At the end of each quarter, I asked my students to write an overall reflection of their work. They were not at all surprised at this request, as they were, by now, quite comfortable with writing in our math class. As was often the case, they had much to say.

Jessi wrote the following reflections on her work:
*"So far in geometry, we have learned a lot. Mostly, we learned
proofs, which was a hard concept to grasp at first. But little by
little, with help from the teacher and each other, we began to
understand proofs better and better. Geometry so far this year has
probably been one of my harder classes, but I think it is my
favorite. At first, I hated proofs. They were the most annoying
things in the world. They just didn't make sense to me. What was even
more irritating was that my classmate Blaine was always going on and
on about how proofs were fun! But after awhile I began to get the
idea of how to do them, and I totally understand them now."*

Blaine was right - they are sort of fun, for Geometry. I started to think of them as puzzles, like the "Can You Get Blood From A Stone" game. Come to think of it, if I could have my way, all the tests would be nothing but proofs! When you understand concepts, it makes you appreciate them more. I feel that the student's attitude in any endeavor is just about the most important part of learning. I encouraged the students to try a variety of interesting "extra credit" projects, all of which were related to our studies, and which were designed to show them interesting (and sometimes unusual) facets of geometry. I was very interested in their opinions, and also their thoughts and feelings about what they were learning. Near the end of each quarter I asked them to write a half-page or more, telling me their opinions on what they had learned, and reflections on the process, methods, and assignments.

They said that no teacher had ever asked them these things before, and they were surprised and also quite pleased that they had the opportunity to voice their feelings. Their comments covered a very wide range, from complaints about timed tests to pride in their accomplishments; from enthusiasm regarding the variety of assignments to candid appraisals of their achievements.

I was alternately amused, fascinated and humbled by their thoughtful comments, their candid appraisals, and their enthusiastic responses. I have included some below, just as they were written:

*"Geometry has been one of my harder classes
this year, but I think it's my favorite. At first, I hated proofs. I
thought they were the most annoying things in the world. They didn't
make much sense to me. But bit by bit as time went by, I started to
think they might not be THAT bad. Then we did that project called
"Getting Blood from a Stone" (see chapter 16). I thought that project
was really fun, and then I realized that what it was all about was
writing proofs! From then on I looked at each proof as a puzzle to
solve, and after a while I started to get the hang of it and even
found proofs to be interesting, and even fun. (I never thought I'd
ever say that!) I really like them now. Come to think of it, if I
could have my way, all the tests would be on is proofs. When you
understand concepts, it makes you appreciate them more."*
Jenna

"*Proofs! Oh no! I had heard that geometry was
all about proofs, and I was not looking forward to that at all
because everyone said proofs were a real pain in the ...oops! Blain,
who sat next to me, kept saying that they were fun and I thought he
was nuts, But after trying a few (with some help from my group!) I
decided that they weren't so bad after all. And after a while I
really began to get it, and even started liking them! When you
understand concepts it makes you appreciate them more.*"
Lane

"*One of the things that I want to write about
here is the struggle that I had with proofs, At first I just didn't
get it at all! Fortunately, we were allowed to work in groups of 3 or
4 students on our first proofs, and this really was great. When I
completed my first successful proof, with just a little help from
Ben, in my group, I was so proud of myself I felt I could conquer the
world! This gave me a positive outlook on geometry. I actually
started to like it, which surprised me a bit, but it's true!*"
Karl

Sometimes I would add a bit of humor to the lessons, and the students enjoyed the occasional joke or cartoon. Renata really liked the cartoon below, and wrote a short essay about how it applied to learning:

"*I really loved this cartoon. I think it's
actually a sort of math joke, because just like there are different
ways to interpret the meaning of the words 'half-way', there are
different ways to do proofs, and some are very straightforward, some
are long and complicated, and sometimes it depends on how you look at
it. So there can be different meanings to 'half -way'! (Also, some
proofs are impossible! Ha ha, just kidding.)*"
Julia

Andrew made the following comments in his
reflections on the second quarter: "*I thought this quarter was
challenging, but also interesting and fun. In writing this summary of
my work for the quarter, I would say that the assignments and
projects were challenging, but actually sometimes they were fun! In
looking back on my work this quarter, I would say the assignments
were always interesting, proofs or constructions, word problems or
computer geometry.*"

The students seemed to enjoy the idea of keeping a
portfolio of their work, "*just like in an art class*" as one
student said. Another student commented that she was glad to have
something to show for all her hard work, and enjoyed showing her
portfolio to her parents.

Our geometry class was quite different from most
geometry classes, and sometimes I wondered if the students thought
that this approach to teaching and learning was helpful to them. But
whenever I read their comments and reflections, I was pleased to see
that they liked the methods used in this class. In her overall
reflection at the end of the semester, Terri wrote the following
comments: "*All of the pieces I chose to go into this portfolio are
very significant to me. When I first started geometry, I thought it
was going to be just like every other math class I had taken:
calculations, formulas, homework, tests and then more calculations;
all the simple no-brainer things I'd always learned in math classes.
But as this course progressed, I found myself struggling at times to
comprehend all of the new material - proofs, geometric diagrams, even
3-dimensional figures, and then writing about it all! It took a lot of
work, but when I finally completed each set of assignments, I got
this great feeling of satisfaction, and was really proud of the work
in my portfolio. This class really introduced me to a lot of new
experiences!*"

Mark had this to say: "*This approach is great
because it allows students to have a better understanding of math. I
think this class was better than the traditional approach because
usually in a math class you just have to memorize information and
then regurgitate it all on endless tests. The strengths of this class
for me were that we were expected to really think about things,
figure them out, and then write explanations too. If you have to
explain something to someone else, you really have to know what it is
all about. It might be harder, but this is what learning is really
all about. Not just memorizing. When you struggle, try different
approaches, figure it out and then try to explain it to someone else,
your really learn it for good!*"

Elaine wrote the following in her overall
reflection: "*I think in all other math classes that I've been in,
we would do assignments from the textbook, drill worksheets, and tests. I'm
glad this class is different because we show different kinds and
levels of thinking through our homework, worksheets, writing
assignments (both explanations and reflections), group work on
projects and homework, and tests. I think that if this course had
been strictly rote assignments I wouldn't have liked it or learned as
much because since it was balanced with creative projects and writing
assignments. All of this helped me to really understand what we were
doing, not just memorize facts. I really liked this portfolio type of
class because we learned the material in a variety of ways instead of
just reading the book and listening to the teacher. It was the first
time I realized that the answer to the question 'Who's in charge of
my learning?' is ' I am!' And that was really
cool!"*

On the topic of proofs (which some student seem to
relish, while others find them the bane of their existence), the
students had many interesting things to say. The quads project
required a great deal of exploration, discovery, and writing. Adam
wrote the following essay about this topic: "*I chose this project
for my portfolio because I am proud of the way my group collaborated
and worked well together in order to figure out what the properties
of each quadrilateral were, and how to prove them. I thought it was
cool that everyone in the group contributed their share, which made
everyone's burden easier. It was exciting to solve even the more difficult challenges! And by arguing and then agreeing
on each property and each proof, we all learned a lot about different
approaches and ideas. And spending so much effort arguing,
questioning, and eventually proving each property, the information
really stuck in our heads! Another thing I liked was that I got to
know my group-mates better, through each mistake and each success. So
I fell this project was a great success, from both a mathematical and
social perspective! And in the end, I was really proud of what we had
accomplished, and was really glad to have had the chance to do this
project*."

Keenan wrote: "*This class was unique because it
allowed me to have a better understanding of math. The strengths of
this class were its projects, group work, and writing assignments
instead of just regurgitating information from the book. Also, by
writing and reflecting we operate on a higher level of learning.
Instead of just memorizing things, we had to really understand why,
and how, and why not. Life is not only tests and if you want to teach
kids about applying logic and geometric thinking you must use
creative approaches to learning. Even though it's not always easy,
students have to learn to **think**.*
"

Christopher wrote the following comments on the
subject of the writing assignments in general: "*I feel that these
in depth writing assignments were among the most valuable work that I
accomplished in this class. They pushed me to really think carefully
about what I was learning, and how to improve. For while they require
a lot of time and thinking to accomplish, the results were well worth
the effort.*"

In her reflection on the whole semester, Eileen
wrote the following essay: "*Geometry isn't very simple. Why? I
don't know. Proofs are one of my least favorite things to write. It
seems that we take the scenic route in proofs instead of going
direct. Everything is logical geometry. This isn't saying geometry is
hard. Although math in general is not my favorite subject, I can see
a point to it. It isn't always what we learn that's important, it's
that we learn from it, and how we learn it. Although I might not
remember the exact structure of a two-column proof 10 years from now,
I will remember the logical way one is written. And I will remember
the pride I felt when I wrote a perfect proof, every once in a while.
So, in summary, I think that the work in my portfolio is important
not necessarily for its content, but more for what I learned from it.
Diligence, patience, understanding, hard work, and even tolerance
were all involved. This is my portfolio*."

*"Geometry enlightens the intellect and sets
one's mind right. All of its proofs are very clear and orderly. It is
hardly possible for errors to enter into geometrical reasoning,
because it is well arranged and orderly. Thus, the mind that
constantly applies itself to geometry is not likely to fall into
error. In this convenient way, the person who knows geometry acquires
intelligence." *

**bn Khaldun (1332-1406)**

*
*

**Go To Homepage** **Go To Introduction**
**1) Constructions** **2) Clock Problem** **3) Test Corrections** **4) ASN Explain** **5) Thoughts About Slope** **6) What is Proof?**

**7) Similar Triangles** **8) Homework Corrections** **9) Quads Midpoints** **10) Quads Congruence** **11) Polygons**

**12) Polygons Into Circles** **13) Area and Perimeter** **14) Writing About Grading** **15) Locus** **16) Extra Credit Projects**

**17) Homework Reflections** **18) Students' Overall Reflections** **19) Parents' Evaluate Method** **20) In Conclusion**