2015 Geometry Workshop: Day 1

We had a fantastic first day of our Geometry workshop today.  Here is a brief run-down of everything we did, including materials discussed and requested.

Block 1: Precision in Communication

We started with Say It, Draw It (you can download the activity) to get us paying attention to how language choices affect others’ understanding in a very concrete way.  We then launched into a discussion of the van Hiele levels (and the CCSS versions), which provide some framework for different ways students can understand geometric ideas.  The Wikipedia page on van Hiele levels is sufficiently  thorough with a good list for additional reading.  We also discussed that the best way to develop higher understanding is through well-structured experiences.   One take-away is when people of differing levels of understanding try to communicate (such as teacher and student), they may be using the same words but have different mental processes.

The same word can conjure different meanings.

Block 2: The Role of Definitions

The goal of block 2 was to recognize (1) how definitions define categories, which can overlap, and (2) how carefully constructed definitions are.  Definitions exist to ensure that the author and reader, or speaker and listener, are understanding key terms in the same way.

We discussed identifying “triangles” and “diamonds” among a list of candidates.  In the end, identifying diamonds proved problematic because there were several opposing viewpoints of what a diamond “should” be.  Andrew brought up the point that there is no mathematical use for “diamond” (except as a synonym for rhombus) because mathematics has found no need for it.

We then moved into the “Writing good definitions” task, based on defining what a Widget is given examples and non-examples.  One take-away message is that objects that are examples where the term does not apply proved to be just as useful as examples of when the term does apply (e.g., the “not Widget” cards helped to inform what a Widget should be).

In “Inclusive vs. Exclusive Definitions” we discussed why inclusive definitions (e.g., one that would allow for squares as rectangles) are preferred by mathematicians over exclusive definitions (e.g., one that would exclude squares from being rectangles).  For example, every statement that is true about rectangles is also true about squares, so it is more efficient to let squares fit within the broader category.  We saw the similar phenomenon explain why rectangles fit within the parallelogram category.  Perhaps the most contentious of these is how trapezoid is defined: should a trapezoid be a quadrangle with “at least one pair of parallel sides” or “exactly one pair of parallel sides.”  The former is the inclusive definition and allows for parallelograms to qualify as trapezoids.  The latter is the exclusive definition and excludes parallelograms from being trapezoids.  The exclusive definition is commonly seen at elementary grades, while the inclusive definition is preferred at the college level.

Block 3: A hierarchy of attributes

Block 3 in practice got admittedly rushed.  We discussed simple, closed, and polygonal curves.   We spent some time talking about convex and concave polygons, as well as equilateral, equiangular, and regular polygons.  In particular, we noted that equilateral-ness and equiangular-ness are independent from each other (you can have one without the other), except in the case of a triangle.  We skipped the discussion of parallelism, but may return to it on Day 2.

We spent time assessing student responses for shapes that fit many categories.  The most commonly recommended course of action was the ask the student to clarify their language.  Andrew has prepared a version with his comments on each: Shapes That Fit Many Categories, with commentary.

Block 4: The Power of Examples

This block got even more rushed than Block 3.  The main point is that there are many instances where a single counter-example can dispel a false statement.  For example,, a statement like “all birds can fly” can be disproven by a bird that cannot fly, such as a penguin.  Handouts 4-4 through 4-7 explore this in more detail, although we cut them for time.

When we present examples of shapes in class, we often draw examples that have incidental properties.  For example, the prototypical parallelogram has two horizontal sides and two slanted sides but that is not a necessary part of being a parallelogram.  Similarly, you may find that you always draw your parallelograms with four equal sides, which gives the impression that parallelograms are always rhombuses.  We skipped Handout 4-3 which repeats this situation with isosceles triangles.  As teachers, we should be mindful of the examples we provide and whether we are stuck in ruts that suggest shapes have certain properties that they do not always possess.

The last part of Block 4 is perhaps the most valuable: a portfolio of counter-examples.  We discussed some in class, you worked on more, and you’ve been asked to try even more for homework.  We will talk about some or all of these on Day 2, and possible answers will be posted with tomorrow’s recap.

Miscellany

A list of the geometry-related Common Core Standards: http://www.corestandards.org/Math/Content/G/

The (draft) progression for Geometry in K-5 according to the team that wrote the Common Core Standards: http://commoncoretools.me/wp-content/uploads/2014/12/ccss_progression_gk6_2014_12_27.pdf  (all of the progressions can be found at http://ime.math.arizona.edu/progressions/