## CSETMathGuru: THE Site for Single Subject Math

**Random Musings on the CSET Singe Subject Math Tests**

**TOPIC:**Style of CSET SS Math Qs

**TOPIC:**Recommended websites for CSET preparation

**TOPIC:**Studying for the CSET Single Subject Math via online tutorials?

**TOPIC:**Graphing Calculator Online Tutorial

**TOPIC:**Could it be Better to Fail by a LOT than by a SLIVER?

**TOPIC:**Single Subject or Multiple Subject?

**TOPIC:**Tackling Test Anxiety and repeated Failure

**TOPIC:**A Pass is a Pass, right?!

**TOPIC:**Preparing for the Praxis II Mathematics Pedagogy (0065)Exam

**TOPIC:**Key Subtest III concepts

**TOPIC:**Score Verification for 219?

**TOPIC:**How Good in Math must one be?

**TOPIC:**Are the CSET Subtests

*valid*

**TOPIC:**Preparing for the Praxis II Mathematics: Proofs, Models, and Problems (0063) Exam

**TOPIC:**Taking the CSET repeatedly

**TOPIC:**Errors on the CSET?

**TOPIC:**Studying for Subtest I from a Precalculus book

**TOPIC:**Algebra topics for Subtest II

**TOPIC:**Studying hard, yet feeling unprepared?

**TOPIC:**Clueless, yet Taking the Test?

**TOPIC:**Grading the Free / Constructed Response section

**TOPIC:**Test Strategies for Passing the CSET

**TOPIC:**Advice for Middle School Math teachers

**TOPIC:**Style of CSET SS Math Qs

Here's the "magic" of the CSET: straightforward Qs that one might encounter in a regular text are routinely re-fashioned to cleverly "model" algebraic and geometric situations. As it should be... so that the myriad real-life applications of Math may be illumined. For instance, here's a "worldly" Geometry problem that I've contrived to appear more complex:

*Three cows are tied to posts and permitted to graze in a circle such that each circle is externally tangential to the other two. The length of the ropes that tether the beasts [?!] to the posts are 10m, 20m and 30m. Find the Area of the region left ungrazed in-between the 3 circles.*

In general, there are 2 likely deficiencies in candidates' preparation.

First, inadequate CONTENT knowledge. Translation: there are concepts the individual does not know at all. Of course, if you're really weak in fundamental concepts, it would help to take courses at a local Community college, say!

Second, inadequate PRACTICE with Qs at a CSET-level difficulty. After all, the Qs on the test are somewhat twisted, unlike those in a regular textbook!

The pass rate for the Math Subtests [especially Subtest I] is ~20%! Because as opposed to Qs from a text, there's always a subtlety involved, and as I always advice candidates, you can expect to be assessed on

* a few concepts that are unfamiliar

* familiar concepts in unfamiliar ways!

My detailed and exhaustive Q Banks for each Subset that seek to narrow this "bridge", and do so

*very*ably, addressing each / every CSET "domain" / topic area. Additionally, I've put together NOTES that are indeed quite exhaustive. Still, I do expect / recommend that candidates have a grasp of the foundations of each topic area since, both - the CSET and my materials - are at a level considerably higher than a conventional textbook!

As an aside, I continue to be staggered by the number of non-Math / -Science majors that sign up to SS Math Subtest I

*and*II and subsequently grouse about the inadequate time after consuming 4+ hours for Subtest II which is

*always*taken first].

Depending on the concept, real-life applications DO make their way into the CSET Qs. Naturally, some concepts - like quadratic functions, systems of equations, etc. lend themselves more readily than others. Here are a couple of examples, for purely illustrative purposes:

MCQ Subtest I:

*A fighter pilot is flying along a route described by the curve y = f(x). If a anti-aircraft battery is located at P(x1, y1), then at what position shall the plane be nearest to the battery?*

Free Response Subtest I:

*A window is in the shape of an equilateral triangle atop a square / rectangle. What should the dimensions of the window be that maximum light might be allowed?*

MCQ Subtest II:

*There are*

**x***T-shirts in a pile of which*

**p***are Large,*

**q***are Medium and the rest are Small. A fellow wants*

**n***T-shirts, one of each kind and delves into the pile extracts one T-shirt after another. What is the probability that he shall be unsuccessful in his quest?*

Free Response Subtest II:

*A child is constructing a cardboard decoration which is in the shape of a flat n-sided regular convex polygon to be cut from a circle of radius, r. How much material shall be discarded?*

I regularly tutor several chaps for the CSET, and when we discuss CSET-like Qs that I've contrived, the most frequent initial reaction is "

*Er, I wouldn't have known how to proceed...*" or "

*Is that ALL that you're asking for? Then why don't you be direct about it, confound you*?!!" Mental dexterity and interpretive fortitude are 2 attributes that the MCQs demand in dollops...

**TOPIC: Studying for the CSET Single Subject Math via online tutorials?**

In my opinion [borne of experience, naturally!], learning entire subjects - like Precalculus or Geometry - and even specific topics - for instance, graphing parabolas - from the Internets [?!] is not terribly prudent on account of the infinitude of material that one needs to sift through. The

__duotrigintillion__dollar question [what the deuce?!!] - especially for test-preparation purposes - is, um, has one secured all the pertinent information one needs?

A textbook would be a more reliable, not to mention, efficient, option.

To illustrate, consider parabolas [for Subtest I], which, according to a High School Algebra II / Precalculus textbook can be represented in 3-4 forms:

1. Standard Form

2. Vertex Form

3. Intercept Form

4. Conic Section Form [which, in certain circumstances, is really a variation of #2].

Now, simply googling

*Parabola*- or employing any search engine, for that matter! - yields sites dealing in Form #4 mostly, and, to a lesser extent, Form #1. In short, then, considerable information is being "denied", especially to one without adequate knowledge of the expanse of the topic / subject. Particularly since the syllabi are most decidedly cryptic:

*Analyze properties of polynomial, rational, radical, and absolute value functions in a variety of ways (e.g., graphing, solving problems*)...that sort of thing, what! Were a novice using online resources only for one's study, I'm not sure if that description would recommend the study of Quadratic Functions...

The really queer thing is that Form #4 alone is irrelevant for Subtest I...and insinuates itself in Subtest II under the topic of Locus.

Likewise, consider the competency:

*Prove theorems and solve problems involving similarity and congruence*[for Subtest II]. Um, how extensive must one's quest be?! I'm not sure if there are sites delineating sound Proof "strategies" while proving sophisticated propositions.

__Here__'s a passable site, and

__here__'s another but very insufficient for readying for an exam as unyielding as the CSET!

__Here__'s one I put together...before I discovered the excellence of

*Geometry: The Easy Way*!

Lest it be misunderstood, I myself use the Intertubes [?!] - especially Wikipedia - for "quick" research quite frequently [see, for instance,

__here__for 78 - yes, 78! - ways to prove Pythagoras' Theorem...about which Wikipedia has decent coverage, too]. It's just that for an endeavour as earnest as the CSET Math, conducting research on the Internetrons [?!] can be numbing...

On the other hand, for certain subject areas, say, US History, World Cultures, even Biology, I imagine "a Google" [

__sic__] would be indispensable!

Websites are likely to be diffuse, and of a supplementary nature ie. with persistence one might encounter a site that explicates a concept well but one's search can be laborious and meandering [greater profit might be had in the quest for the Holy Grail!].

Of course, one may argue that the "search process" itself can be considerably instructive and educational - for instance, read the contents of several sites on, say, spherical geometry or quadrilateral properties, and I'm certain, quite a bit of generalities can be gleaned [to pursue the idiotic gag further, one might not secure the Grail itself, but gets to rescue a rather toothsome lass, what! In passing, might one set one's sights low and seek a "grail" not necessarily holy? Just wondering, yes?!]

Also, online, it is only with hardship that one may extricate advanced-level practice problems of the cunning sort that competency exams like the CSET assesses.

Oh, just use a reliable textbook!

As an aside, I confess to being rather wary - oh, I still am! Paranoia gallops unrestrained in the family like a horse on steroids...or rather, a horse on steroids gallops as unrestrained as paranoia in my family (?!!) - about the usefulness of Wikipedia as a research resource, at times the quality being mottled at best, even downright confounding and misleading.

Be that as it may, I found the Wikipedia content of fairly robust calibre on Abstract Algebra topics of Groups, Rings and Fields. One need grasp only the rudiments of each, and the basic Laws of Commutativity, Associativity, etc. Naturally, as a fellow pickled (?!) somewhat in Math, I didn't the matter too abstruse...but give it a stab and make what you want of it!

**TOPIC: Graphing Calculator Online Tutorial**

Only Subtest II permits use of a Graphing calculator. Click

__here__for a Subtest II-specific TI-83 / TI-83 PLUS / TI-84 / TI-84 PLUS primer.

__Here__'s another smashing TI-83/84 site.

**TOPIC: Could it be Better to Fail by a LOT than by a SLIVER?**

If one misses passing by a scrape, it often leads to unjustified cockiness: Oh, I was this close! Perhaps, I can simply pull it off with just a quick review! And what an infinite folly is that...since, it may have been only the vicissitudes of the test that had led to an outcome as favourably close as that, not at all reflecting ones' true ignorance! After all, failure by a few points might have been simply a consequence of guesswork: um, how does one divine which concepts require extensive study?!

If one fails the test by a large margin [the passing score is 220 and only Pass / Fail is indicated in the transcript!], that would indicate an explicit and substantial need to study intensively.

**TOPIC: Single Subject or Multiple Subject?**

One might safely hypothesize an inverse relationship between Pass Rate and Employability...while the Multiple Subject tests might be easier to overcome - requiring knowledge that is a mile wide but an inch deep - on the downside, I imagine there not being a cornucopia of

*those*jobs

*relative*to the Single Subject ones.

Conversely, English teachers - likewise for Math and the Sciences - don't undergo too much privations securing employment [once competency has been demonstrated...for which, it is indeed true, the success rates are lower].

*That*ought to be a criterion not to be cast off idly.

**TOPIC: Tackling Test Anxiety and repeated Failure**

There are those afflicted with congenital Text Anxiety i.e. they were

*born*that way, feeble and fragile and need to be Handled-With-Excessive-Care [

*of course*, I jest!], and others who owe their nervousness to a chronic absence of success. The former feel the onset of anxiety even when well-prepared,

*en route*to success...and could secure relief via breathing exercises!

The larger endeavour requiring remedy that

*can*be controlled by the candidate is dreading the test, having taken it several times already. Of course,

*repeated*failure likely indicates insufficient knowledge / mastery of content matter and / or a lacuna in test-taking skills [

*part*of which is anxiety-management]. Very likely, a combination. [Here, I shall accept kudos for Excruciating Banality.]

In the former case, one

*ought*to simply master, at the least, come close to attaining it...asymptotically [what the deuce?!], the syllabus as laid out at the CSET website. There are no short-cuts, paved picturesque roads that may be traversed untrammeled and effortless. One should aim for rigour, for it is that that shall mark the able instructor in the classroom.

Yes, it

*is*a platitude to declare that those that fail to pass the SS Math CSET aren't adequately prepared. In my experience tutoring candidates, I've observed characteristics prevalent amongst my own students at school: there are those that DO Math and emerge triumphant, and others who merely LOOK AT Math and worry at the adverse outcomes. The latter chaps sit recumbent with a textbook, reading the examples and nodding sagely in apparent comprehension:

*Oh, it all makes sense!*, they exclaim merrily moving on to "master" the next section.

Query them about their preparation and they shall unintentionally reveal their application, or lack thereof:

*Oh, I've LOOKED AT the material! I've LOOKED AT my Notes!! I've LOOKED AT the California State Math Standards!!! I've LOOKED AT the Practice Qs!!!*and so on. Is it any wonder that their scores aren't quite worth LOOKING AT...several times?

While all subjects call for rigourous application for concepts to be grasped and reinforced, Math is somewhat

*sui generis*in its "requirement" of having to put pencil to paper: one may contemplate the chemistry that underpins one's cooking, say, or the biology relevant to one's garden, but idle rumination on finding the intervals for which one curve lies above another might not yield commensurate dividends. Actually embarking upon to resolve the Q might reveal obstacles not heretofore perceived.

CSET SS Math Subtest I, in particular, is quite challenging. One's efforts should, ergo, reflect the

*purpose*of the test: is one sufficiently equipped to

*teach*a High School Precalculus class. Even if Middle School is really one's destination.

In my case - however anomalous that might be! - given as I have countless tests [um, I imagine I

*could*count them, come to think of it, what!], I have indeed found myself prey to the usual apprehensions that bedevil others - fed as they are by uncertainty about outcomes - which I have

*then*sought to banish through robust preparation.

On test days, I am the very soul of equanimity - giving the Dalai Lama a run for his

__Renminbi__! - ...since I am seldom under-prepared.

**TOPIC: Preparing for the Praxis II Mathematics Pedagogy (0065) Exam**

The Pedagogy exam is all about how one'd go about teaching myriad Math concepts of a fundamental nature. The weird thing, of course, is that till one is actually in a classroom setting, one doesn't give much thought about instruction...after all, why would one ponder over it?!

Here's how I'd go about preparing for the pedagogy Praxis - I've examined the Syllabus & Sample Qs on the ETS site quite robustly! - and the following might offer a sliver of guidance.

I'd examine a Middle school and/or 9th grade Algebra I & Geometry textbooks, and spend time [~ 10/15/20 min!] contemplating Instruction and Assessment for each section.

I'd just write down a few teaching ideas with regards to:

* INTRODUCING the topic,

* CONNECTING the concept to the students' PRIOR KNOWLEDGE,

* EXTENDING the concept,

* ANTICIPATING the COMMON ERRORS that students might be prone to committing

* devising progressively sophisticated illustrations

* EVALUATING student UNDERSTANDING and APPLICATION [q.v. Bloom's Taxonomy]

Naturally, time constraints preclude doing this for every concept area pertaining to Middle School Math / Algebra 1 / Geometry.

But I'd surely examine the fundamentals viz.

Negative Numbers,

Operations on Integers,

Fractions,

Operations on Fractions,

Decimal Representation,

Ratio & Proportions,

Percentages,

Absolute Value,

Probability,

Linear Equations,

Relations & Functions,

Graphing and Writing Equation of Lines,

Parallel and Perpendicular Lines

Systems of Equations

Inequalities

Exponents,

Polynomials,

Factoring,

etc.

The mental exertion expended on even a sample of those areas shall be of tremendous profit, providing a "template" for rumination on the Test.

It may even be useful to talk to Middle School teachers about their lessons! [Inquire from the Principals of local schools about their highest-performing / reputed Math teachers and get their input. In my experience, most teachers are happy to give "those who seek" their time freely!]

**TOPIC: Key Subtest III concepts**

In general, focus on Applications [nothing "hard-core"!].

- Graphing trig. functions and identifying trig. functions from a graph / word problem

- Proof and Applications of the Laws of Sines and Cosines

- Proof of the 3 Pythagorean Identities [an old favourite is about sin^2 x + cos^2 x = 1 for x, say, in the 2nd Quadrant...oh, it's the same bloody principle!]

- Determining the Area / Perimeter of a regular convex n-sided Polygon, and its Relationship with a inscribed / circumscribed circle.

- Finding the coordinates of a Point - in terms of trig. ratios - on the vertex of a regular convex polygon circumscribed / inscribed on a circle of radius, r, say[oh, just use: x = r cos theta; y = r sin theta!].

- Existence of a Limit & Finding Limits

- Continuity / Differentiability of a Function [the distinction thereof!]

- Derivatives as instantaneous rate of change & Finding Derivatives of Simple Functions using the Definition [Do not waste time finding derivatives of complicated functions: familiarity with polynomial, rational, irrational, trigonometric and logarithmic / exponential differentiation would suffice!]

- Applications of Derivatives:

* Maxima / Minima of curves, Concavity / Convexity, Increasing / Decreasing Intervals, Intercepts, Curve-plotting...in general!

* Related Rates

* Maximizing / Minimizing a Function subject to constraints [Surface Area / Volume]

- Implicit Differentiation

- Basic Differentiation Rules [Chain Rule; Power Rule; Multiplication & Quotient Rules]

- Intermediate & Mean Value Theorems

- Definite Integral & Properties: Finding the area under a curve using Riemann Sums [Do not waste time finding integrals of complicated functions: familiarity with very basic polynomial, rational, irrational, trigonometric and logarithmic / exponential differentiation would suffice!

- Fundamental Theorem of Calculus: Proof

- L'Hospital's Rule

- Infinite Series: Convergence Tests; Power, Taylor, MacClaurin Series; Radius of Convergence [Basic Understanding]

- Application of Integrals: Finding lengths of Arcs; Areas / Volumes of Solids [Basic Understanding]

- Applications of Exponential and Logarithmic Functions: Exponential Growth / Decay Situations

Again, if one finds oneself getting involved in some complex manipulation, one has gone too far! For those with more sophisticated Math skills, Subtest III is the easiest of the lot. Far more rigourous Algebraic manipulation has been accomplished in Subtest I! Subtest III is - in my opinion - less austere than the equivalent of a 1st Semester College Calculus course. An AP Calculus BC student would be over-prepared!

**TOPIC: Score Verification for 219?**

219 is a "common" FAIL score - I've personally known miserable fellows that secured 219 4-5 times! [Yes, it's a bloody pity!]

Otherwise, NO, it's not worth submitting to a re-scoring. The grading procedures, in general, are quite rigourous and consistent: considerable training precedes the actual process wherein the scorers are apprised about the myriad ways to approach a problem, and the partial credit to be apportioned.

Furthermore, customarily, each Free Response Q is graded by 2-3 scorers so that in case of discrepancies, a "supervisor" shall break the tie. Finally, the scorers "specialize" in particular Qs: one might ONLY grade the 1st response for, say, 100 candidates. This enables "mastery" in grading that Q!

It's a lot akin to grading AP tests.

**TOPIC: Preparing for the Praxis II Mathematics: Proofs, Models, and Problems (0063) Exam**

All I offer is a template for study: what I'd do were I to accoutre myself capably.

Proofs could be Precalculus [Algebra / Trigonometry]-based or Geometry related.

a) For proofs that have Precalculus as their underpinning, I'd suggest mastery of the following derivations / propositions:

- Irrational Numbers [Prove that sqrt(2) is irrational, that sort of thing!]

- Quadratic Formula

- Derivation of the Vertex of a Parabola

- Factor and Remainder Theorems

- Rational Roots Theorem

- Conjugate Pairs Theorem

- Log Rules

- Trigonometric Identities

- Law of Sines and Cosines

- Equations of Conic Sections in Standard Form

- Binomial theorem

- Mathematical Induction Propositions

- Divisibility Rules for 3, 4, 6, 8, and 9

Additionally, it is a sound notion to be proficient in various definitions and properties such as:

- Even and Odd Functions

- Increasing and Decreasing Functions

- Inverse and Composite Functions

and so on. The Index of a Precalculus textbook would be a masterful resource. The rationale for this is that examining the ETS's Test Guide for the Exam, an elementary Proof Q based on Even Functions is shown, and er, if one is innocent of the very definition, well, the obstacle shan't be surmounted, yes?!

Another variety of Proof is simply a regular problem with the Q itself containing the Answer wherein one is required to "prove" that such is the case. For instance,

*Prove that*- as in "Show that"! -

*the function f(x) has irrational roots between 2 and 4*. These Qs, of course, are relatively straightforward!

For study resources, any solid College Precalculus text should prove equal to the task; recommendations are

__here__.

b) For Proofs calling forth knowledge of Geometry, well, there is no limitation there! I'd recommend using texts suggested

__here__, especially a) Barron's

*Geometry: the Easy Way*By: Lawrence S. Leff; OR d) Houghton Mifflin's

*Geometry*By: Ray Jurgensen, Richard G. Brown and attempting every bloody proof they cast!

Modeling is a more delicate concept, relating to applying algebraic / geometric principles to real-world problems and applications. I would suggest trying out many of the word-problems at the rear of Q sets in each section of a robust text. The following areas afford capital opportunities for modeling:

- Functions: Quadratic, Composite, Piece-wise, Exponential and Logarithmic

- Conic Sections

- Trigonometric Applications

- Systems of Equations, especially Linear Programming

- Statistics [Least Squares Regression, Normal, Binomial & Poisson Distributions]

**TOPIC: Taking the CSET repeatedly**

An unfunny joke is that I've encountered candidates that rather fatuously believe that if one were to give the tests enough times [what the devil?!], one might be exposed to the entire gamut of Qs!

Alas, but were time more lucratively devoted to studying for the bloody tests! When one budgets for the money gambled on this foolhardy venture - for there are no guarantees and it costs ~$75 per test - the folly is staggering...and what horrid teachers these shenanigans make for!

**TOPIC: Errors on the CSET?**

Hardly ever, if at all!

NES does permit the candidate who suspects a mistake to describe the issue and file the grievance with the Proctor. Apparaently, NES will review the question based on the information provided and make a judgment to throw the question out or not. Of course, one never knows as the decision is not broadcast!

But don't mistake your ignorance for an error on NES's part, [which, I hasten to add, doesn't seem to be the case in this situation!].

Just clarifying, so that less-than-prepared blokes don't wantonly scream "Bloody murder!" each time they encounter something unfamiliar! But alas, that does not preclude the possibility of there indeed being a mistake. Akin to the logic of Just because you're paranoid, doesn't mean they're not out to get you!, I imagine that among the pitfalls of ignorance is the verity that one might not know if errors really bestrew the test document!

**TOPIC: Studying for Subtest I from a Precalculus book**

Almost all concepts in Precalculus need to be mastered for Subtest I. The following alone need be

*skipped*:

* Trigonometry more sophisticated than ordinary ratios and knowledge of sin / cos / tan of 0-30-45-60-90

* Permutations & Combinations;

* Probability

* Conic Sections

* Parametric & Polar Functions

In general, problems requiring tedious calculations / use of a calculator may be avoided - albeit the underlying concepts may be vital!

**TOPIC: Algebra topics for Subtest II**

The following Algbera concepts ought to be studied from a Precalculus textbook:

- Lines: Graphing, Writing Equations [of Parallel and Perpendicular lines], Finding Points of Intersection

- Distance & Mid-Point Formulae

- Right Triangle Trigonometry

- Law of Sines and Cosines

- Conic Sections: Circles, Parabolas, Ellipses, Hyperbolas

- Permutations, Combinations & Probability

**TOPIC: Studying hard, yet feeling unprepared?**

I was examining the

__CSET Subtest descriptions for Subtest I__, and here's something to mull over by candidates who felt that they'd studied extensively, but in retrospect, suppose their efforts to have fallen short:

the descriptions are

*extremely*pithy and cryptic [translation: brief, oblique and mystifying!]. There's

*substantial*content

*implied*in each statement.

For instance, under

**004 Linear Algebra (SMR 1.4)**the description states:

...

*c. Understand apply the basic properties and operations of matrices and determinants (e.g. to determine the solvability of linear systems of equations)*

Simple and straightforward, yes?

NO!

This is equivalent to

*mastering*about ~70+ pages of content from a regular textbook (er, depending on how "regular" one's textbook is, naturally!).

And don't get fooled by the dissimulating phrase "basic properties and operation". The meat of the topic is about Properties of Determinants, Matrix Multiplication, and critically, solving Systems of Equations using Matrices [in 2 ways: using Inverses (a most excruciating procedure!) and by using an Augmented Matrix (less so!)] and Determinants [Cramer's Rule], and Classifying systems of equations as Consistent / Inconsistent.

The aforementioned topics

*can*be rather challenging to a novice candidate so that one might, understandably, be disposed to merely "skim" through the content to "get a feel", wonder about its relative importance in the larger scheme of things ["Who am I?", "What am I doing here?", "Whither did I come from?" "Thither am I going?" and "What is this rattlesnake doing in my bed?" "Shall I have time to scream, Aaah?"! OKaaaaay, enough tomfoolery!] and be tempted to gloss over it, since the problems are indeed long-winded, tedious and cumbersome to work out!

Here's a sobering thought: while relative emphasis differs from one edition to the other, Matrices and Determinants are assessed extensively on the CSET Subtest I!!

And so on...for every succinct description in the CSET guide for Subtests I, II

*and*III.

I wonder if this "disconnect" is what does most inexperienced candidates in!

Mentioned repeatedly before and worth belaboring again is the First Principles: the reference to the California Content standards. For a correspondence of Subtests to Content Standards click

__here__for Subtest I,

__here__for II and

__here__for III...

Hope this penetrates the veil of ambiguity a trifle...

Also, the MCQs in particular, tend to evaluate familiar concepts in unfamiliar ways - much like the SATs! For instance, instead of asking "Find the roots of f(x)" [a direct textbook Q!], they might demand: "At what points does the graph of f(x) intersect the x-axis?" and to one unaware of the equivalence of the terminology "root of equation" ~ "x-intercepts of corresponding function", it might prove frustrating...though it's an absurdly trivial Q!

It's a good idea to examine / review terminology

a) at the end of each chapter: good texts shall have a

**Things to Know**or

**Terms to Remember**section just before the

**Chapter Review**exercises!

b) from the index at the back of the book - scan each term and write brief descriptions!

**TOPIC: Clueless, yet Taking the Test?**

Candidates often choose to take a stab at CSET Subtests if only to gauge their competence - or, er, lack thereof!

Honestly,

*I*don't see anything inordinately valuable emerging from it: if, as professed, the intimacy with Math is marked with a significant gap, and assuming that not much yet has been done to fix the impoverishment in skills, well, unless the situation is substantially remedied, one shan't even be able to recognize the specific concepts that the Qs seek to evaluate, especially if it's from some topic like Inverse Functions or Solving Systems of Equations using Matrices that are simply glossed over in High School.

I am compelled to recall the words of the irrepressible, Mr. Rumsfeld:

"...as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns -- the ones we don't know we don't know..."

Aye, it shall be that last that shall trip you: the unknown unknowns!

On the other hand, were one to have a reasonable Math background, then, unless a refund has been inveigled off those NES buggers OR one finds onself enveloped by a miasma of malaise (?!) on test day, PLEASE do NOT forgo the priceless opportunity of taking a dekko at the tests! Even under the worst-case scenario of believing yourself woefully under prepared, the peek shall enable you to discern what you are really against. [And perhaps you shall be pleasantly surprised, yes?] This shall enable you to direct / re-direct your efforts in a more apt direction ...

Next, in such narrow-specialty circumstances i.e. one where the Math knowledge is quite specific, one is almost always inclined to denigrate one's abilities and competence [I imagine it's different when one goes to war...oh, I jest, you fellows!] - and the CSET tests are indeed a strange beast, and every encounter seems to engender fresh tales of terror among those inadequately equipped. So feelings of acute anxiety and a propensity to crawl under things (?!) are quite natural.

The point is - OK, I was rambling there...! - chaps that've done advanced Math in the past, with strong mathematical / analytical / manipulative skills - say, engineers - well, these fellows, despite the passage of time and notwithstanding their apprehensions, retain extraordinary amounts of residual knowledge / skills!

And I speak earnestly having personally interacted with / assisted countless chaps with robust Math backgrounds who were deciding to shift to a teaching career from a "professional" one. And oddly enough, these blokes tend to be more wary of their chances of success than others!

**TOPIC: Grading the Free / Constructed Response section**

The Free Response Qs are graded based on the following parameters, with the letters in the transcript denoting "codes" that were

*lacking*in the reponse:

*p*~ Purpose: indicates that the response didn't show adequate evidence of the test taker having grasped the overall object of the question.

*k*~ Knowledge: indicates the candidate ought to have employed more buzz words [!]. As someone described it, "CSET, unlike CBEST, is designed for Practitioners In The Field and is an invitation to Talk Math with all the terminology one can muster".

*s*~ Support: suggests an absence in bolstering one's answer with evidence, showing steps, referring to Theorems, stating formulae, that sort of thing.

*d*~ Depth and Breath: um, I wonder what this would mean!

In case of scoring, first, of course, is the grader's perceptions of one's responses lacking in Purpose, Knowledge, Support, and Depth and Breath of understanding. As any teacher shall attest, students routinely contest their grading, arguing how certain answers merit more points.

Differences always exist, of course, between 2 graders (we are confining ourselves to the Free Response Qs, naturally!) but tests like the CSET - and likewise, the AP Exams & the SAT English Essay - have VERY explicit "rubrics" or templates about how EACH response is to be graded. Specific details / steps are looked for, and in their absence points are taken off! Simple. Since the graders undergo considerable "orientation", for practical purposes, there is little / no variation between them. In all such cases - where subjectivity MIGHT play a role - every response is scored "blind" by 2 or 3 chaps: each is unaware of the points awarded by the others. Ordinarily, there is complete consensus (yes, it's really remakable, but then again, maybe not!), but in instances of a disparity, the assessment is "appealed" to a supervising grader who makes the final decision.

So, on the whole, it's a fairly rigorous process.

Second, it is NOT the quantity, but the quality of your answers that matter. I distinctly remember composing some VERY succinct responses to a few Qs, but since they were thorough in their resolution to the problem, they weren't penalized at all, manifesting as they did both Knowledge and Depth / breath of understanding [I got ++++s virtually across the board for all 3 Subtests...].

So, were one to take an arbitrary stab at the Qs and the graders perceive that the respondent is merely groping his / her way around with no sense of direction - it can be obvious at times if one is merely "experimenting" - then depending on the direction taken ie. the objective isn't conclusively reached, one might receive austere penalties.

In the AP Stat Exam, for example - similar situations hold for the other tests too... - students are implored to "take a stand and justify it" rather than indulge in ambiguity: "on the other hand..." even if their hemming and hawing actually points them in the right direction. Furthermore, if a fellow supplied a perfect answer, but at the end exhibited even the remotest "ambivalence" by "straying from" / "contradicting" what transpired before, he is marked down HEAVILY, as it is deemed to indicate an inadequate mastery of the concepts!

Anyway, would I advice score verification? Let me just ask you to invest that money in a more productive enterprise, like, um,

*Amway*!

**TOPIC:**

**Test Strategies for**

**Passing the CSET**

If there was but ONE strategy for the FR Qs that was demanded of me, this would be it: do NOT leave ANY Q blank!

In my assessment, the grading is not stringent at all, and effort, however, miniscule, is rewarded magnanimously. In the past, I've had several "new" candidates report that they got a 219, and I suspect they had left a couple FR questions unaddressed!

Eve if they felt totally at sea with the Qs, sustained contemplation may have led to ideas and concepts that build on one another to furnish a response, which, while incomplete, might not suggest rank ignorance! Partial credit is easily vouchsafed!!

Offhand, "studying more" is not the same as "studying right" [yes, yes, it's easy for me to pontificate thus, and, er, that is precisely why I, um, pontificate thus!]...

The point is that a gimlet-like focus needs to be directed at the CSET domains [shall we all agree to ostracize the blighter that struck upon this utterly pompous term?!] as outlined on the official CSET site as well as "fine-tuned" and expatiated upon

__here__for Subtest I,

__here__and

__here__for Subtest II and

__here__for Subtest III.

Next, it isn't a terrible idea to accoutre oneself with quality study resources. Numerous threads on this site have dwelt on this realm: while some swear by Schaum's Series, and others swear

*at*Schaum's Series [oh, just an idiotic gag! seriously, the books are quite serviceable!], my own favourites - texts I've used myself quite lucratively or those I've browsed upon others' endorsements - are summarized

__here__for Subtest I,

__here__and

__here__for Subtest II, and

__here__for Subtest III.

Then, either one teaches oneself and has, both, the capacity and the disposition to do so, or one hires a professional tutor, perhaps, or, if time is not o.t.e. one might consider enrolling into an Intermediate Algebra, and then, Precalculus class at a local Community College, as the need may be...

In my experience with countless candidates that've got a 200+ or 200-, fundamental stuff like Quadratic Functions and Equations, Systems of Equations, etc. have usually been mastered. It's the "hard-core" Precalculus concepts like Mathematical Induction, Linear Programming, Logarithmic Functions, Solving Equations with Matrices, Hyperbolas, etc. that cause one to trip up. It is the latter that ought to be cynosure of one's attentions...

Finally, subjugation to a severe, unsparing, even forbidding [what the devil?!] study regimen is much recommended [oh it's just the simulacrum of the monk that resides in me speaking...]. Of course, there shall be resolute opinions to the contrary; but one needs to realize that there is no room for frivolity and disport here! There is a notion - to which I readily vouchsafe my imperial imprimatur [?!]; everybody can relax now! - that each Subtest requires ~ 100 hours of diligent study, untrammeled by sundry diversions. Much merit resides there!

Do realize that most of the CSET Qs are calculated to befuddle, perplex and discombobulate. The capacity to make "conceptual leaps" is an enviable attribute to own.

"

*Oh, if *that* is what you seek, why in the blazes can you*

Not say so in syllables of 2?!"

Not say so in syllables of 2

Alas, the very pedestrian recommendation remains: try problems and exercises from a variety of resources.

Anyway, chaps that didn't pass: don't lose it, yes? This is all infernally frustrating, to be sure, but agonizing self-recrimination is self-defeating. There are sure to be concepts / topics on the test that made the eyes glaze over: make a list, and take a vicious stab at those, for a start!

And don't tarry!

I've known chappies that've taken the test 8-9 times. Not a very savoury porspect, what? Imagine the time and money disspipated!

Well, I've always maintained that the CSET SS Math skills are NOT of the elementary or even intermediate calibre! There is so much complacency about these tests primarily on account of bogus courses / materials - poor sods paying for Math Qs devised for the Multiple Subject Credential test: what a beastly scam and a veritable crime upon the credulous...- and county / district tutorial programs caculated to reinforce basic concepts than equip one with more advanced ones. One would imagine that practically ALL aspirants are accoutred with rudimentary skills, yes? I've had blokes declare that they knew more than their error-prone stammering instructors, and that Qs on topics like factoring and quadratic formula were agonized over. This sort of balderdash is INDEFENSIBLE!!

Anyway, a "loose" touchstone for one's state of preparedness is this: could you open a College Algebra / Precalculus book to ANY page (excluding the Trig stuff, naturally) and solve ANY problem? Repeat, ANY problem? Especially, the usual bugaboos of candidates:

* Mathematical Induction,

* Linear Programming,

* Vectors,

* Graphing Rational / Logarithmic / Exponential Functions and

* Solving Equations Using Row Transformations re Matrices

* Modelling Quadratic Situations

When I speak to candidates to gauge their skill-level on the aforementioned areas, it produces no little bemusement to hear, "I don't think I would have any serious difficulty doing them!" This sort of convoluted "double-negative-ish" hedging phraesology immediately communicates that they DEFINITELY do NOT possess the requisite competencies, and are simply embarrassed to admit the lack of virtuosity. Understandable, of course, under the circumstances, but reflective of a state of denial, to explode which, would call for the services of that Genius of Scolds, Dr. Phil.

My prescription remains:

* Get good books / reliable prep resources / materials

* Do ~ 2 hours of study on weekdays, ~ 4-6 hours on weekends

* Banish from your existence any dalliances with flippancy and frivolity till you emerge victorious. There shall be a moment for wanton celebration, later.

**TOPIC: Advice for Middle School Math teachers**

The duration for preparation for Subtest I, unsurprisingly, depends on one's familiarity with advanced Algebra and middle-school Math teachers [current / prospective] aren't terribly proficient in High School / College level Precalculus - the assumption, after all, is:

*why would I need it if I'm teaching Middle School*?!!

That said, if studying individually is a frightful chore, form a study group if possible (though it's hard to find chaps in the neighbourhood / area in a like situation... I would have imagined that "advertising" on this Forum to seek fellow pilgrims worshiping at the same CSET shrines would be more widespread)!

Break the entire course / syllabus into chunks, prepare a detailed plan to cover a certain number of sections each week, and come hell or high water, resolve to accomplish your objectives. Needless to say - and yet I say it! - sacrifices shall have to be made: get up earlier / sleep later, study when others are yet to awake! Put in ~2-3 hours daily and ~4 on weekends. Curtail customary diversions: fun and frolic can be had once the Subtests have been surmounted...

Alternately, hire an accomplished tutor or get somebody from the state university in advanced Math classes as an instructor. You shall make much quicker progress!

But, as I've said before till I've got blue in the face - and you should see my face: it's decidedly

*blue*, and not merely because it was struck recently by a violent blunt object... - there *are* NO shortcuts, NO elixir to magically transform the unprepared bugger to an intellectual X-Man. So guard against complacency.

Anxiety - unless it is marked by unseemly panic that requires blokes in masks poring over you and yelling "CLEAR!" - is OK: it's a natural reaction that usually ceases once the test begins. I passed all 3 subtests at one shot, but even I was more nervous than a Mafioso appearing as witness for the prosecution.

If you don't pass this time, examine the back of your Score Report and remedy the deficiencies. Gird your loins to "lucubrate", which, my bedside Oxford informs me, refers to "studing assiduously into the night"...

I'm sure there shall be chaps out there that shall vociferously disagree, snort their disdainful protestations and be willing to slap my countenance and challenge me to to duel at sundown if not actually start Holy Wars and begin turning keys for the launch of nuclear missiles at my peremptory declaration. But the inflexible Tauran that reposes in me refuses to concede this point of no short-cuts even upon threat of exposure of past crimes and misdemeanours. (Three Cheers for the Statute of Limitations!) So there: those that don't relish it can stuff in their collective pipes and smoke it!

But remember: there's nothing like success to put the flush to one's cheeks, a scabrous song to one's lips, a swagger to one's steps, and dionysian mischief in one's intentions!