grammar for the last 40 years is foolish. The benefits that are provided to all the others are unmistakeably clear. The farther down on the list yougo, the more fundamental and the more abstract is the study. Grammar has no formal propositions or arguments. It is thinking stripped to its barest elements. It is likely that these skills are learned in reverse order each one motivated by a desire to better at the previous. COULD THERE BE A BENEFIT TO ALL AREAS THROUGH STUDYING ANY ONE? YES. WOULD STUDYING GRAMMAR NECESSARILY BE OF VALUE TO ALL THE OTHERS? YES. GOING THAT DIRECTION ON THE HEIRARCHY WOULD IMPLY THAT THOUGH THERE MIGHT BESOME TIME LAG AND IT MIGHT BE NECESSARY FOR SOMEONE TO POINT OUT HOW GRAMMAR SKILLS HELP LOGIC HOW LOGIC HELPS RHETORIC HOW CRITICAL THINKNIG HELPS PROBLEM SOLVING TO GET THAT THROUGH.

The more familiar one is with the various steps on the hierarchy the shaper will be his intellectual skills and the more able willhe be to avail himself of any of the skills as necessary to solveproblems. The more abstract the problem the more he is going to need to consult the higher levels of the hierarchy. The more concrete the skill the more he can rely on the lower levels.

A pun such as Karl Marx has no class requires little thinking to be enjoyed, but if one puts into an argument grammatical, and logical knowledge is required to arrive at a proper assessment.

If Karl Marx has no class, then he is likely to be seen withhis shirt tail hanging out.

What is the relation of grammar to the rest of these skills:

Column A begins with no images and preceeds to problem solving where there are real world images as conveyed through linguistic representations. Conversely Column B begin with out any words and moves to a point where words and images are mixed as spiritual or psychological rather than real world problems. Column B can be analyzed by column a because A is focused on a distinction between language and the images being viewed. Columnb cannot because Column A is deductive and consciousn, Column B is inductive and intuitive.

Due to the level of abstraction required for grammatical analysis, its role as a good tool for sharpening critical thinking is apparent: critical thinking in fact is of progressively less value the more concrete is the evidence and situation in which one must work. If A then B, B. Requires morecritical judgment if that orange is not taken, I will eat it. The orange is not spoken for. Critical thinking is a skill that is called for to sort through the implications, ambiguities, and similarities of complex abstract principles, concepts, ideas andso forth. It is not much required in “watch your step, there is ice all over the walk.”

WE can conclude then that any work with grammar is of value because it sharpens critical thinking skills. Whether or not it sharpens or dulls the DISPOSITION to use such skills is another matter. In fact, if grammar work was presented in such a way as to cause students to dislike all critical thinking results such as those of the NCET 1963 report could be understood. Perhaps too the study of grammar had resulted in more active, but immature critical thinking which appeared as a loss in writing skills. In any case, to conclude that teaching grammar damages writing skills is much like concluding some forms of sharpening analytical thinking are deleterious to writing while others are not. Such a conclusion opens the door to testing whether not

math, science or legal thinking improves or lessens ones writing abilities.

It is also not surprising that a team of experts focusing on the core tenets of critical thinking arrived at a description that conforms to basic grammatical analysis.

APPENDIX 2: REQUIREMENTS OF BASIC GEOMETRY

NCTE 1995 Slate Starter Sheet.

A sentence must have a subject and a verb. A verb must agree with its subject in person and number. A pronoun must have a specific, stated antecedent. An introductory prepositional phrase must refer to the subject of the

clause which contains the phrase. A modifer must be placed as close as possible to the word it describes.

Euclid based his geometry on five fundamental assumptions:

Postulate I For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.

Postulate II For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE.

Postulate III For every point O and every point A not equal to O there exists a circle with center O and radius OA.

Postulate IV All right angles are congruent to each other.

Postulate V If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

 

No one seemed to like this Fifth Postulate, possibly not even Euclid himself--he did not use it until Proposition 29. The reason that this statement seems out of place is that the first 4 postulates seem to follow from experience--try to draw more than

one line through 2 different points. The Fifth Postulate is unintuitive. It does come from the study of parallel lines, though. Equivalent to this postulate is :

Playfair's Postulate Given a line and a point not on that line, there exists one and only one line through that point parallel to the given line. 

Euclid's

Ptolemy followed with a proof that used the following assumption:

For every line and every point P not on , there exists at most one line m through P such that m is parallel to .

We will show at a later date that this statement is equivalent toEuclid V, and therefore this did not constitute a proof of EuclidV.

http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node4.html

  Axiomatics and Euclidean Geometry

Objectives

After completing Chapter 11 the student should be able to:

describe the historical evolution and origins of the axiomatic approach to geometry and the power as well as thelimitations of the axioms of Euclid’s Elements.

identify and use the axioms for incidence and betweenness in Euclidean geometry, and show examples of other geometries that satisfy these axioms.

apply the congruence axioms and explain why these axioms, together with the axioms for incidence and betweenness, arenot sufficient to completely describe Euclidean geometry.

explain the roles of the continuity and parallel axioms in Euclidean geometry, including the identification of properties dependent on, or equivalent to these axioms.

explain why the Cartesian coordinate plane provides a model

for the axiom system for Euclidean geometry constructed in the text, and use this Cartesian model to deduce theorems of Euclidean geometry.

explain what is meant by a categorical set of axioms for a mathematical system, and what that term means specifically for Euclidean geometry.

http://wps.prenhall.com/esm_usiskin_mathhstch_1/0%2C6625%2C488291-%2C00.html

The Axioms of Incidence The following axioms set out the basic incidence relations between lines, points and planes. They also characterise the concept of ``dimension'' that we associate with these notions.

1. Incidence between points and lines: 1. There are at least two distinct points. 2. There is one and only one line that contains two

distinct points. 3. Every line contains at least two distinct points.

2. Incidence between points and planes: 1. There are three points that do not all lie on the same

line. 2. For any three points that do not lie on the same line

there is a one and only one plane that contains them. 3. Any plane contains at least three points.

3. Incidence between lines and planes: 1. If a line lies on a plane then every point contained in

the line lies on that plane. 2. If a line contains two points which lie on a plane then

the line lies on the plane. 4. Dimensionality of space:

1. If two planes both contain a point then they also contain a line.

2. There are at least four points that do not all lie on the same plane.

The first four axioms (which do not refer to planes) are called the plane geometry axioms, while the remaining are the space axioms. Out of the various Theorems that can be proved we note

Theorem 1   Given a line and a point not on it there is one and only one plane that contains the line and the point.Theorem 2   Given a pair of lines which meet in a point there is one and only one plane that contains the lines.Theorem 3   Given four points that do not all lie on a plane, there is no line containing three of these points.Exercise 1   There is a ``geometry'' consisting of 4 points, 6 lines and 4 planes that satisfies these axioms.Exercise 2   Which of the above axioms can be omitted? For those that are necessary construct a ``geometry'' that satisfies the chosen axiom and defies the others.

Axioms of Order These axioms were almost ignored by Euclid except the second one below. Their importance was noticed by M. Pasch who saw how they were implicitly being used in many proofs. This is one problem with ``evident truths''; we often forget to state some of the axioms and then the geometry is incomplete without them. The following axioms make clear the notion of a point lying between two other points.

1. When B is between A and C then, A, B and C are distinct points lying on a line and B is between C and A.

2. Given a pair of points A and B there is a point C so that B is between A and C.

3. If B lies between A and C then A does not lie between B and C.

4. Let A, B and C be three points on a plane and a be a line on that does not contain any one of these points. If thereis a point D on a that is between A and B then either a contains a point between A and C or a contains a point between B and C.

The Line l contains a point on one of the other two sides The first three axioms allow us to introduce the notion of a half-line or ray. Given a pair of points A and B the half ray starting at B and pointing away from A consists of all points C so that B is between A and C. Similarly, the last axiom allows usto introduce the notion of a half plane. Given a point A and a

line a the half-plane bounded by a and opposite to A consist of all points B so that a contains a point lying between A and B.

In spite of the axioms of order being ignored for so many hundreds of years they are so important that one can entirely replace the axioms of incidence by giving an extended set of axioms of order. Think of it this way. If a straight line is to be the shortest path from a point to another then we must at least be able to say what are the points ``on the way'' or in-between.

The following theorems can be deduced from the axioms of Incidence and Order.

Theorem 4   Given any two point A and B there is a point C that lies between A and B.Theorem 5   Given three points A, B and C that lie one a line exactly one point that lies between the other two.Theorem 6   Given four points on a line they can be labelled A, B, C and D so that B is between A and C and between A and D and C is between B and D and between A and D.Theorem 7   Given any finite set of points on a line they can be labelled A1, A2, ..., An so that the points are in that order.An important theorem that can be deduced from the order axioms was first discovered by G. Desargues: Theorem 8   Given points A, B, C, A', B' and C' so that the lines AA', BB' and CC' all pass through a point O. Further, let AB and A'B' meet in a point C'', AC and A'C' meet in a point B'', BC and B'C' meet in a point A''; moreover, let us assume that the 10 points considered are distinct. Then the points A'', B'' and C'' all lie ona line.Proof. In case the plane containing the points A, B and C does not contain all of the points A', B' and C' then the line containing the points A'', B'' and C'' is just the line of intersection of with the plane determined by A', B' and C'. Thus, the theorem needs only to be proved under the assumption that all the points lie in a plane. In this case we shall show how to construct A''', B''' and C''' that do not lie in the plane and so that the points A, B, C, A''', B''' and C''' also satisfy the hypothesis of the theorem. Moreover, A''', B''' and C'' are collinear

and so on cyclically. Thus, the planar version will then follow from the non-planar version.

Axiom of Parallels The axiom in this section caused the most controversy and confusion of all. The axioms of parallels (which is also an incidence axiom) is Axiom of Parallels

Given a line and a point outside it there is exactly one line through the given point which lies in the plane of the given line and point so that the two lines do not meet.

Note that, while asserting that there is a line through the givenpoint that doesn't meet the given line, it also says there is only one such line. In other words, it also asserts that all the ``other'' lines co-planar with the given line meet that line. This motivates the introduction of the following (stronger and stranger) version of the Axiom of Parallels: Projective Axiom of Parallels

Any pair of lines that lie in the same plane meet. The idea behind this axiom is that even (apparently) parallel lines appear to meet at the horizon. We can demonstrate that thisaxiom is consistent with the axioms of Incidence by means of Linear Algebra as in the examples below.

APPENDIX 2: A Curriculum for High School Algebra

Browse High School Basic AlgebraStars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:    Solving simple linear equations.    Positive/negative integer rules.    Mixture problems.    Quadratic equations.    Absolute value.    Completing the square.    Direct and indirect variation.    Inequalities and negative numbers.

Acid Concentration, Heating Bill [7/6/1996] 1) A pharmacist has 8 liters of a 15 percent solution of acid. How much distilled water must she add.... 2) By installing a $120 thermostat, a family hopes to cut its annual heating bill...

Adding and Subtracting Polynomials [12/08/1996] How do you add and subtract polynomials?

Algebra and Trig Equation [4/20/1996] How can I solve this equation: 1 = sin(3x) - cos(6x) ?

All About Functions [11/06/1996] Could you please explain functions?

Completing the Square [09/08/1997] Please explain 2x^2 = x + 5 .

Derfs and Enajs: Algebra and Venn Diagrams [03/09/2003] All Derfs are Enajs. One-third of all Enajs are Derfs. Half of all Sivads are Enajs. One Sivad is a Derf. Eight Sivads are Enajs. The number of Enajs is 90. How many Enajs are neither Derfs nor Sivads?

Diagonals and Tiles [11/17/2001] Jay tiled a 15x21' rectangular ballroom with 1 ft. sq. tiles. Then he drew diagonals connecting opposite corners ofthe room. How many tiles did the diagonals pass through?

Direct/Indirect Variation [6/12/1996] Please explain direct and indirect variation.

Distributive Property [04/25/2002] What's the best and simplest way to teach pupils that a-b = -(b-a)?

Distributive Property, Illustrated [09/28/2001] How can I show (12*2) + (12*3) using the distributive property?

Doubling Sequence [8/24/1996] On Jan 1st it snowed one centimeter; on Jan 2, 2cm; on Jan 3, 4 cm...

Equation of a Line Parallel to an Axis [06/19/2003] How can I get the equation of a line if the line passes through a point and is parallel to the x- or y-axis?

Equations of a Line given Two Points [8/31/1996] How do you find the equation of a line going through two points if you only know the two points?

Equations: the "Elimination" Method [11/20/1996] Is there an easier way to solve a system of two equations than by the substitution method?

Explaining Algebra Concepts and FOIL [08/04/1998] Some ideas for learning algebra that may help prevent frustration.

Exp, Log, and Ln Functions Explained [7/31/1996] What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?

Factoring Polynomials [11/14/1996] Is there a logic behind the factorization of polynomials?

Factoring Polynomials [11/07/2001] Factoring polynomials like 12x^3y^9 + 20x^5y^4.

Factoring to Solve a Problem [7/31/1996] I have problems understanding the solution I get when I factor an expression to solve: x (x + 1) = 132...?

Finding the Equation of a Line [01/08/1997] Given either a point and a slope or two points, how do you find the equation of a line in both point-slope and standardform?

Geometric Objects and Properties using Algebra [06/29/1998] Why does y = mx+b, and what does it mean?

Graphing Coordinates [1/15/1997] How can you tell if you have a straight line just by lookingat the numbers?

Horizontal and Vertical Asymptotes [12/5/1995] I am confused about horizontal and vertical asymptotes. Please tell me how to find them, and their definition.

An Introduction to Parabolas [02/04/1999] What is a parabola? What are some of its properties? What are some examples of parabolas in real life?

Linear Expressions [08/17/1999] What are linear expressions, and how will I use them in my life?

Linear Programming [01/30/1997] Maximize the profit of a nutrition center given information about the energy trail mixes they sell.

Locus and Equations of Lines [01/10/1999] Describe the locus of points that are 3 units from the line x = -1...

Long Division of Polynomials [6/26/1995] How do you use long division to solve polynomials with remainders?

Loosening the Earth's Belt [1/18/1996] The earth has a belt around the equator that is perfectly fit. If a measurement of 1 foot is added to the belt, theoretically the belt will no longer touch the earth at anypoint. What is the new distance between the surface of the earth and the belt?

Man and Train on a Bridge [2/1/1996] When a man is 3/8 of the way across a railroad bridge that goes from point A to point B, he hears a train approaching at 60 mph. If he runs back he will meet the train at A, and if he runs forward the train will overtake him at B. How fast can the man run?

Meaning of Irrational Exponents [03/17/1997] y^x is the product of x factors of y. Where do irrational exponents fit in this? How can you have an irrational numberof factors?

Mixing Milk and Butterfat [09/04/2001] Milk that has 5% butterfat is mixed with milk that has 2% butterfat. How much of each is needed to obtain 60 gallons of milk that has 3% butterfat?

Negative Ratios and Dividing by Zero [6/8/1995] How can 1 : -1 = -1 : 1 ? Is x divided by 0 = infinity? Are there different infinities?

No Solution, Infinite Solutions [01/03/2002] Solve: 8(2x-3) = 4(4x-8); -3(x-3) is greater than or equal to 5-3x.

Order of Operations Puzzle [09/22/2002] What is _ + _ x _ - _ = 22? We're given the numbers 2, 3, 4,8.

The Particular and the General Case [11/06/2001] If a+b+c=3 and a(^2)+b(^2)+c(^2)=5 and a(^3)+b(^3)+c(^3)=7 what is a(^ 4)+b(^4)+c(^4)? And how is it done?

Polynomial Inequality Problem [2/7/1996] A student asks for help with solving algebraic inequalities.

Proof by Induction [07/13/1997] Prove by induction on n that |A^n|=|A|^n.

Purpose of Algebra [10/15/1996] What is the purpose of algebra and who came up with the idea?

Pythagorean Theorem Proof: Four Right Triangles [10/7/1996] I don't understand the Pythagorean Theorem.

http://mathforum.org/library/drmath/sets/high_algebra.html

Bickerton, 1991 pp 4-5

And yet the true source of our difference has been lying all the while, like Poe’s purloined letter, hidden in plain view. There are not merely two things, consciousness and power over nature, that distinguish us from other species, there is a third thing: language. While it would be absurd to suppose that language in and of itself provided everything that differentiates us from apes, language was not only the force that launched us beyond thelimits of other species but the necessary (and perhaps even sufficient) prerequisite of both our consciousness and our uniquecapacities.

If this is so, why have people looked elsewhere for explanations?All along there have been those who recognized that language musthave played an important role. Consider for instance the quotation from Darwin that serves as the epitaph for this book. But the precise nature of that role remained obscure because a number of factors conspired to make language itself an elusive and slippery object.

Language is, of all our mental capacities, the deepest below the threshold of our awareness, the least accessible to the rationalizing mind. We can hardly recall a time when we were without it, still less how we came by it. When we could first frame a thought, it was there. It is like a sheet of glass through which every conceivable object in the world seems clearlyvisible to us. We find it hard to believe that if the sheet wereremoved, those objects and that world would no longer exist in the way that we have come to know them.

That, in turn, is because for most of us language seems primarily, or even exclusively, a means of communication. Ratherit is a system of representation, a means for sorting and manipulating the plethora of information that deluges us throughout our waking life. How such as a system came to be, howit functions, and what it accomplishes will form the themes of this book.

Also p 185-187Also 58-59 (noun, verb, and predication)

The most basic experience even for animals is inner or “I” and outer or “other”. Animals cannot articulate it and do not have much consciousness with which to reference it, but there is a sense of a center of the senses and an outside that is being sensed. This sense of I and other is what drives the animals. The inner state of “I” and the outer state of “other” direct all activity and experience and are the core elements of instinctual behavior. I and other are the first two nouns. The first verbs is “experience” The first predication is “I experience”, the second is “I experience other”. The first is intrnasitive the second transitive. The first adjectives are hot cold hungry thirsty and these can be predicated of either I or other. I can be projected onto all others. All others can be indentified withas I. These four parts of speech, noun, verb, adjective, and adverb correspond to the four basic mental functions (cf. Carl Jung Psychological Types): thinking for nouns (objectification of I and other which is later spread to all other objects that can be experienced in time and space), feeling (more like experiencing than having emotions) for verbs; sensing for adjectives, and intuition for adverbs (making distinctions such as very, seldom quickly and so forth have to do with an intuitive assessment of asensed situation. For example, the animal runs is noun (object/thought) and experience (feeling), the animal runs quickly is known through an intuitive assessment of the sensed situation. The first sentence is “inner experiences”. The second is “inner experiences outer” the third, due to the processes of projection and identification is “outer experiences” “outer experiences inner” “outer1 experiences outer2” and so on. These experiences exist long into the past and probably exist in other animals as well. However, when through the process of abstraction, inner becomes a category (agent) and outer becomes a category (patient)and experiences becomes a category (Verb), language can begin as the early beast/human says I sleep, I walk or transitively, I eatvegetables or I take the stick. His ability to name objects and

experiences quickly becomes a large number of basic sentences much like those described by Bickerton for Proto-Lanuage or the early speech of children. Though nouns and verbs seem at first sight to be the first two grammatical categories, they more likely occur at the same time with adjectives and adverbs which are the result of sensations of experience and expressions of intuitions about experience.

This stage of language can go no further without a further level of abstraction the creation of linking words that serves as the first bit of grammar. The first were most likely prepositions which are half verb and half noun. They provide both informationabout experience (between, near, on, under) and information aboutlocation in time and space. They talk about what is in time and space and their relation through time and space. IT is actually quite a creative link of intelligence to create a half verb half noun element such as a preposition. What is both in time and space and through time space can be only two things: position information between near on and so forth . That is is this is either an amazing leap of intelligence or just a kind of fall outfrom familiarity with nouns and verbs Perhaps derived from a deminoun or a demi verb, the prepositon could easily have arisen fromsituations su. Other aspects of grammar have to develop as well.

The core verbs of be have and do is probably the first breakout of the initial verb “experieince” be refers to states have refersto possession and do to actions. All three are aspects of immediate experience and can be easily developed from that first verb: be refers to intransitive actions or basic descriptions of inner experience, have is used for the experience of outer objects that have been incorporated into the world of inner whileactions are activities of inner acted on outer. The tree types of verbs that Bickerton mentions (186) can also be derived transitive verbs are the Ditransitives arise from

Why are there only two objects in a di-transitive (why are three or more impossible). Perhaps projected and indentified with simultaneously where the direct object is the projected outer andthe indirect object is the identified-with outer. There are only

three possible relations: inner, projected outer, and identified with outer as a kind of outer inner. Thus, there are only three possible sorts of verbs: intransitive, transitive, and ditransitive. All other arguments in a sentence are non-required; that is, they … Things like instrument, location, time,and so forth do not have to do with the core experience of inner,projected inner, or identified with outer. We can not have an indirect object as a sole required argument because that would apply an outward identification without an outward object. We must project out to something before we can identify thus the theobject that we identify with is always in reference to a projected object (not unlike jealousy).

Is projecting out a core ability for abstraction and the discovery of language: do animals not project?

What is this long diversion into Bickerton for? 1) to help understand what is difficult about grammar, 2) to help demonstrate the role of grammar in critical thinking.

The first bit of predication occurs when “I” can say something ofitself as an intransitive such as I sleep or says something of other (transitive) I took a rock.

which only later becomes “be”, “do” and “have”.

I and other are the first two categories of nouns.

It is not likely that pronouns existed at this early stage but some sense of a self that wanted, needed, liked,

P. 6 of Neumman’s The Origins and History of Consciousness. 1954.

“The dawn state of the beginning projects itself mythologically in cosmic form, appearing as the beginning of the world, as the mythology of creation. Mythological accounts of the beginning

must invariably begin with the outside world, for world and psyche are still one. There is as yet no reflecting, self-conscious ego that could refer anything to itself, that is, reflect. Not only is the psyche open to the world, it is still identical with and undifferentiated from the world; it knows itself as world and in the world experiences its own becoming as a world-becoming, its own images as the starry heavens, and its own contents as the world creating gods.”

Parts of speech as elements

And sentences as molecules and compounds.

References  Does this man think through his beliefs before they harden into unwavering principles? Is he open to countervailing evidence? Does he test his beliefs against new evidence and outside argument? Does his understanding of a subject go any deeper than the minimum amount needed for public display? Is he intellectually curious? Does he try to reconcile his beliefs on one subject with his beliefs on another?

  It's bad if a president is incapable of the abstract thought necessary for these mental exercises. If he is capable and isn't even trying, that's worse. It becomes a question of character. When a president sends thousands of young Americans to kill and die halfway around the world, thinking about it as hard and as honestly as possible is the least he can do.

From Truth out “is george bush a dope.

We are not conscious of our language facility though we are conscious of our language output. We are resistant to looking atwhat we are unconscious of. We are conscious of statements aboutmath, but we have to look at the unconscious faculty of language to look at nouns and verbs. We don’t say

p. 49 of Bickerton

“The putting together of words, once they have been selected is carried on so unconsciously, so automatically, that we are quite unaware of the mechanisms involved or even that such mechanisms exist. Yet if we did not have those mechanisms at our disposal, our vaunted ‘system of communication’, for all the richness of its conceptual map [dictionary], would not be very much more expressive that the grunts and snarls of beasts.”

Entity/N Affect/Adj = first subjects and predicates

In time and spacethe state of something in time and space

Through time and space

variation on something through time and space

A Proposal for a coordinated view of grammar and grammar teachingThe core of this is the division of grammatical studies into operative and speculative (following Mulroy’s use of the term), where the former includes all those aspects of grammar that are needed to describe aspects of writing (subject verb agreement), English grammar for its own sake, foreign language study, and logic and rhetoric.

Operative SpeculativeWords: Parts of Speech required argumentsPhrases: Parts of Sentence complementsSentences: Types of sentences ergativity

Conjoined coreferenceTransformations transformations

CONNECT BICKERTON’S PROTO LANGUAGE WITH LANUAGE BETWEEN NEANDRATHAL AND ERECTUS VIA NEUMANN/KLEIN

It was the discovery of the self/other distinction that led from proto language to language. Once this discovery was made subcategorization and empty categories fell into place. Other grammatical categories came from a recognition of relativity and

1. I other2. I other other I3. he/she

When living in a primitive “participation mystique” world and self are one allresponses are instinctive. The SRS makes this more tenable and leads toward Iand other, but with proto-language it is a largely unconscious recognition of I and other

Freud Klein Neumann – in the paranoid schizoid position there is only I and other and relative states of chaos or calm and response to those states: with the recognition that other is both good and bad and can be manipulated, grammar arises and man sinks into a depressive position

Pre-language paranoid schizoid position (I/exper)intransitive only PRS

Proto-language depressive position (I/exper/other)transitive and intransitive only SRS

Language resolve depressive position (I/exper/prs other/absnt other) ditransitive is added SRS

Subject/predicate exits from beginning as a experieince experiencer – topic???Grammar = subcategorizations exist at all three levels but empty categories donot until level 3.

Prepositions = relativity of time and space (absent other allows absent time of past and present) Tense=Case = others exist in their own right as subjects objects and there are more than oneGender = still a ways offPronouns=experiencer1(I) present other(the present mother)(2) not present other(the negative mother)

Tools artifacts and attempts to propitiate the universe appear at this time asa result of the knowledge of self and present other and absent other.

Rapid spread of language would occur through participation mystique rather than procreation

Complementizers: arize from pronouns: 3 become six fixed word order or modifications of pronouns quickly becomes an issue.

What about sov, svo etc distinctions

Necessary for Grammar:C-CommandCase AssignmentGovernmentBindingEmpty categoriesRequired argumentsOptional argumentsGrammatical Elements do not refer at all (unlike the substantive parts of speech) rather they reflect structural elements bickerton90 p. 10 (lexical items vs grammatical items

p. 11 he says, “Language, on the other hand, talks mainly about entities (whether other creatures, objects or ideas) and things

predicated of entities (whether actions, events, states, or processes).”ProtoGrammar Full grammar

Velocity distance = static dymamic where dynamic for adverbs modifies verbs and static/distance modifies nouns

Bickerton refers to the first verbs as behaviors and notes they are dynamic where entities are static. Category of categories: the set from which all others can be derived and which themselves can be no further broken down.

SRS SRS

Resolution of depressive position or connection in (participationmystique) with individuals or culture that as resolved the depressive position.

One-world intransitiveParanoid schizoid transitive (multiple outer but no relation between outers

They are either good and bad e.g one at a time)

Depressive Position ditransitive good and bad our one. Outer is one indvidual sometimes good sometimes bad (two at once) – noticing relations between others in the observed world.

Now union and separation are the only two processes thus the ONE self (intransitive) becomes TWO (transitive) and returns to ONE. The TWO becomes One and TWO and RETURNS TO ONE – the agent is the projector; the patient is recipient, the subject is the experiencer the predicate is the experience of the experiencer.

The resolution of the depressive position is the change from Proto-to Real Language and the explanation for subcategorized arguments. It allows one to project two objects rather than one at at time. The two can return to transitive, the transitive can return to intransitive. But there are no other possibities

Prepostions do not represent space and time – nouns and verbs do that. Rather they relate velocity and distance.

Case 1, 2, or 3 (subject object or indirect objectTenseNoun = in time and spaceVerb = through time and spacePrep= relative time and space or relative velocity distanceAdv = intuitionAdj = sense infoPossession

The boy runs in the school yard

Resolution of depressive position must also allow generation of preps, case, tense, comp and so on. Relation of one object to another prep to noun or prep to verb. Boy run yard boy run

Much the same as peripheral vision there is now a more

With the resolution of the depressive position and the discovery of ditransitivity relations between others arrive. This is the first hint that prepositions can be and that of stating the relation between others through case … Prepositions represent both time and space relations between nouns and verbs and now velocity and distance between nouns and verbs. It is like the squaring of a number. And hierarchy arises at this point

ON THE INTERCONNECTEDNESS OF ALL ASPECTS OF SYNTAX AS AN ARGUMENTFOR THE ALL OR NOTHING HYPOTHESIS

All or nothing thinking of the paranoid schizoid position disappears and a relativity takes it place.

The recognition that outer is not sometimes good sometimes bad, but one thing with more than one attribute; the world of syntax comes into place:The relativity of time and space and velocity and distance makes prepositionsThe recognition of roles rather than an unconscious acceptance of“other” brings in subcategorizationsThe need to maintain relations brings in case and fixed word orderEmpty categories are assumptions about fixed word order.

There is a blending of noun and verb that occurs to create prepositions that directly reflects the blending of good and bad into one. 2=1 means 2 have been discovered. InOnAt are all attempts to freeze things in time and space like “idea” “enthusism” and so fFrom forth; however, like “be” or “exist” they exist throughtime and space as well. With They are demi-verbs and demi-nouns at the same time. As such they cannot

receive case or tense. However, in their demi-ness they can relate nouns to verbs

or nouns to other nouns. A preposition tells you a little about how a noun is located in time and space and through time and space. They can also attach to verbs singly to expand the nature of that verb. Preps are both in and through time and space. You can say the “idea” I had yesterday, but in-ness or from-ness are not things that can be

LOGIC IS NOTHING OTHER THAN SENTENCES STRIPPED OF THEIR MEANING AND INTERPRETED SOLELY IN TERMS OF THEIR GRAMMARJohn is a lawyer. Becomes J is an a lLawyers have big houses L have b H

And so on. If the grammatical patterns match additional patterns that quarantee truth, then arguments are said to be valid statements are said to be true an false.

Verbs and nouns are representatives of our higher order functions: thinking and feeling. Adjectives and adverbs precedeas the representatives of our preceiptions: adjectives represent sensations and adverbs intuitions (judgments on things through time and space such often, regularly, slowly quickly, and so forth).