If we live in a 3-dimensional world, then why are students struggling
in High School Geometry with 2-dimensional sheets of paper,
2-dimensional drawings on the whiteboard, and 2-dimensional computer
screens? I wanted to do Action Research that we determine if a 1-hour
lesson that involved Realia, hands-on construction of certain
3-dimensional shapes, guided instruction emphasizing self-discovery,
and minimal lectures that attempted higher levels of Bloom’s Taxonomy
to relate this to other sensory modalities, would tap into the sense
of Beauty in students, engage them, provide and enjoyable experience,
and change their self-assessment and their feelings about
Mathematics.

Through the 1950s in America, geometry was often taught based on
Euclid’s geometry, the most successful textbook of all time, having
gone through over 1,000 editions and translations. In particular,
students all learned Solid Geometry. In the late 20th and early 21st
Century, geometry was “dumbed down” despite new technologies that
should have allowed the content to be “beefed up.” The group projects
in my Action Research lesson plan gave today’s students as good – or
better – a tactile and visual and kinaesthetic grasp of certain key
3-dimensional shapes as students of past centuries attained by more
abstract means.

These shapes are also BEAUTIFUL. [This connects to my lecture
emphasis on Truth and Beauty in Mathematics, and as summarized by the poet John Keats].

Students were familiarized to the vocabulary using realia (defined
below) as well as by a table that they constructed by peer teaching on
the white board. They BUILT several of the shapes discussed out of
paper and glue, , to make visually attractive realia which were passed
from hand to hand, rotated, manipulated, and compared.
Realia is a term used in library science and education to refer to
certain real-life objects.

The concept was introduced to the learners utilizing a situation that
they may encounter in their present lives. The class used the same
word problem to build their mathematical understanding, moving from the scenario, to values, and finally to terms, thus gradually establishing a solid foundation. In this manner, the teacher/lesson-planner expected that students will learn not only the new mathematical knowledge but also a connection of how these mathematical concepts are important in their everyday lives.

The emphasis on group learning, dialogue, and self-discovery as
opposed to current practice is based on philosophical changes in the
foundations of mathematics.

Paul Ernest wrote:

“Currently, there is a move in some quarters to reconceptualise
mathematics and the philosophy of mathematics in fallibilist,
human-centred and even social terms [Davis and Hersh, Kitcher,
Lakatos, Tymoczko, Ernest 1991a].

This reconceptualisation represents a break from the traditional
absolutist views of mathematical knowledge which see it as monological
in character.

Monologicality is a central assumption of Cartesian rationalism and
the modernist outlook based on it. Mathematical knowledge is presented as if it is God-given, not uttered by human voice, let alone by a one of several voices (albeit a dominant one) in a dialogue or conversation.

Instead, my argument is that mathematics is dialogical, and that
conversation permeates mathematics in deep and multiple ways.”

I used a class of 13 male and female 9th, 10th, and 11th grade
Geometry students at Stern Math and Science School, on the edge of the
Cal State L.A. Classroom.

In the end, I knew more about the extent to which these students
found Geometry fun, or beautiful, and whether they thought that they
could increase their own intelligence, thus, increasing my ability to
relate to such students and to become a better, more efficient Mathematics educator through this new understanding. I was also
interested to see if the regular teacher whose students and classroom
I’d borrowed assessed the lesson similarly to the way that I did and
the students did. This Action Research Project could potentially help
me meet the goals I am setting out to accomplish as an educator in
this content area.

I have a series of YouTube videos which I hope will entice people back into thinking about geometry more seriously, it’s the WildTrig series on Rational Trigonometry.

I believe that young people actually yearn for a subject that makes logical sense, in which they are not just told to memorize stuff, but learn how to follow an argument.

I don’t see much point in teaching high school geometry without proofs. Sure, it’s useful to be able to compute areas and volumes etc. But you learn those things before you get to high school (I hope). The whole point of geometry at that level should be rigorous proofs.

Am I correct that the only other branch of pure mathematics to use the manipulations of visual images as a proof-method is Category Theory?

And I can say that I teach with physics and engineering in mind - not the proofs. How to manipulate the equations, that’s my focus. In part because of my 20 years as an engineer before becoming a teacher, and my degree in physics.

I’m pretty sure none of my students could do a proof by induction - I certainly haven’t taught them. There’s nothing like that on the AP Calculus tests that I’m trying to prepare them for.

This has given me pause, and I’ll think about how I might want to make math more “mathematical” as opposed to simply a great problem solving tool.