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Quebec High School Mathematics Education (English Version of)
his folder has a tree like structure. The child, same level and parent level webpages for this webpage follow.. More Links: 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 |
Secondary IV
Deficiencies – math 436 only: ·
Mathematical Reflections 436 Book 1, Guy Breton et al. 1997, ISBN
2-89127-426-1 ·
Mathematical Reflections 436 Book 2, Guy Breton et al. 1997, ISBN
2-89127-427-X http://www.mels.gouv.qc.ca/dfgj/dp/programmes_etudes/secondaire/pdf/mata436.pdf Cruel Final
Examination Practice: As a calculus lecturer in the Department of Mathematics and Statistics
here at McGill, a long time ago, my teachers and colleagues suggested that a
final examination in a course have 60% accessible to students [who] had come and
attended, 20% accessible to students who did the homework, and 20% to represent
a challenge. But the mathematics 436 examinations, I have examined nine from the
period 2000-6, are 85% challenging. The latter statistics combined with use of
Guy Breton et al., textbooks in 436 and the previous one or two years of
instruction is cruel. Note in the Front cover of the textbook for mathematics
436, books I and II, there
contribution of a McGill Professor of Mathematics Education is acknowledged.
Whence the Faculty of Education is an accessory to the materials in books
I and II. This association may have
prevented the Faculty or the Professor in Question from performing their duty to
raise standards for course material and hence course delivery in Quebec English
schools, secondary II to V, where Guy Breton textbooks appear.
Sets
appear in the discussion of relations and functions in Secondary IV, Book 1 for
mathematics 436.
More
About Book 1: [B]
On page 38, Book 1, the green box which explains mathematically the concepts of
increasing and decreasing is poorly put. The meaning of the for
all sign "
needs to be inferred from the text. The following sentence is gibberish or too
cryptic. : Strict increases and decreases also exist, which
include the possibility of equality in each case The assertion that a function that is both increasing and decreasing on an interval is necessary constant is correct, but there is no explanation of why in the text. The definition of what it means for a function to be constant comes after the use of the concept. [C] On page 41, Book 1, the green box uses the word image instead of the word value. The same box employs the symbol ó for if and only [if] ahead of their formal introduction or explanation in Book 2. In the explanation of relative maximum, the use of the word peak represents non-standard mathematics terminology, [D]
The Green box on page 43, Book 1, includes gibberish. The last statement “A function f is
strictly positive or negative when f(x) cannot equal zero” is incomplete.
[E]
Chapter 1 of the Guy Breton Book 1, for secondary IV mathematics 436 ends with
the LexiMath page 86 in which students are expected to know and understand
several expressions, presumably from the previous pages
Government
Objectives related to Book 2. Page 3 of the government for mathematics 586-436 in pdf
file says the following. Mathematics 436 differs from Mathematics 416 in
two ways. First,
it covers more material in greater detail and deals with more
complex situations, problems and applications. Secondly, the
students must use advanced terminology and formal notation,
always be rigorous and precise, and justify every step in
their solutions. In addition to preparing the students for science
instruction, mathematics education should provide fertile
ground for the development of skills that will be useful to them
in the future: As Resnick and Klopfer have noted, "Graduates
must not only be literate; they must also be competent
thinkers.” If the Faculty of Education has any evidence that Book 2 [or 1] fosters the ability to use
advanced terminology and formal notation in a rigorous and precise, with
justification for every step please provide or explain it. Page
22 of the government objectives for
416 in the pdf file http://www.mels.gouv.qc.ca/dfgj/dp/programmes_etudes/secondaire/pdf/mata416.pdf give
the following details or directions for instruction that do not appear in the
government objectives for mathematics 436. That
is an inconsistency. Students who have attained Terminal Objective 2.1
will be able to
solve problems involving the concepts of similarity and isometry
by structuring their solutions and, if necessary, justifying
the steps in their reasoning by referring
to relevant definitions, theorems or corollaries. A close connection should be
established with Terminal Objective 2.1 of the Secondary III
mathematics program to ensure that students understand that
the concept of similarity is directly derived from the characteristics
of geometric transformations. By defining the concept
of similarity in this way, we can apply it to any twoor three-dimensional
figure. As a result, cases involving similar
or isometric triangles, which were examined as theorems
in Euclidean geometry, become properties of similarity
transformations in transformational geometry. Given
similar or isometric figures, the students will discover that
there is always at least one similarity transformation or isometry
which maps one figure onto another. The proofs assigned
to the students should be within their capability. For both
solids and similar polygons, the students will be asked to deduce
certain measures or ratios required to solve problems. The
bold faced statement is not bold-faced in the government document.
Since mathematics 436 students are suppose to cover more material in
greater detail, I will assume that
mathematics 436 students will [also] be justifying
the steps in their reasoning by referring to relevant definitions,
theorems or corollaries. That
raises the expectation that the Government approved and required text for
mathematics 436 will provide a clear basis for such reasoning Book
2 for mathematics 436 [1]
The coverage of coverage of conditional statement, biconditional statements and
implication begins on page 2. Page 4 says the following: Quite
often, fulfilling a condition necessarily leads to the consequence. A condition
that meets this requirement is called a logical implication.
If a biconditional has this property in both directions, it is a logical
equivalence. That
is awkward. In particular, there is no need to say if. [2]
Multiple attempts to explain what is a definition in geometry (not in general)
appear in pages 6 to 8. However,
definitions in this text are few and far between, and not always clear.
Moreover, before pages 6 to 8, in Book 1 for mathematics 436 and in the
Guy Breton books for secondary II and III, the word define has appeared
many times correctly and/or in phrases where the word determine would
have been a better choice.
The
explanation above of what is a definition is not satisfactory, and there are no
clear definitions, or very few clear ones, in the Guy Breton texts for
mathematics 216, 314 and 436. In
other math textbooks, I have seen formal definitions with the format: Definition:
A blah is a bleh if and only
if …. [The existence of an object that is a blah is not assumed!] Lip
Service:
The Guy Breton textbooks for secondary II to IV do not provide formal
definitions. Instead, they introduce terms and phrases by talking about them and
putting the terms or phrase in sentences that may or most likely will not be
definitive. However, the phrase by definition and defined by do appear. Whence
the definition of many concepts is vague justifying the steps in their reasoning by
referring to relevant definitions, theorems or corollaries.
[4]
Pages 13 to 15 cover the concept of properties.
The coverage on page 13 begins as follows. To define an object, we use its essential characteristic(s). (Sounds
good except for the lip service). Many geometric objects, however, have other
properties that cannot be described as essential.
These properties are usually expressed as conditional statements and are
logical implications. But page 39 says properties are: statements of observable facts about
objects. The government objectives call for properties of transformations to be
employed in arriving at conclusions. It is not clear that text views
transformations as a geometric objects. Page
15 continues Drawing diagrams and studying them carefully
can be helpful where memory fails. Geometry
is much more about reasoning than it is about memory. Furthermore, by analysing the properties of a figure, you will begin to
notice that some of them are interrelated and that one property can often be
deduced from one or other properties. The
text here is heading into uncharted territory.
As a mathematics student 1965-83 and as a college or university level
mathematics instructor 1979-89, I have never encountered any similar description
of what is a property. This
description, while grammatical correct, is mathematical babble.
The text here represents a poor commentary, an incomplete exposition, and
not a full explanation. Pages 16 and 17 ask students to recall properties of
polygons and transformation in a manner that ironically requires some memory –
exercises are given in which students are expected to fill in the blanks. [5]
Page 18 introduces the concept of axiom
in a manner that is not out of date but also unclear. Page
18 says Axioms are
statements that are considered to be obvious and true.
The
modern mathematics viewpoint is as follows: axioms are assumptions used as a
starting points for the development of a theory; and for the sake of
consistency, the avoidance of contradictions, a theory should depend on a
minimal set of assumptions. The page 18 axiom Given two isometric figures, at least one isometry
exists that maps one figure to the other is not obvious, or it within the Guy Breton framework of saying two
figures are isometric when and only when there is isometry that maps into into
the other, it is a tautology.
Page
19 begins: Axioms are
added to the definitions and properties of geometric objects to form deductive
geometry. The most obvious properties can even be considered to be axioms. The
text is inviting students to form their own axiom systems, or
the reader has to determine which properties met in exercises are obvious
enough to be called axioms. There-in
lies an echo and also large departure from axiomatic structure of Euclidean
Geometry and modern mathematics (say
set theory) in which the text states axioms clearly. Proofs now become
subjective. Page
19 continues It is possible to perceive properties or
relationships that are not at all obvious and can even prove false. Such
statements are called conjectures. Page
20 says ·
In mathematics, and
especially in geometry, conjectures that are not immediately obvious have to be
proved. But in order to encourage creativity and the thrill of discovery, you
have the right to state any conjecture be it true or false. In order not to
mislead people, however, you are obliged to prove or disprove these conjectures. ·
To be true, a conjecture
must apply to all cases. It is only necessary to find one case that contradicts
the conjecture to prove that it is false. This case is called a counterexample. ·
It is customary to
illustrate the figure mentioned in the conjecture and to express the hypotheses
and conclusions by geometric symbols whenever possible. Page 22 says the following: Justifications (for assertions) are usually a definition, a geometric or algebraic property, an axiom or a theorem. Recapitulation:
Secondary II , III and IV Guy Breton textbooks use the word define
properly or in the nearly correct but wrong sense of determine. Then Book
II in the secondary IV textbook package for mathematics 436 gives multiple
explanations of the word define or what is a definition in terms that are
confusing or not simple. The text
does not mention the use and assumption of undefined terms in geometry or
mathematics to avoid the dictionary paradox of how defining one word in terms of
others, successively, may ultimately lead the word being defined in terms of
itself. The description of the role of deduction in geometry or mathematics with
the aid of definitions, properties and axioms is unclear.
A
Last Thought: The
Guy Breton, Mathematics 436, Chapter on statistics talks about several kinds of statistical
analysis: a study, a census and
a survey. In English, a
survey always refers to a sample and not the whole population. But the chapter
or its falls into the linguistic
trap of calling each statistical analysis, a survey. From that arises on page
270, the phrase sample
survey to distinguish a survey (old sense) from the surveys the text
indicates a present in a study or a census. The foregoing points to a poor
translation or a poor comprehension of statistical terms as employed in English.
The emphasis on statistics in secondary IV mathematics as a path to critical
thinking strikes me as odd or misplaced in a mathematic[s] program that fails to
provide most students with fraction sense and a mastery of arithmetic operations
on fractions in a repeatable, reproducible and hence verifiable manner.
Documents introducing the new or current reform in the Quebec
Mathematics Program continue the use of the phrase Sample
Survey. |
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