Mathematics 314
A Secondary III Course
This is a third year high school mathematics course taught in Quebec. An abridged and sometime paraphrased version of the Quebec government course objectives in this pfd file follows. The aim is to provide a clear, accurate and full image of those MEQ content objectives as is, apart from any criticism I may have.
The MEQ approved textbook for this course (English language instruction) is not a source of enlightenment. Join a group to discuss how to survive or bypass it.
Relative Importance of the Objectives
- use algebra to generalize situations. 45%
- apply knowledge of geometric figures. 40%
- analyze statistical data. 15%
Observe Geometry is put second, not first. I would put it last and arrange matters so that with this placement of geometry, topics not necessary for further courses or distractions there in (my opinion) are emphasized the least.
The prerequisite to this course is or should be mastery of fraction
sense and skills, and the ability to solve one equation in one unknown or
to solve systems of equations in essentially one unknown. However
most secondary III students enter secondary III with weak fraction sense
and skills and hence weak algebra sense and skills. Those prerequisites
need to be reviewed and ensured at the start of mathematics 314.
Otherwise students are going to continue through secondary III, IV and V
without the very skills the MEQ says are needed for algebra. The
lack will undermine and sabotage their learning and teaching.
First, fraction skills and sense are prerequisite to algebra. Here I recommend the following sequence to build or rebuild fractions skills:
During steps 1 , 2 and 3 emphasize divisibility rules for recognizing multiples of 2, 3, 5 and 10 and employ in simplification.
Second, continue the development and maintainance of fractions skills and introduce or re-introduce algebra with the solution of linear equations in one unknown with or without stick diagrams. Inclusion of stick diagrams and fractional operations on them, if your students permit, reinforces fractions sense (the concept of what is a fraction) and the notion of balancing equations.
Third, emphasize that every formula in high school mathematics and science will be used forwards and backwards (directly and indirectly). Given numerical how - include arithmetic solutions for the backward problems or backward use, and point your better or more able students to the literal or algebraic solution of backward problems. And every time you use a formula or proportionality relation backwars in a arithmetic and/or algebraic (literal) manner repeat the message: every formula and proporitionality relation in high school mathematics and science, technology too, will be used forwards and backwards, or directly and indirectly. Expert Advice: If you do not review and develop fraction skills and sense, and algebra sense too, the rest of the course will be a formality, and both learning and teaching will have more difficulty through-out. |
Connection with earlier studies
As they acquire new knowledge, the students will review the following skills and concepts acquired in previous programs:
-
number and operation sense;
-
the habit of estimating;
-
sense of proportionality;
-
understanding of the concept of a variable;
-
translation from one mode of representation to another;|
w: in other words, numerical situations can be described in different ways. -
definitions, properties, theorems or corollaries related to
-
different geometric concepts;
-
organization of statistical data;
-
simulation of random events and the concept of probability.
Moreover, it should be noted that in the Secondary III program, arithmetic content has been incorporated into the terminal objectives relating to algebra, geometry and statistics.
W: In other word, the numbers skills and sense should maintained and developed through out this course
Objective 1 (45%)
use algebra to generalize situations
Since we are now living in the information age, students should be equipped to handle, process and interpret the information they will encounter.
(w) what does the latter mean in practice.
In Secondary II, student should have learnt that algebra was a powerful and useful language or communication tool.
(w) see the notes in Math 216 notes here on developing algebraic thinking skills.
In Secondary III, students will expand their knowledge in this area by using algebra to derive general rules from a number of specific situations.
(w) Need examples
Conversely, they will apply these general rules to individual cases.
Like any other language, algebra has its own rules and syntax, and it is important that the students observe them.
(w) But poor notation on final examinations is allowed according to the marking rubric. That leads students and some teachers to conclude syntax or notation is not important.
Improper syntax or notation (including improper use of the equal sign) demonsrate an improper comprehensions. Confusion over notation leads student to mean one thing, write what they mean incorrectly , so it means a second thing, and then later read it as a third thing - all that departs from mastery of algorithms for arithmetic and algebra in a repeatable and reproducible manner.
After performing operations on arithmetic expressions given in exponential form, they will generalize the properties of these operations and then apply these properties to algebraic expressions. .. In this case, literal expressions do not have to represent a specific value and are used as mathematical objects.
(w) If students are uncomfortable with the ideas of a letter representing a number as indicated by the phrase Let q be a number, say with tongue in cheek, apparent to the students, Let q be the secret number in this envelope. and/or say let p be the length of this line segment. Explain we can talk about numbers or describe their role in calculations without needing to know their values. There is more to mathematics than just doing arithmetic. The Logic & Algebra area and Solving Linear Equations with Stick Diagrams of this site contains ideas for introducing the algebraic way of writing and reasoning. Chapter 14 and 15 in the Logic & Algebra area give and contrast arithmetic and algebraic solutions to problems.
The Pythagorean theorem links algebra and geometry.
(w) Chapter 17 in the Logic & Algebra area presents the Chinese square proof of the theorem algebraically. The MEQ approved text indicates the proof without the algebra.
The theorem will even help them understand the concept of irrational numbers, thereby allowing them to complete their study of the set of real numbers.
(w) Enriched sections of 314 might be shown the assumption that sqrt(p) is rational is inconsistent with unique prime decomposition of natural numbers when p > 1 is prime. The Number Theory. site area should include the proof (if not its addition is on the site to do list)). The Logic & Algebra has several chapters on direct and a postscript on indirect reason The latter gives an explanation of proof by absurdity or contradition.
|
w - How to Develop Algebraic Reasoning Skills. With the use of letters to denote quantities or numbers, expression involving those letters become meaningful. They describe calculations that could be done. By using letters to denote lengths or non-negative numbers, the commutative law for multiplication represents the notion that two different ways to compute the area of a rectangle should provide the same result, the distributive law and the foil method represent two different ways to calculate the areas of a rectangle as a whole or as the union of subrectangles. The commutative law for addition represents the ideas that the order in which two line segments are placed or measured does not affect the overall length. The distributive law can also be associated with the notion that a change of units (change of currency) should not affect a sum. Geometric significance here provides a scaffolding for the introduction of algebra with positive or non-negative quantities. By algebra in the first instance, we mean the role of shorthand notation in denoting numbers and quantities, and beyond that in describing the calculation of numbers and quantities, named or not, and the equality of calculations - when one calculation can replace another because both give the same result.. The simplest context for introducing algebra appears before or apart from the use of negative numbers and lengths and areas are non-negative. The site area Solving Linear Equations with Stick Diagrams may be used to introduce and re-enforce the skills and concepts in class.
The use of letters as abbreviations for lengths and areas in polygons and circles provides an easier introduction to algebraic ways of writing and reasoning than the context-free phrase: Let x, q and r be numbers. The novice may react in an offended manner to this phrase and say give m the numbers. Yet less offense will be taken, if we say Let x, q and r be the lengths of three line segments or Let s be the number of units in the area of that circle, or Let y denote the number or amount of money in this container. The geometric or physical or monetary significance of the letters turns them into placeholder or pronouns for numbers and quantities easily visualized. Again, it is easier for students to accept the height of a rectangle and to say it is h units or h is the number of unit in its length, than it is for them to say let h be a number. The introduction to algebra will come more easily if letters are introduced as abbreviations or shorthand for number or quantities, or their longer descriptions, and algebra is done in the first instance with letters that have a more concrete meaning than the phrases let x be integer or suppose a, b and c are real numbers. The abstract meaning of these phrases leaves student asking for and insisting being given the numbers. They see not the need to describe calculations in general. Letters with meaning are more understandable even though they may denote an unknown or unspecified quantity. |
Objective 1.1
illustrate the type of dependence characterizing the relationship between the variables in a situation
In Secondary II, the students .. developed their ability to reason proportionally.
Develop and verify the ability to use different ways (MEQ modes of
representation) to analyze a situation.
The students will be given situations they can understand and in which the
variables are directly or inversely proportional or in which one of the
variables is proportional to the square of the other.
(w) A quantity Y is proportional to X if Y = KX. The task becomes to find the K from some data and then use that value to compute Y or X in another situation.
(w) A quantity Y is inversely proportional to X if Y = K/X. The task becomes to find the K from some data and then use that value to compute Y or X in another situation.
(w) Now if work done W is proportional to the number of workers N and the time T each spends working, assuming they all work the same length of time and the same pace, then W = KNT is jointly proportional to N and T. The task then is to find K and then apply its values to find one of the numbers or quantities W, N and T when the other is given. BUT there is connection with inverse proportionality. If W = K NT then when both W and K are given or fixed then, then N T are inversely proportional to each other. In particular T = (1/K) W/N is jointly proportional to W and 1/T There in lies a inverse proportionality relation.
(w) Newton's Law of Gravitation provides an inverse square law. Light density or illumination provides another.
(w) The area of a sphere (and the cost of painting it) is proportional to the square of the radius. The density of a fixed amount of substance spread over the area of a sphere is inversely proportional to the square of the radius.
(w) The area of a disk (and the cost of covering it) is proportional to the the square of a radius. The constant of proportionality is p
Intermediate Objectives 1.1
- Determine the dependent variable and the independent variable in a given
situation.
W: The area A of a rectangle is given by and has the same value as the product of its width W and height H. So A = WH. In the direct use of this formula, the forward use, the value of A is obtained from the values of W and H. In this situation, the value of A is dependent on the values of the dimensions W and H, and we say the latter are independent, to contrast with the dependence of A. On the other hand, if we use the equation A = WH backwards or indirectly to find the value of H from values of the area A and width W, the value of H depends on those of A and W. So in this situation, in this use of the equation A = WH, the quantity H is dependent while A and W are independent.
W: Textbooks have a habit of using the label variable for letters which denote or stand for numbers and quantities in formulas That is fine. But you should understand the notion of variable is more general. A number or quantity that may vary in one way or another may be called a variable.
- Represent a rule that applies in a given situation, using a table of
values.
(w) Represent a rule a formula should be here as well, alone or with the above.
- Represent a situation and its corresponding rule by means of a graph,
given a table of values.
(w) The word rule here may mean a formula.
- Express in their own words the relationship between the variables in a specific situation, given the description of that situation, a table of values or a graph.
page 18, Objective 1.2
Solve problems related to situations in which a linear relationship exists between the variables
In Secondary II, the students acquired certain skills that enabled them to write a first-degree equation for a given situation. In studying ratios and proportions as earlier in secondary III, they explored situations involving direct variation.
Develop and verify the ability to solve problems involving direct or partial variation.
In a situation involving direct variation, the dependent variable is expressed as a single term consisting of the product of the independent variable and a constant.
(w) Seems like proportionality again. So here is a repetitive and redundant part of the curriculum.
In a situation involving partial variation, the dependent variable is expressed as the sum of two terms, one being the product of the independent variable and a constant and the other being a constant. (w: why did not they say a linear expression?)
(w) In secondary IV, student will learn how to solve linear equations in or two unknowns. I do not see the advantage of emphasizing here the different between direct and partial variation. They will not need this terminology. And if it ever needed, it can be quickly taught in or after secondary IV discussion of linear equations in one or more unknown. Students have or should have also learnt how to solve equations ax+b =c in secondary II.
The objective here is to have students examine situations represented by a straight line.
w: I would start by graphing equations y = ax + b, and explaining how the change in y is proportional to the change in x, and identify b as the initial value of y. That being said, students may be given many numerical and then algebraic excises to find a in the cases where (i) b = 0 and (ii) b is non-zero. The foregoing provides numerical experience - much needed. And the foregoing may lead to a discussion of the point slope form of a linear equation or relation and how to obtain the values of a and b. After this general case, the direct and partial variation may be mentioned briefly. The course or course textbook here makes a mountain out of a mole hill. When and where students have fully and properly developed fraction and algebraic skills, see the suggestions for development above, the whole discussion of direct and partial variation becomes redundant for the most part. And when students do not have full and properly developed fraction and algebra sense, the lengthy discussion of variation, direct or partial, becomes a distraction from development of the missing skills and sense. See if the aims of this objective can be met in a way that supports and extends fraction & algebra skills and sense rather being a digression or distraction of the development and maintenance of those skills and sense.
The students will analyze these situations in greater detail by associating each type of variation with a specific form of equation, describing the role of the rate of change and determining how a parameter change will affect the situation, the graph and the equation.
The students must learn that straight lines are sometimes used in situations involving only integers, or that only a part of a line is significant.
(w) The program should leave this to later.
They will also study situations in which the rate of change is constant within a given interval and then assigned a different value within another interval, as is the case with broken-line graphs.
Functional notation should not be used, because it is important that students be able to observe and explore situations without being distracted by overly complex symbolism.
(w) Rightly or wrongly the foregoing statement is contradicted by the practice in the MEQ approved textbooks for secondary II in the function or function-like notation for translations, rotations and reflections.
Activities in which the students learn how to interpret graphs and to
understand the relationships between symbolic, graphic and numerical
representations are encouraged. It would be useful to employ a
variety of learning aids and methods (i.e. mental calculations, "paper and-
pencil" exercises, graphic display calculators, computers).
Intermediate Objectives 1.2
- Translate a situation involving direct variation or partial variation into an equation.
- Translate an equation involving direct variation or partial variation into a word problem.
- Determine the rate of change in a situation involving direct variation or partial variation, given the corresponding equation or graph.
- Provide a qualitative description of how a parameter change will affect a graph, given the equation for a situation involving direct variation or partial variation.
W: Suggestion: Meet and unify these objectives by talking in general about the graph y = ax +b - how to tabulate and plot points on the graph, and how b may represent the initial value or charge or set-up cost, and a may represent a slope or rate of change.
page 20, Objective 1.3
Convert an arithmetic or algebraic expression into an equivalent expression
In the Secondary I program, the students gradually developed their understanding of the four operations and their ability to perform these operations on rational numbers.
(w) In other words, mastery of arithmetic with fractions is assumed from secondary I (math 116).
In Secondary II, they added and subtracted expressions containing one variable and constants and multiplied and divided this type of expression by a constant.
Develop and verify the ability to convert an arithmetic or algebraic expression into an equivalent expression.
They must be able to perform operations on numerical and algebraic expressions containing exponents. They must not only find the simplest expression, but also recognize and give other equivalent expressions. Appropriate visual aids may be useful in introducing the relevant algebraic manipulations.
Activities in which the the students gradually expand their knowledge of the structure of algebra while developing their ability to perform certain operations are encouraged.
The focus should be on the students' understanding of these operations rather than on the mechanics of performing them. The students will also be able to apply the skills they have developed with respect to number and operation sense. By establishing the properties of exponents, the students continue to learn how algebra can be used ...
(w) The Logic & Algebra in discussing how a box volume formula V = hA and V = h (WL) can be transformed into each other illustrates and may introduce the notion of equivalent expressions. The law applied here is A = WL is a geometric law rather than an algebraic law (like the distributive law). None, the idea that an expression represents a number or quanity and that there may be more than one ways to compute the number or quantity is key to the notion of equivalence.
page 21, Intermediate Objectives 1.3
- Apply the properties of exponents in transforming arithmetic expressions.
- Apply the following properties in transforming algebraic expressions:
- Add and subtract polynomials.
- Multiply a monomial by a polynomial and a binomial by a binomial.
- Divide a polynomial by a monomial.
Objective 1.4
Solve problems using the Pythagorean theorem
In Secondary I and II, the students learned how to calculate the measure of one of the dimensions of a polygon given sufficient information, while assimilating the concept of square root.
Develop and verify the ability to solve problems in which they can apply the Pythagorean theorem. The goal is to ensure that the students can state this property using words and algebraic symbols. They will also use the converse of the Pythagorean relation to establish whether a triangle is right-angled.
(w) A proof of the latter depends on the cosine law met in secondary IV or V. Is there another quick proof, more accessible to students in secondary III. In the absence of proof, we have here a case of rote learning.
They will also learn about irrational numbers and locate them on the number line using a ruler and compass.
W: Students may know about finite decimals, which fraction have finite decimals, and which fractions have infinite decimal expansions which (eventually) repeat. I do not know if conversion of the latter is part of the course. That being said, the number theory area of this site explains how 0.999 (9 recurring) represents a sequence of approximations to the number 1, a sequence that has the limit 1. From there, from a coordinate and measurement viewpoint of the location of points on a number line, one may emphasize that infinite decimal expansions represent a sequence of approximate coordinates or locations on the number line which converge - there-in lies an assumption. The pythagorean theorem provides us with right triangles where say the legs (sides not opposite the hypotenuse) have whole number lengths while the length of hypotenuse is not given by a fraction but can be approximated by a sequence of decimal multiples of a unit length. The fact that sqrt (2) and other square roots are not fractional multiples can be shown by contradiction - the inconsistency of the supposition that sqrt(2) is a fraction with the expression of fractions in reduced form. See discussion of direct and indirect reason in Volume 1A, Pattern Based Reason (A copy of that should be moved to the Volume 2 site area as a postscript.
page 24, 1.4 Intermediate Objectives
- To express the Pythagorean relationship between the measures of the sides of a right triangle, using variables.
- To justify an assertion used in solving a problem that involves applying the Pythagorean theorem. See appendix.
- To locate some irrational numbers on a number line.
The appendix provides two examples:
|
page 25, General Objective 2:
To enable students to apply their knowledge of geometric figures
Geometry helps us represent and describe our world. In Secondary III the students continue to develop their perception of space in the world around them.
The students progress through a hierarchy of levels in developing their geometric thinking skills. They first learn to recognize shapes and then analyze the different properties of these shapes before establishing relationships between the properties and making simple deductions. Through various active exploration and observation activities, the students establish a system of relationships pertaining to triangles, quadrilaterals, circles and regular polygons. In the Secondary III program, this system will be expanded to include solids.
The students live in a three-dimensional world which they began to
understand through orientation and observation activities in elementary
school. In Secondary III, it is essential to continue developing the
students' spatial sense, which is a mental skill that makes it possible to
create and manipulate the images of objects.
page 26. Terminal Objective 2.1Solve problems involving isometries or dilatationsIn Secondary I and II, the students constructed figures resulting from
isometries or dilatations and explored the properties of these
transformations. They also constructed these figures in a Cartesian plane.
Prescribed Teaching Methodology:
page 27. 2.1 Intermediate Objectives
W: I would put this topic or some subtopics last in the course. Given first and higher priority to counting, fraction and algebaic skills and sense that appear in the first objective. Focus on their development and perfection. Anything less undermines and sabotages secondary IV and V mathematics. |
page 28. Objective 2.2
Solve problems involving three-dimensional objects
In elementary school, the students established spatial relationships between objects by locating them or by making networks, frieze patterns or grids.
Develop and verify (?)_ the ability to perceive three-dimensional objects and represent them in different ways.
Given a set of cubes, students must give a verbal (w: written too) description of what they see, draw sketches of the front, side and top views or describe the various layers of a given object. (W: Good Concrete Directions)
An introduction to basic techniques such as the Cavalierian perspective or the use of isometric dot paper will help the students produce better representations of three-dimensional objects.
(w: How can this be tested in a course or at the final.?)
The students will also be asked to construct three-dimensional objects on the basis of descriptions or two-dimensional representations.
page 29. 2.2 Intermediate Objectives
- describe three-dimensional objects, using words or drawings.
- represent three-dimensional objects in two dimensions. •
- build a three-dimensional object on the basis of a description or a
drawing. See ideas for 314.
Terminal Objective 2.3
To solve problems involving solids
In the first cycle of elementary school, the students discovered certain characteristics of solids. In the second cycle, they explored certain ways of arranging plane figures to construct solids and identified the characteristics of these solids. In Secondary I, they created plane figures through isometric transformations.
Develop and verify the ability to apply their knowledge of solids in order to solve a variety of problems.
- They will create solids through rotations or translations of plane figures or by splitting a cube into sections. (W: why bother?)
- They will analyze the elements of solids (e.g. edges, sides, vertices, diagonals, altitudes, slant heights (apothems)) and use these elements to classify solids.
- The students must also be able to deduce a measure in a solid or in a
figure they have already created.
W: For this, students will have to use formulas for perimeters, surface areas and volumed not only directly but also indirectly, that is, backwards. Time to repeat the message: all formulas and equation in high school mathematics and science are or will be used directly and indirectly, that is, forwards and backwards.
Intermediate Objectives 2.3
- generate a cone, sphere or cylinder by rotating a figure 360
degrees about an axis.
w: Appears in the study of integration in calculus. Appearance here may be premature. - generate a prism by translating a polygon.
w: Appears in the study of integration in calculus. Appearance here may be premature. - Represent solids in two or three dimensions.
(w) What does this mean? - classify solids.
w: Here is connection to the study of the shape of crystals, and the first two intermediate objectives for 2.3. - split a cube into sections to obtain a solid with one triangular or
quadrilateral face.
w: Why is this important? - Deduce the measure of a segment from an appropriate definition or property. See appendices.
- Justify an assertion used in solving a problem involving solids. See
Appendices
page 32, Objective 2.4.
solve problems related to the area or volume of certain solids
In elementary school, the students estimated and measured volume and even calculated the volume of certain simple solids. In the first two years of secondary school, they calculated the perimeter and area of the following plane figures: triangles, quadrilaterals, circles and polygons.
Develop and verify the ability to establish (w: use?) relationships between the dimensions of a solid and its area or volume in order to solve problems.
Consolidate knowledge of the concepts of area and volume, establish connections between the characteristics of solids and their measures,
W: Here another opportunity to repeat the message that all formulas and equations may be used directly and indirectly, forwards and backwards.
Use the appropriate units of measure
W Reporting correct units for area and volume, density too, is needed in practice. Here students can be shown how to carry units through calculations, with conversion if need-be, all in an algebraic manner.
Apply knowledge to solids that can be broken up into several parts.
W: Perimeter, area and volumes are additive - can be computed from the sum of disjoint parts, or parts that overlap in 1D, 2D or 3D sets of measure zero for length, area and volume.
page 33, 2.4 Intermediate Objectives
- distinguish between situations in which the area should be calculated and those in which the volume should be calculated.
- calculate the area of solids that can be broken up into right solids or hemispheres.
- express the relationship between the volume of a solid and some of its measures.
- calculate the volume of objects that can be broken up into right solids or hemispheres.
- convert a measure of volume from one unit to another.
- establish the connection between units of volume and units of capacity.
- calculate a measure of a solid, given its area and sufficient data.
- calculate a measure of a solid, given its volume and sufficient data.
- justify an assertion used in solving a problem involving solids. See
Appendix
Objective 3
analyze statistical data
Students often encounter data presented and interpreted in a variety of ways. A knowledge of descriptive statistics can help them pinpoint the essential facts about a situation that are concealed in a mass of data. The students will then be better equipped to understand situations and assess them critically.
In Secondary I, the students learned how to present data in tables or graphs so as to highlight various aspects of a situation. They continue to develop these skills in Secondary III by constructing histograms that illustrate a distribution of continuous quantitative data. The graph of a distribution provides a brief description of the data in question. The students will acquire other tools for analyzing data by learning how to derive information from measures of central tendency and the range of a distribution.
Statistics provides an excellent opportunity to study simple, concrete situations that show the connections between mathematics and the real world. Students should be encouraged to examine various aspects of a situation, to structure and organize data, and to formulate and test hypotheses. In this way, they will develop their inductive-reasoning ability while playing an active role in their own education. The use of computers and calculators will allow them to focus on the interpretation of measurements rather than on calculation methods.
(w) Where is the manual for this?
page 36.Objective 3.1
Solve problems involving situations represented by a one-variable statistical distribution
In Secondary I, the students learned how to use tables and graphs to present information about a situation. Students who have attained Terminal Objective 3.1 will be able to solve problems by using certain methods for analyzing statistics and interpreting data. While becoming familiar with the relevant terminology, the students will be asked to present data in the form of tables or histograms. They will use measures of central tendency to analyze or describe a distribution.
... The students will learn to approach situations as a statistician would—by thinking critically, analytically and logically about the problem at hand.
(w) Where is the manual for the above?
3.1 Intermediate Objectives
page 37.
- tabulate data.
- present data in the form of a histogram.
- calculate the mean, median, mode and range of a distribution consisting of data that has not been grouped into classes
- derive qualitative information about a distribution from its mean, median, mode or range.
- describe a distribution, given its mean, median, mode or range.
page 41. Appendix.
While studying geometry in this course, the students examine the Pythagorean theorem, geometric transformations and solids. Through learning activities, they increase their grasp of a number of concepts and improve their skills. They should also become familiar with the definitions of the figures they are studying, with some of their properties (see below) and with the properties of geometric transformations. The students will use these definitions and properties of figures to calculate measurements and justify any assertions used in solving problems involving solids.
|
page 42, Geometric Statements Studied in Mathematics 116
|
page 43. Geometric Statements Studied in Mathematics 216
|
Quebec English Mathematics Education
A farce is a farce is a farce.
Area pages represent an effort to follow and understand the objectives of the 1997-2005, the prior reform, and the text books required and used 1997-2005. In retrospect, the objectives and texts in question are too incoherent, too full of nonsense, for rational comprehension and for service as a base for the current reform. A farce is a farce, is a farce