Tending the GreenhouseVertical and Horizontal

Connections within the Mathematics Curriculum

Kimberly M. ChildsKimberly M. ChildsStephen F. Austin State Stephen F. Austin State

UniversityUniversity

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

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. Are equidistant apart

2. Do not intersect

3. Go in the same direction

4. 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 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)