Tending the Greenhouse
Vertical and Horizontal
Connections within the
Mathematics Curriculum
Kimberly M. Childs
Stephen F. Austin State University
Maintaining a Climate for
Mathematical Maturity
Plant
 Fertilize
 Prune
 Re-pot

Mathematics for Teaching

Teachers need several different kinds of
mathematical knowledge – knowledge
about the whole domain; deep flexible
knowledge about curriculum goals and
about the important ideas that are central
to their grade level; knowledge about how
the ideas can be represented to teach
them effectively; and knowledge about
how students’ understanding can be
assessed …

This kind of knowledge is beyond
what most teachers experience in
standard preservice mathematics
courses in the United States.


Principles and Standards for School
Mathematics
National Council of Teachers of
Mathematics
The Vertical Disconnect

Most teachers see very little connection
between the mathematics they study as
undergraduates and the mathematics they
teach. This is especially true in algebra,
where abstract algebra is seen as a
completely different subject from school
algebra. As a result, high school algebra
has evolved into a subject that is almost
indistinguishable from the precalculus
study of functions.
The Horizontal Disconnect

In preservice preparation, teachers are
often focused on the particular topic or
subject matter at hand. Because
individual topics are often not
recognized as fitting into a larger
landscape, the emphasis on a topic
may end up being on some low-level
application instead of on the
mathematically important connections
it makes.
A Burning Question:

How do we foster “connectedness”
and mathematical maturity between
grade levels/courses and within
grade levels/courses?
Some of the “Big Ideas” in
Mathematics

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Composition and Decomposition
(“Doing” and “Undoing”)
Shape and Structure
Generalization
Process and Content
Essential vs. Nonessential Features
Geometric Thinking:
Using the van Hiele Model

Level 1: Visualization (recognition)
At this level, the student views geometric figures
in terms of their physical appearance and not in
terms of their individual parts.
“This is a square because it looks like one.”
So…what is a triangle? An equilateral triangle?

Level 2: Analysis
At this level, the student becomes aware of
characteristics of geometric figures but is
unable to understand the significance of
minimal conditions and definitions.
“This figure is a square because it has four
right angles, equal sides, parallel opposite
sides.”

A student who argues that a figure is
not a rectangle because it is a square is
showing level 2 thinking.
So… what is a quadrilateral?

Level 3: Informal Deduction
At this level, the student becomes aware of
relationships between properties of geometric
figures and minimal conditions; definitions
become meaningful.

For example: Included among the dictionary
descriptions of parallel objects are the
following four different characterizations:
1.
2.
3.
4.
Are equidistant apart
Do not intersect
Go in the same direction
Can be obtained from each other by a
translation.
So… what about a line being parallel to itself?
 Students begin to harness the power of
“if-then” reasoning but are not yet able to
really appreciate the need for formal,
axiomatic geometry.
So… what is your conjecture about the sum
of the measures of the interior angles of a
regular, convex polygon?

Level 4: Formal Deduction
At this level, the student recognizes the
significance of an axiomatic system and can
construct geometric proofs.
This is the level of high school geometry –
axioms, definitions, postulates, theorems,
and so on. Students thinking at this level can
understand and appreciate the need for a
more rigorous system of logic and are able to
work with abstract statements and make
conclusions based on logic rather than just
on intuition.
So …
Prove that the sum of the
measures of the interior angles
of an n-sided convex polygon is
given by the equation
S(n) = (n – 2)·180°
for all positive integers n  3.

Level 5: Rigor
At this level, geometry is seen as abstract
and various non-Euclidean geometries can
be understood and appreciated.
Revisiting the Concept of “Parallel”
So … let’s think about “parallel” again

Parallel Postulate (Euclidean Geometry): In a plane
through a point not on a line, exactly one line is parallel
to the given line.

Riemannian Postulate (Spherical Geometry): Through a
point not on a line, there are no lines parallel to the given
line.

Lobachevskian Postulate (Hyperbolic Geometry):
Through a point not on a line, there are infinitely many
lines parallel to the given line.
Algebraic Thinking from EC - 16
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Primary: Sorting and Classification
Elementary: Counting and Grouping
Intermediate: Properties of Arithmetic
Middle: Properties of Algebra
Secondary: Equations and Classification
College-level: Sorting and Classification
Finally,

How can we enhance mathematical
experiences for prospective teachers in
order that they might understand
mathematics in a broader and deeper
sense?
What if we consider questions like:


Are there other functions that act like
absolute value on the rational
numbers? What does it mean to “act
like absolute value”?
Geometric probability suggests there is
a connection between probability and
area. Is there more than a superficial
similarity?

The formula for standard deviation
looks a lot like the distance formula.
Is that a coincidence?

There is a characteristic equation in
linear algebra and a characteristic
equation used to solve difference
equations.
Are they connected?
(Questions taken from the article, Mathematics for Teaching, Al Cuoco)