The Ontario Ministry of Education is currently conducting a review
of the grade 11-12 mathematics curriculum. Universities provide input
to that review through the Council of Ontario Universities (COU). In
preparation for a set of meetings to be held in July 2005, the two COU
delegates to these meetings met with representatives of many Ontario
universities, at the Fields Institute on June 29, 2005, from 10am-3pm.
As well as Mathematics departments, faculties of Engineering, Science,
and Business were invited to send representatives. Present at the meeting
were the COU delegates Walter Whiteley (York) and Peter Taylor (Queen's),
as well as Anthony Azzopardi from the Ontario Ministry of Education.

The discussions concentrated on three courses, 11U, 12U Advanced Functions
and Introductory Calculus and 12U Geometry and Discrete Mathematics.
The 12U Mathematics of Data Management course seems to be working well
and only minor changes are contemplated. The COU representatives will
meet with Ministry of Education folks July 5-6 along with representative
from other organizations (Colleges, Teacher Associations) to prepare
a revised curriculum draft. This will be available in the Fall (late
Sept. or Oct.) for a feedback process.

Anthony Azzopardi described changes currently being proposed to the
existing courses and changes proposed for the pathways between the courses.
These include adding a trigonometry component to the 12U AFIC course,
requiring 12U AFIC students to have the 11U Functions and Relations
course, changing the 11U/C Functions course so that 11U/C followed by
11U is a reasonable progression for students requiring additional study
before taking Calculus, and moving conic sections and loci from 11U
Functions and Relations into 12U GDM.

The points below address specific suggestions for further changes, that
were raised at the meeting and received widespread support:

**11U Functions and Relations**

1. Remove or greatly deemphasize the financial math strand in 11U and
replace it with some analytic geometry. By this we don't mean an exhaustive
treatment of analytical geometry, but a few nice examples of geometry
problems in 2-D and 3-D (especially 3-D) that are beautifully handled
by introducing a coordinate system or with vector methods. This will
naturally extend and build on the geometry topics covered in Grade 10.
This is important material for students to see who would not normally
take GDM in grade 12. It is also important for students taking GDM as
this bridges the time gap between early and late geometry, created by
the lack of geometry in grade 11.

2. The sequences and series component of the financial math strand is
better covered under the expanded exponential functions strand.

3. Students at this stage need lots of practice with algebraic manipulations,
but done in the context of interesting questions.

**12U Advanced Functions and Introductory Calculus**.

1. The course might better be called Advanced Functions and Calculus
Concepts. We should remove or greatly reduce the last strand on calculating
derivatives except for last section on "Calculus techniques (replaced
by ideas) to analyze models of functions." This section can be
expanded somewhat to provide interesting modeling examples in the physical,
life and social sciences.

2. Support the inclusion of trigonometry (already in the plans). A few
trig identities might be covered here, again done in the context of
interesting questions. Examples: sine sum, half angle, in addition to
sin^2 + cos^2 = 1 which has been in Grade 11. Working with such formulas
goes beyond trig: they provide meaningful practice in algebraic manipulation.

**12U Geometry and Discrete Mathematics**

There was much discussion about removing counting techniques in order
to give more time for an in-depth, investigative approach to geometry,
proof, and vectors. This would be particularly helpful if the plan to
put conic sections into this course is followed. The point is that these
counting techniques are not part of the main-stream requirements for
any math/science/engineering program other than computer science.

Having said this, one of the advantages of this topic is that it ties
in well with notions of probability, a topic that receives too little
attention in the curriculum between K-8 and Mathematics of Data Management.

Our long-term goal is a curriculum in which the basic requirements of
functions, algebra and geometry required for university students are
covered in 4 courses, one in each of grades 9-12. There could be an
extra enriched course for those who wanted more mathematics, but perhaps
the marks in this course would not count as part of the entrance average.
Such a course could however be used for special purposes, e.g. a component
of certain types of scholarships, entrance to special programs within
the university etc. The GDM course was in fact designed as an elite
course but in the present structure in which marks mean everything and
the marks in this course are expected to be lower, students are warned
away from taking it.

Walter Whiteley (York)

Peter Taylor (Queen's)

July 2005

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