216 Book Review

Appetizers and Lessons for Mathematics and Reason ( Français)
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Mathematics 216 (Moot perhaps due to Reform)

The MEQ has approved the following texts for use.

Breton, G. et al.

Instructional Package

Wilson & Lafleur Ltée , Les Éditions CEC inc.

Certificate emitted April 10, 1997

Mathematics Carrousel

Approved components

Pages

Student's Textbooks (2)
Teacher's Guides (2)

607
1742

No other texts have been approved.

Mathematics Carrousel 2, Secondary II, Book 1, Guy Breton & al, 1994, ISBN-289127-362-1

I prefer textbooks which develop concepts one at a time and one after another in compact groups, and which provide clear definitions. I prefer a textbooks which provide direct explanations rather than textbooks activities to develop comprehension followed by summaries that may or may not be understandable. Some of the destination pages contain explanations or summaries in substandard English.

These textbooks while providing activities to indirectly develop or reinforce concepts in should employ clear language when and where it attempts to be direct. While there are many mathematically correct elements in this first chapter, I suspect students and teachers who have not read the MEQ objectives will be mystified by the chapter title and will not understand nor benefit from all the prose in the chapter. The MEQ approval of Book 1 is incomprehensible to me. Book chapters vary in depth of incoherency. A few are good but my overall impressions is that this work is very difficult for students and teachers to learn and teach mathematics. Several comments about book content and flaws, almost certainly incomplete, follow.

This textbook begins with a chapter called various modes of representation. It then proceeds in a dozen different directions, and ends with a summary or destination page page 48 first emphasizes the use of coordinates (matter seen in the chapter) and then the representation of situations. However, the whole account is unclear.

Some quotes from the summary or summary or destination page page 48 follow.

Plane	a flat surface that extends forever or an infinite set of points that can be shown on a sheet of paper
Cartesian Plane	a grid formed by two perpendicular lines called axes, by which the position of any point in the plane can be described.
Graphic Representation	Illustration of a situation by a Cartesian Plane

I disagree with calling Cartesian Plane as a grid. In my opinion, the above explanations are not helpful.

The destination page 48 also employs the following phrases:

Giving a comprehensive description, Representing a situation comprehensively, Giving a comprehensive description

whose meaning is not obvious to me.

The same chapter includes drills and exercises in arithmetic and mental arithmetic (or estimation). The arithmetic theme runs through this and the next two chapters without any mention of arithmetic in their summary of destination pages stating what student should have learnt from each chapter. So students and teachers may not recognize the importance of this arithmetic theme.

Page 47 gives a graph of distance versus time for ski trip. The graph violates the vertical line rule for functions. The challenge of explaining why the graph is wrong is set as an exercise. I do not see any early context for this challenge or enrichment in the text or its activities. The topic is out of sequence - it is secondary IV topic.

Many people learn about different ways to represent or model situation, one at a time and one after another. There is place for discussing different ways to define functions, but the discussion here of different modes of representation strikes me as not needed and most likely meaningless to students, if not most teachers.

The MEQ course objectives besides specifying (with insufficient detail) how to teach talk about modes of representation, a concept that appear in the mathematics education literature, but it is not part of the vocabulary of any mathematician, accountant, engineer or scientist developing or applying mathematics. The observation that mathematics employs different ways to model or represent situation is correct, but before I had read the MEQ documentation the chapter title Modes of Representation in a secondary II textbook,

namely Mathematics Carrousel 2, Secondary II, Book 1, Guy Breton & al, 1994, ISBN-289127-362-1

in my opinion is too abstract for students, their parents and teachers to fathom.

Online Reference: The standards of the National Council of Teachers of Mathematics has a page on Representations

The emphasize on different modes of representation in the MEQ objectives is best left I think to research article in mathematics education journals, journals I henceforth avoid.

The letters and formulas in pages 90-91 travel logs on ratios and their properties are mathematically correct, but I do not understand how students are prepared by the text to understand the shorthand roles of letters or literals in this situation.

In the destination page 108, the explanations or summaries of ratios, rates, proportionality are not clear to me. A rewrite is recommended.

Chapter 3 on similarity transformations emphasizes dilatations (the radial magnification and shrinkage from a point) .Here we gone back to the coordinate free version of geometry just after introducing coordinates. The text is faithful I presume to the Quebec objectives for mathematics 216, it could be fun for students, but the objective it supports in the Quebec curriculum should be cancelled. The time and will of students is being spent on an objective that is not essential, an objective that reoccurs in secondary IV. The mathematically appealing transformation theme in the MEQ objectives is overdone.

The Travel Log on page 170 says a symbol used to replace a number is a variable. The symbol p replaces the number given by the infinite decimal expansion 3.1415 ... and the p denotes a constant. A clearer definition, more precise, is found in the next years work in the Carrousel Series, namely

Mathematics Carrousel 3, Secondary III, Book 1, Guy Breton & Jean-Charles Morand, 1997, ISBN 2-89127-397-4

In Carousel 3, Book 1, the green box at the top of page 112 says that "Quantities with changing values are called variable quantities or variables."

The algebraic shorthand description of the properties of addition and multiplication of numbers (which numbers?) on page 177 comes before the explanation page 179 onward of the algebraic use of letters to describe calculations. The description appears to me to be a sudden jump in the level of algebraic way of reasoning.

Small Typo, page 177.. The commutative laws for addition and multiplication says a+ b = a + b and ab= ab instead of saying a+b = b +a and ab = ba.

On page 188, we have a brief and isolated appearance of set theory concepts first in the explanation of the domain or replacement set of a variable, and second in the notation for the natural numbers and integer. I do not do not see any earlier context for this introduction of sets in the exposition.

In the Destination or chapter summary page 219, we read that a numerical value of an algebraic expression is a value found by replacing one or more variables by the elements of their domain. Here again I do not see any preparation for this technical. set-theory language in the preceding text

Page 224 does not appear in my eyes to give a clear concept of equality and equations. The page includes the following.

An equation is formed by hiding one or more numbers in an equality. The best way to hide a number is to replace it with a variable. In equations, this variable is often called an unknown quantity.

I do not like the prose. This should represent the first draft, not the final, in my opinion.

The activity 2A on page 227 advocates solving equations by substitution of possible values. That provides an arithmetic trial and error view of solving equations which lead students, some I have met, to assume trial and error is the best way to solve an equation. That is not in my opinion a good way to promote algebraic thinking. The last sentence on the page 227 points out that this substitution method is limited, but students who do not read the text carefully, do not take the text literally, may not get the message.

The cross-multiplication rule for solving equations involving fractions is given in problem 11, page 247. as an algorithmic method for solving proportions (a type of equations). The emphasis of that rule in my view shorts or bypasses the algebraic discussion of equation solving. The rule may be the result of previous activities in the text that students may or may not have grasped. Ouch.

Mathematics Carrousel 2, Secondary II, Book 2, Guy Breton & al, 1996 ISBN-289127-363-X,

In the destination page 62, the single or multi-variable function notation for rotation, reflection and similarity transformation rules are also at least two year ahead of their time. In particular, we meet the modified multi-variable function notation t_(a,b)(x,y) = (x+a, y+b) in the discussion of translations via the shift (a,b). We also meet technically correct function notations for rotations, reflections and similarity transformations (dilatations) But all seem to appear without any prior explanation of function notation.

The MEQ objectives if they sanction multivariable function notation should also sanction the use of the word function in in secondary II.

In the destination page 62, a Transformation on the the plane is described as a rule or relation that associates each point on a plane with another point on the same plane. Mathematical conventions in the English language would put the word function here. I have the feeling here of reading the words that are written or translated by a unfamiliar with the English terminology of modern mathematics.

The destination page 253 characterizes Concave and Convex Polygons in terms of the extension of their sides. That characterization is correct. Previously, I seen a set in the plane be characterized as a convex when and only when (or if and only if) for each pair of points in the set, the straight line segment joining the points belongs to the set. Then a set is concave if and only if it is not convex. The destination page 253 is using a characterization that works only for polygons, different from ones that will in use later. Is that pedagogically sound?

Mathematics 216 (Moot perhaps due to Reform)

Mathematics Carrousel 2 ©1996

Mathematics Carrousel 2, Secondary II, Book 1, Guy Breton & al, 1994, ISBN-289127-362-1

Mathematics Carrousel 2, Secondary II, Book 2, Guy Breton & al, 1996 ISBN-289127-363-X,